DEVELOPMENT OF A FRACTAL ADVECTION-DISPERSION 
EQUATION AND NEW NUMERICAL SCHEMES FOR THE 
CLASSICAL, FRACTAL AND FRACTIONAL ADVECTION-
DISPERSION TRANSPORT EQUATIONS 
 
Amy Allwright 
 
Submitted in fulfilment of the requirements for the degree 
Doctoral Scientiae in Geohydrology 
 
in the 
Faculty of Natural and Agricultural Sciences 
(Institute for Groundwater Studies) 
at the 
University of the Free State 
 
 
Supervisor: Prof Abdon Atangana 
 
January 2019 
 
2019 
 
DECLARATION 
 
I, Amy Allwright, hereby declare that the dissertation hereby submitted by me to the Institute for 
Groundwater Studies in the Faculty of Natural and Agricultural Sciences at the University of the Free 
State, in fulfilment of the degree of Doctoral Scientiae, is my own independent work. It has not 
previously been submitted by me to any other institution of higher education. In addition, I declare 
that all sources cited have been acknowledged by means of a list of references. 
I furthermore cede copyright of the dissertation and its contents in favour of the University of the Free 
State. 
 
 
 
_____________________________ 
Amy Allwright (2010083661) 
January 2019 
  
i 
 
ACKNOWLEDGEMENTS 
 
First and foremost, I thank my supervisor, Prof Abdon Atangana. It has been an honour learning 
from you, your patience and encouragement is at the center of this thesis. Your ideals on academic 
excellence and passion for Africa will surely influence my future path. What I have learnt from you is 
truly invaluable and immeasurably appreciated.   
I am eternally grateful to Prof Toufik Mekkaoui and Prof Zakia Hammouch, at the University of 
Moulay Ismail, who graciously hosted me and took time out of their busy schedule to give me an 
introduction to mathematical modelling. The value this work added to the thesis is immense, and your 
patience in teaching me is greatly appreciated.  
I thank Prof Gerrit van Tonder, in his absence, who first believed that I could be an academic, and 
Prof Danie Vermeulen, who took me under his wing and sent me for training to achieve this goal. The 
training in groundwater modelling ultimately lead to this research and I appreciated the influence you 
have had on my life.  
To my family for their encouragement and support through the years of this research, and for the 
foundation upon which all my achievements are based. Special mention to my husband, Craig, who is 
the most proud of me and has been by my side through the ups and downs of completing a thesis. 
Your unfailing support is the glue holding it all together.  
My heavenly Father, who has given me everything I needed to complete this thesis. You have 
been a constant presence through all the trials and highlights. Your hand has guided this endeavour 
from the beginning to the end, and all the glory is Yours.  
  
ii 
 
ABSTRACT 
 
Groundwater is a vital source of fresh water to many people across the globe, but it is prone to 
contamination by human activities. In the last few decades, great strides have been made in legislation 
protecting and controlling the quality of groundwater, creating awareness of potential groundwater 
contamination and the importance of prevention and mitigation. The accurate representation of 
contaminant movement within a groundwater system is important, because misrepresentation could 
increase the environmental impact due to inadequate mitigation or remediation measures. However, 
groundwater transport is particularly complex due to the inherent heterogeneity of aquifers. Where, 
predicting the movement of contaminants within groundwater systems, especially in fractured 
systems, is prone to discrepancies between modelled and observed. There are two general 
approaches to improve the simulation of groundwater transport: develop the physical 
characterisation of the heterogeneous system, or redefine the formulation of the governing 
equations. The focus of this research is to advance the simulation of groundwater transport by 
examining the formulation of the governing advection-dispersion equation. To achieve this aim, 
improved numerical approximation schemes for the classical advection-dispersion equation are 
developed, fractal and fractional derivatives are incorporated into the formulation, and fractional and 
fractal derivatives are combined.  
Augmented upwind finite difference numerical approximation schemes, which are better suited for 
advection-dominated systems, are applied to the solution of the classical advection-dispersion 
equation for fractured groundwater systems. The simulation of anomalous transport in fractured 
aquifer systems is improved by providing a fractal advection-dispersion equation with numerical 
integration and approximation methods for solution. The fractal advection-dispersion equation is 
proven to simulate superdiffusion and subdiffusion by varying the fractal dimension, without explicit 
characterisation of fractures or preferential pathways. To improve the governing equation for 
groundwater transport modelling, the Caputo and Atangana-Baleanu in Caputo sense (ABC) fractional 
derivatives are applied to the advection-dispersion equation with a focus on the advection term to 
account for anomalous advection. Appropriate numerical approximation methods for the fractional 
advection-dispersion equations are provided and analysed for stability requirements. A fractional-
fractal advection-dispersion equation is developed to provide an efficient non-local, in both space and 
time, modelling tool. The fractional-fractal model provides a flexible tool to model anomalous 
diffusion, where the fractional order controls the breakthrough curve peak, and the fractal dimension 
controls the position of the peak and tailing effect. These two controls potentially provide the tools to 
improve the representation of anomalous breakthrough curves that cannot be described by the 
classical model.  
A modest step is taken forward to advance the use of fractional calculus, achieve the collective mission 
of resolving the difference between modelled and observed, and to increase the comprehension and 
management of natural systems. 
iii 
 
 
 
 
 
 
 
 
Academic journal articles published from thesis: 
Allwright, A. and Atangana, A. (2018). Fractal advection-dispersion equation for groundwater 
transport in fractured aquifers with self-similarities. The European Physical Journal 
Plus, 133(2), 48. Impact factor 2.240. 
 
Allwright, A. and Atangana, A. (2018). Augmented upwind numerical schemes for the 
groundwater transport advection‐dispersion equation with local operators. 
International Journal for Numerical Methods in Fluids, 87, 437-462. Impact factor 
1.673, 
 
Book chapter from thesis (published): 
Allwright, A. and Atangana, A. (2019). Upwind-based numerical approximation of a space-time 
fractional advection-dispersion equation for groundwater transport within fractured 
systems. In Fractional Derivatives with Mittag-Leffler Kernel (pp. 309-341). Springer, 
Cham. 
 
Academic journal articles from thesis (accepted): 
Allwright, A. and Atangana, A. (2020). Augmented upwind numerical schemes for a fractional 
advection-dispersion equation in fractured groundwater systems. Discrete & 
Continuous Dynamical Systems – Series S, 13(3). Impact factor 0.561. 
 
Academic journal articles from thesis (submitted): 
Fractional-fractal advection-dispersion model: simulation of a dual non-local system for 
anomalous transport in fractured systems 
  
 
  
iv 
 
TABLE OF CONTENTS 
 
Declaration ............................................................................................................................................... i 
Acknowledgements ................................................................................................................................. ii 
Abstract .................................................................................................................................................. iii 
List of figures .......................................................................................................................................... ix 
List of tables ............................................................................................................................................ x 
1  Introduction ........................................................................................................................ 1 
1.1 Anomalous diffusion and transport in heterogeneous systems ............................................. 2 
1.2 Non-local approaches ............................................................................................................. 4 
1.2.1 Fractal geometry and derivative ..................................................................................... 4 
1.2.2 Fractional derivative ....................................................................................................... 6 
1.3 Aims and objectives ................................................................................................................ 8 
1.3.1 Local advection-dispersion equation .............................................................................. 8 
1.3.2 Fractal advection-dispersion equation ........................................................................... 8 
1.3.3 Fractional advection-dispersion equation ...................................................................... 8 
1.3.4 Fractional-fractal advection-dispersion equation ........................................................... 8 
1.4 Structure of thesis ................................................................................................................... 9 
2 Classical advection-dispersion equation .............................................................................. 10 
2.1 Upwind finite difference scheme for local operator ............................................................ 11 
2.2 Upwind finite difference schemes for the classical advection-dispersion equation ............ 13 
2.2.1 Explicit upwind .............................................................................................................. 13 
2.2.2 Implicit upwind ............................................................................................................. 14 
2.2.3 Upwind Crank-Nicolson scheme ................................................................................... 14 
2.2.4 Upwind advection Crank-Nicolson scheme .................................................................. 15 
2.2.5 Explicit upwind-downwind weighted scheme .............................................................. 16 
2.2.6 Implicit upwind-downwind weighted scheme .............................................................. 16 
2.3 Numerical stability analysis ................................................................................................... 17 
2.3.1 Explicit upwind .............................................................................................................. 18 
2.3.2 Implicit upwind ............................................................................................................. 21 
2.3.3 Upwind Crank-Nicolson scheme ................................................................................... 24 
2.3.4 Upwind advection Crank-Nicolson scheme .................................................................. 26 
2.3.5 Explicit upwind-downwind weighted scheme .............................................................. 29 
2.3.6 Implicit upwind-downwind weighted scheme .............................................................. 32 
2.4 Comparison of numerical schemes stability conditions ....................................................... 36 
2.5 Numerical simulation ............................................................................................................ 39 
2.6 Chapter summary.................................................................................................................. 45 
v 
 
3 Fractal advection-dispersion equation ................................................................................. 46 
3.1 Fractal differentiation ........................................................................................................... 47 
3.1.1 Fractal derivative........................................................................................................... 47 
3.1.2 Fractal integral .............................................................................................................. 48 
3.2 New groundwater transport model in a fractured aquifer with self-similarities ................. 51 
3.2.1 Fractal formulation of the advection-dispersion transport equation ........................... 52 
3.2.2 Qualitative properties ................................................................................................... 54 
3.3 Numerical approximation methods (fractal derivative) ....................................................... 58 
3.3.1 Forward finite difference scheme ................................................................................. 58 
3.3.2 Crank-Nicolson finite difference scheme ...................................................................... 60 
3.4 Numerical integration methods (fractal integral) ................................................................. 61 
3.4.1 Simpson’s 3/8 Rule of numerical integration ................................................................ 62 
3.4.2 Boole’s Rule of numerical integration .......................................................................... 64 
3.5 Numerical simulation for different fractal dimensions......................................................... 67 
3.6 Investigation of fractal velocity (𝑉𝐹𝛼) and fractal dispersivity (𝐷𝐹𝛼) ................................. 71 
3.7 Chapter summary.................................................................................................................. 75 
4 Fractional Derivatives: Singular and Non-Singular ................................................................ 76 
4.1 Special functions in fractional definitions ............................................................................. 76 
4.1.1 Gamma function ........................................................................................................... 77 
4.1.2 Power Law function ...................................................................................................... 77 
4.1.3 Exponential function ..................................................................................................... 77 
4.1.4 Mittag-Leffler function .................................................................................................. 78 
4.2 Riemann-Liouville fractional definition ................................................................................. 79 
4.3 Caputo fractional definition .................................................................................................. 82 
4.4 Atangana-Baleanu fractional definitions .............................................................................. 84 
4.4.1 Properties ...................................................................................................................... 85 
4.4.2 Boundary analysis of fractional order 𝛼 ....................................................................... 87 
4.5 Analysis of kernels associated with fractional derivatives .................................................... 88 
4.6 Fractional derivatives application to the Advection-Dispersion Equation ........................... 97 
4.7 Chapter summary.................................................................................................................. 98 
5 Fractional advection-dispersion equation with Caputo derivative ........................................ 99 
5.1 Upwind numerical approximation schemes ....................................................................... 101 
5.1.1 Explicit upwind ............................................................................................................ 102 
5.1.2 Implicit upwind ........................................................................................................... 104 
5.1.3 Upwind advection Crank-Nicolson scheme ................................................................ 105 
5.1.4 Explicit upwind-downwind weighted scheme ............................................................ 106 
5.1.5 Implicit upwind-downwind weighted scheme ............................................................ 107 
5.2 Numerical stability analysis ................................................................................................. 108 
vi 
 
5.2.1 Explicit upwind ............................................................................................................ 109 
5.2.2 Implicit upwind ........................................................................................................... 116 
5.2.3 Upwind advection Crank-Nicolson scheme ................................................................ 122 
5.2.4 Explicit upwind-downwind weighted scheme ............................................................ 130 
5.2.5 Implicit upwind-downwind weighted scheme ............................................................ 136 
5.2.6 Evaluation of numerical stability results ..................................................................... 143 
5.3 Chapter summary................................................................................................................ 143 
6 Fractional advection-dispersion equation with ABC derivative ........................................... 145 
6.1 Advection-focused transport model with Atangana-Baleanu (ABC) derivative ................. 146 
6.1.1 Picard-Lindelöf theorem for existence and uniqueness ............................................. 146 
6.1.2 Semi-discretisation stability ........................................................................................ 149 
6.2 Upwind numerical approximation schemes ....................................................................... 153 
6.2.1 Explicit upwind ............................................................................................................ 155 
6.2.2 Implicit upwind ........................................................................................................... 156 
6.2.3 Upwind advection Crank-Nicolson scheme ................................................................ 157 
6.2.4 Explicit upwind-downwind weighted scheme ............................................................ 157 
6.2.5 Implicit upwind-downwind weighted scheme ............................................................ 158 
6.3 Numerical stability analysis ................................................................................................. 159 
6.3.1 Explicit upwind ............................................................................................................ 160 
6.3.2 Implicit upwind ........................................................................................................... 165 
6.3.3 Upwind advection Crank-Nicolson scheme ................................................................ 171 
6.3.4 Explicit upwind-downwind weighted scheme ............................................................ 176 
6.3.5 Implicit upwind-downwind weighted scheme ............................................................ 181 
6.3.6 Evaluation of numerical stability results ..................................................................... 187 
6.4 Chapter summary................................................................................................................ 187 
7 Fractional-fractal advection-dispersion equation ............................................................... 189 
7.1 Fractional-fractal advection-dispersion equation ............................................................... 190 
7.2 Numerical approximation ................................................................................................... 191 
7.3 Relationship between fractional and fractal dimensions ................................................... 191 
7.4 Simulation ........................................................................................................................... 199 
7.5 Chapter summary................................................................................................................ 205 
8 Discussion ........................................................................................................................ 206 
9 Conclusions ...................................................................................................................... 208 
9.1 Local advection-dispersion equation .................................................................................. 208 
9.2 Fractal advection-dispersion equation ............................................................................... 209 
9.3 Fractional advection-dispersion equation .......................................................................... 209 
9.4 Fractional-fractal advection-dispersion equation ............................................................... 209 
References .......................................................................................................................................... 211 
vii 
 
Appendices ............................................................................................................................................... I 
Appendix A – Scilab codes ....................................................................................................................... I 
Appendix B – MAPLE codes .................................................................................................................... VI 
 
 
 
  
viii 
 
LIST OF FIGURES 
 
Figure 1-1 Aperture fields measured by fractal fractures and Gaussian fractures (top), and the associated 
simulation results (bottom). Modified from Yang et al. (2012). ......................................................... 3 
Figure 1-2 Simulated hydraulic head contours and plume migration for a homogeneous aquifer (top) and for a 
heterogeneous aquifer (bottom). Modified from Qin et al. (2013). ................................................... 3 
Figure 1-3 Example of a fractal geometry with self-similarity. Modified after Mandelbrot (1982). ...................... 5 
Figure 1-4 Natural fracture networks demonstrating fractal geometries. Taken from Shokri et al. (2016). ......... 5 
Figure 1-5 Measured concentration outputs from an experiment performed by Benson et al. (2000) and the 
predicted concentration by the traditional Fickian approach (top). The fractional models generated 
by varying the fractional order (𝛼) to better represent the experimental system (bottom). Taken 
from Benson et al. (2000). ................................................................................................................... 7 
Figure 2-1 Generic one-dimensional transport problem, where an initial contaminant concentration in an aquifer 
is simulated along a single line in the x-direction.............................................................................. 39 
Figure 2-2 Simulated breakthrough curve of concentration (concentration against distance) for the First-order 
upwind explicit numerical scheme .................................................................................................... 41 
Figure 2-3 Simulated breakthrough curve of concentration (concentration against distance) for the First-order 
upwind implicit numerical scheme .................................................................................................... 41 
Figure 2-4 Simulated breakthrough curve of concentration (concentration against distance) for the First-order 
upwind Crank-Nicolson numerical scheme ....................................................................................... 42 
Figure 2-5 Simulated breakthrough curve of concentration (concentration against distance) for the First-order 
upwind advection Crank-Nicolson (explicit) numerical scheme ........................................................ 42 
Figure 2-6 Simulated breakthrough curve of concentration (concentration against distance) for the First-order 
upwind advection Crank-Nicolson (implicit) numerical scheme ....................................................... 43 
Figure 2-7 Simulated breakthrough curve of concentration (concentration against distance) for the First-order 
upwind-downwind weighted (explicit) numerical scheme with variable theta (𝜃) values from 1 
(black), 0.9 (blue), 0.6 (dark blue), and 0.1 (green) ........................................................................... 43 
Figure 2-8 Simulated breakthrough curve of concentration (concentration against distance) for the First-order 
upwind-downwind weighted (implicit) numerical scheme with variable theta (𝜃) values from 1 
(green), 0.5 (dark blue), and 0.1 (black) ............................................................................................ 44 
Figure 3-1 Simulation results for the simple transport problem for variable fractal dimensions (0.6 ≤ 𝛼 ≤ 1), 𝑥 
is distance in meters, and 𝑡 is time in days. When 𝛼 = 1, the equation simplifies to the traditional 
advection-dispersion equation and forms the basis of comparison. Changing the fractal dimension 
increases the extent of the transport plume along the one-dimensional line in the x-direction, 
representing superdiffusion. ............................................................................................................. 68 
Figure 3-2 Simulation results for the simple transport problem for variable fractal dimensions (0.1 ≤ 𝛼 ≤ 0.5). 
Fractal dimensions below 0.5 result in significant contaminant transport along the line. ............... 69 
Figure 3-3 Simulation results for the simple transport problem for variable fractal dimensions (1 ≤ 𝛼 ≤ 2). 
When 𝛼 = 1, the equation simplifies to the traditional advection-dispersion equation and forms the 
basis of comparison. Fractal dimensions greater than 1, impede transport along the x-directional 
line, representing subdiffusion. ......................................................................................................... 70 
Figure 3-4 Fractal velocity and dispersivity over space for varying fractal dimensions (0.7 ≤ 𝛼 ≤ 1) on the same 
scale. .................................................................................................................................................. 71 
Figure 3-5 Fractal velocity and dispersivity over space for varying fractal dimensions (0.1 ≤ 𝛼 ≤ 0.6) on the same 
scale. .................................................................................................................................................. 72 
Figure 3-6 Fractal velocity and dispersivity over space for varying fractal dimensions (0.5 ≤ 𝛼 ≤ 1) on a local 
scale for each plot. ............................................................................................................................ 73 
Figure 3-7 Fractal velocity and dispersivity over space for varying fractal dimensions (0.1 ≤ 𝛼 ≤ 0.4) on an 
individual scale for each plot. ............................................................................................................ 74 
Figure 7-1 Fractional-fractal velocity over space for a fractional order 𝛼 = 1 (simplifying back to the local 
equation in time) and varying fractal dimensions (0.1 ≤ 𝛽 ≤ 1) on a variable plot scale. These plots 
solely evaluate the influence of the fractal derivative in space, and thus form a base of comparison, 
and the plot of 𝛼 = 1 ; 𝛽 = 1 represents the traditional-local model in both time and space. ..... 193 
Figure 7-2 Fractional-fractal velocity over space for a fractional order 𝛼 = 0.9 and varying fractal dimensions 
(0.1 ≤ 𝛽 ≤ 1) on a variable plot scale. ........................................................................................... 194 
Figure 7-3 Fractional-fractal velocity over space for a fractional order 𝛼 = 0.7 and varying fractal dimensions 
(0.1 ≤ 𝛽 ≤ 1) on a variable plot scale. ........................................................................................... 195 
ix 
 
Figure 7-4 Fractional-fractal velocity over space for a fractional order 𝛼 = 0.5 and varying fractal dimensions 
(0.1 ≤ 𝛽 ≤ 1) on a variable plot scale. ........................................................................................... 196 
Figure 7-5 Fractional-fractal velocity over space for a fractional order 𝛼 = 0.3 and varying fractal dimensions 
(0.1 ≤ 𝛽 ≤ 1) on a variable plot scale. ........................................................................................... 197 
Figure 7-6 Fractional-fractal velocity over space for a fractional order 𝛼 = 0.1 and varying fractal dimensions 
(0.1 ≤ 𝛽 ≤ 1) on a variable plot scale. ........................................................................................... 198 
Figure 7-7 Simulation results illustrated for distance (m) in the x-direction along a line over time (d) for the 
fractional order 𝛼 = 1 (simplifies to a local order), and varying fractal dimensions (0.5 ≤ 𝛽 ≤ 1)
 ......................................................................................................................................................... 200 
Figure 7-8 Simulation results illustrated for distance (m) in the x-direction along a line over time (d) for the 
fractional order 𝛼 = 0.9, and varying fractal dimensions (0.5 ≤ 𝛽 ≤ 1) ....................................... 201 
Figure 7-9 Simulation results illustrated for distance (m) in the x-direction along a line over time (d) for the 
fractional order 𝛼 = 0.8, and varying fractal dimensions (0.6 ≤ 𝛽 ≤ 1) ....................................... 202 
Figure 7-10 Simulation results illustrated for distance (m) in the x-direction along a line over time (d) for the 
fractional order 𝛼 = 0.7, and varying fractal dimensions (0.7 ≤ 𝛽 ≤ 1) ....................................... 203 
Figure 7-11 Simulation results illustrated for distance (m) in the x-direction along a line over time (d) for the 
fractional order 𝛼 = 0.6, 0.5, and varying fractal dimensions (0.8 ≤ 𝛽 ≤ 1) ................................ 204 
 
LIST OF TABLES 
 
Table 2-1 Traditional upwind numerical schemes for the one dimensional advection-dispersion equation with 
associated assumptions in the numerical stability analysis and the resulting stability conditions ..... 37 
Table 2-2 New upwind schemes for the one dimensional advection-dispersion equation with associated 
assumptions in the numerical stability analysis and the resulting stability conditions ....................... 38 
Table 2-3 Computational times (in nanoseconds) for each numerical scheme investigated ............................... 44 
Table 4-1 Summary of kernels associated with each fractional derivative definition .......................................... 91 
Table 4-2 Summary of results obtained by Tateishi et al. (2017) on influence on different fractional derivative 
definitions on the Diffusion Equation .................................................................................................. 96 
Table 5-1 Summary of the assumptions made and corresponding stability condition for each numerical 
approximation scheme for the fractional advection-dispersion equation with Caputo fractional 
derivative ........................................................................................................................................... 144 
Table 6-1 Summary of the stability conditions for the upwind-based numerical schemes for the fractional 
advection-dispersion equation with ABC derivative.......................................................................... 188 
Table 7-1 Tabulation of the fractional order (α) and fractal dimension (β) valid combination options for the 
fractional-fractal advection-dispersion equation. Valid combinations (x), Non-valid combinations 
(grey), recommended combinations (green). .................................................................................... 204 
  
x 
 
1  INTRODUCTION 
 
Real world systems are complex, by definition a complexity by which any one method is not able to 
capture all the nuances of that system. It is this that compels science to improve and continuously 
strive for new methods and approaches, ever endeavouring to reconcile the difference between 
modelled and observed.  
Simulating transport with the advection-dispersion equation is a real world problem, where the 
general discrepancy between modelled and observed is particularly large. This discrepancy lead to the 
development of the term anomalous diffusion (non-Fickian diffusion), especially when using 
traditional methods. In groundwater systems, this discrepancy can be attributed to the highly 
heterogeneous nature of aquifer systems, especially in fractured systems; and the understanding on 
which the governing equation was formulated upon. Therefore, the solution of this problem from a 
groundwater perspective logically involves either improving the characterisation of heterogeneity in 
the system, and/or improving the formulation of the governing equation.  
Improving the characterisation of heterogeneity has been historically topical; from some of the first 
comprehensive groundwater books (Freeze and Cherry, 1979) up to the present. In 2015, the MADE 
challenge for groundwater transport in highly heterogeneous aquifers: insights from 30 years of 
modelling and characterisation at the field scale and promising future directions was held to assess 
the progress made from the original tracer series at the Macrodispersion Experiment, in the late 1980s 
to 1990s, which sparked the consideration of new approaches for solute transport in highly 
heterogeneous media (Gómez-Hernández et al., 2017). While extensive research has been done to 
1 
 
improve the characterisation of heterogeneity, ranging from new measurement devices, new 
parameter estimate methods, to discrete fracture network models, the challenge of how best to 
characterise a heterogeneous system, to incorporate that heterogeneity into a model, and to simulate 
transport within that system, remains.  
On the other hand, from the perspective of improving the formulation of the governing equation, 
numerous non-local approaches have been applied, ranging from multiple-rate mass transfer method 
and rate-limited mass transfer, stochastic averaging, continuous-time random walk, to fractal and 
fractional differential equations (Koch and Brady, 1988; Schumer et al., 2003a; Schumer et al., 2003b 
and b; Berkowitz et al., 2006; Singha et al., 2007; Zhang et al., 2009; Neuman and Tartakovsky, 2009; 
Zhang et al., 2013; Sun et al., 2014). 
For this research, the latter approach will be perused, where the formulation of the advection-
dispersion equation can be improved to better simulate groundwater transport without a detailed 
characterisation of the heterogeneity of a system, especially a fractured aquifer system.  
1.1 Anomalous diffusion and transport in heterogeneous systems 
The traditional advection-dispersion equation is used to describe the transport of non-reactive 
contaminants or tracers, which is founded on an analogy to Fick’s Law of diffusion. Non-Fickian 
transport or diffusion, also termed anomalous diffusion, is any transport that is not adequately 
described by the traditional advection-dispersion equation. Anomalous transport can be defined by 
non-Gaussian leading or trailing tails of a plume or break-through curve from a point source, or 
nonlinear growth of the centered second moment. Subdivisions of anomalous diffusion include 
superdiffusion and subdiffusion, superdiffusion is characterised by a growth rate faster than linear 
growth (i.e. faster movement found in preferential flow pathways), and subdiffusion is characterised 
by a growth rate slower than linear growth (i.e. slower movement in low-permeability zones) (Zhang 
et al., 2009; Neuman and Tartakovsky, 2009; Zhang et al., 2013; Sun et al., 2014).  
Anomalous diffusion is not anomalous in the traditional sense of the word, but rather a term coined 
to describe natural phenomena that cannot be modelled accurately using traditional modelling 
approaches and equations (Klafter and Sokolov, 2005). Examples of anomalous diffusion range from 
signalling of biological cells to foraging behaviour of animals, from the operation of photocopier 
machines to transport of contaminants in groundwater. Anomalous diffusion in fractured 
groundwater systems have been well documented (Liu et al., 2004; Klafter and Sokolov, 2005; 
Meerschaert et al., 2008; Pablo et al., 2009; Cello et al., 2009). Yang et al. (2012) considered two-phase 
flow in fractured media with an adaptive circle fitting interface, where the fractured system modelled 
producing highly irregular contaminant plume (Figure 1-1).  
2 
 
Fracture aperture fields 
Fractal fracture Gaussian fracture 
Simulation results 
 
Figure 1-1 Aperture fields measured by fractal fractures and Gaussian fractures (top), and the associated 
simulation results (bottom). Modified from Yang et al. (2012).  
 
Figure 1-2 Simulated hydraulic head contours and plume migration for a homogeneous aquifer (top) and for 
a heterogeneous aquifer (bottom). Modified from Qin et al. (2013). 
3 
 
The resulting contaminant movement shows the potentially complex transport patterns in fractured 
media. The influence of heterogeneity on transport in groundwater was considered by Qin et al. 
(2013), where the transport of a solute within a dipping heterogeneous, confined aquifer is used to 
determine plume shapes under the various conditions (Figure 1-2). Figure 1-2 illustrates the potential 
influence heterogeneity has on the transport of contaminants within the subsurface, and highlights 
the importance of correctly including this aspect to ensure the correct prediction of contaminant 
movement for remediation and mitigation.  
The main limitation in accurately predicting the movement of contaminants in groundwater is that 
spatial heterogeneity will never completely be characterised, regardless of the amount of data 
collected (Gómez-Hernández et al., 2016). Thus, investigating the reformulation of the governing 
equation to simulate variable transport without a complete characterisation of the system is a valid 
pursuit. 
1.2 Non-local approaches 
Numerous non-local approaches exist and have been applied to the problem of anomalous diffusion, 
as discussed. For this research, specific non-local approaches will be considered, including fractal and 
fractional derivatives.  
1.2.1 Fractal geometry and derivative 
Geometry is traditionally considered rigid, because classical geometry is unable to define the shape of 
a cloud, mountain or coastline. These natural phenomena and objects do not conform to the Euclid or 
standard geometry of smooth spheres, cones, circles and straight lines. Mandelbrot (1982) found that 
nature exhibits a higher-degree of irregularity and fragmentation when compared to standard 
geometry, where it can actually be considered a different level of complexity altogether. Mandelbrot 
(1982) describes these irregular and fragmented patterns by identifying a family of shapes termed 
fractals, forming the foundation for fractal geometry, and the fractal derivative (Figure 1-3). The term 
fractal was selected by Mandelbrot (1982) for the original Latin meaning, “to break, to create irregular 
fragments, describing a general fragmented and irregular nature (Mandelbrot, 1982; Le Méhauté, 
1991; West, et al., 2012). 
Fractal geometry is based on the principle that irregular objects in nature tend to exhibit inherent 
patterns that repeat themselves at different scales, termed self-similarity. Field observations have 
demonstrated that multiple length-scales exist in a variety of naturally fractured media (Acuna and 
Yortsos, 1995; Shokri et al., 2016). Shokri et al. (2016) document natural fracture networks that 
demonstrate fractal geometries (Figure 1-4). Fractal heterogeneity is not spatially periodic, but rather 
exhibit a pattern that is independent of the scale of observation (Wheatcraft and Tyler, 1988).  
4 
 
 
 
 
Figure 1-3 Example of a fractal geometry with self-similarity. Modified after Mandelbrot (1982). 
 
Figure 1-4 Natural fracture networks demonstrating fractal geometries. Taken from Shokri et al. (2016). 
 
5 
 
Developments in the theory of fractal geometry have resulted in numerous applications, to many 
branches of science, especially to problems stemming from unstable phenomena (Acuna and Yortsos, 
1995; Rocco and West, 1999; Schumer et al., 2003b; Chen, 2006; Santizo, 2012; Liang et al., 2016). 
Fractal geometry incorporates the complexity of the natural system by describing an object by the 
repetition of a given algorithmic rule or pattern over a multitude of separate length scales. Fractured-
rock aquifers have proven to be an appropriate candidate for a fractal description (Acuna and Yortsos, 
1995), and the ability of a fractal geometry to describe complexity on different scales provides the 
ability to potentially describe scale-dependent dispersivity. Currently available transport models for 
fractured systems are not able to capture the property of self-similarity found in fractured systems. 
In response to the current limitations of groundwater transport modelling in fractured systems, a 
fractal groundwater advection-dispersion transport equation is deemed an appropriate and 
worthwhile investigation. A fractal advection-dispersion equation has the potential to simulate the 
full range of observed non-Fickian behaviour, from subdiffusion to superdiffusion, which is related to 
the well-established fractal model for fractured systems, without the detailed characterisation of the 
heterogeneity of the system. A fractal in space advection-dispersion equation has not been developed 
before, and therefore will be considered in this research.  
1.2.2 Fractional derivative 
Complexity from the perspective of fractional calculus is explored by West (2016), where fractional 
differential equations are one approach to improve the simulation of real world problems. Fractional 
calculus is not a new topic, having its original inception in the late 1600s, but the application of 
fractional derivations to practical problems has steadily increased since the 1970s.  
The inception of fractional calculus has its roots in the relationship between Leibniz and L’Hopital, 
where Leibniz ignited an idea within L’Hopital that has gone on to diversify knowledge. L’Hopital posed 
the seemingly inapt question, what would happen if the derivative order were a fraction? Leibniz’s 
famous response was “it will lead to a paradox from which one day useful consequences will be 
drawn”. Perhaps, Leibniz should have paid greater attention to the seemingly inapt question at the 
time because; the non-integer fractional derivative has proven to be exceptionally useful (Oldham and 
Spanier, 1974).  
From this original conversation, the practical application was only realised much later and resisted 
direct application for many years. Oldham and Spanier (1974) found the earliest systematic studies on 
the fractional derivative were performed by Liouville (1832), Riemann (1953), and Holmgren (1864), 
although Euler (1730) and Lagrange (1772) made even earlier contributions, as cited by Oldham and 
Spanier (1974) (Debnath, 2004; Loverro, 2004; Petráš, 2010; Herrmann, 2011; Tarasov, 2013).  
6 
 
Fractional calculus can be described as an extension of the concept of a derivative operator from an 
integer order (𝑛) to an arbitrary order (𝛼) (Herrmann, 2011): 
𝑑𝑛 𝑑𝛼
→  
𝑑𝑥𝑛 𝑑𝑥𝛼
where, 
𝛼 denotes a non-integer order that is a real, complex value or even a complex-valued 
function 𝛼 = 𝛼(𝑥, 𝑡). 
The fractional advection-dispersion equation was first developed by Benson (1998), and an application 
demonstrated by Benson et al. (2000) (Figure 1-5). In an experiment, Benson et al. (2000) 
demonstrated an anomalous diffusion system and the inadequacy of the Fickian or traditional 
equation to model the measured concentrations. Furthermore, by incorporating the fractional 
derivative and adjusting the fractional order (𝛼), a model can be developed to improve the 
representation of the system.  
With the endeavour to continually improve simulation methods, a progression of fractional derivative 
definitions have been developed over the years, with definitions including Riemann-Liouville, Caputo, 
Caputo-Fabrizio, and the latest Atangana-Baleanu (Oldham and Spanier, 1974; Herrmann, 2011; Li et 
al., 2011; Atangana and Baleanu, 2016). Following on from the work performed by Tateishi et al. 
(2017), which found that the new fractional derivatives could be effective in modelling anomalous 
diffusion, the new fractional derivatives are applied to the advection-dispersion equation to model 
not only anomalous diffusion, but potentially anomalous advection in the form of preferential 
pathways in fractures within the groundwater system.  
 
Figure 1-5 Measured concentration outputs from an experiment performed by Benson et al. (2000) and the 
predicted concentration by the traditional Fickian approach (top). The fractional models generated by 
varying the fractional order (𝜶) to better represent the experimental system (bottom). Taken from Benson 
et al. (2000). 
7 
 
1.3 Aims and objectives 
The aim of this research is to improve the simulation of groundwater transport in fractured system 
using the advection-dispersion equation, following the approach of reformulation of the governing 
equation. The objectives are sub-divided between improvements for the local advection-dispersion 
equation, fractal advection-dispersion equation, and fractional advection-dispersion equation.  
1.3.1 Local advection-dispersion equation 
The objectives for this research to improve the simulation using the local advection-dispersion 
equation includes: 
o Identify numerical approximation methods for the specific advection-dominated fracture 
systems 
o Consider possible improvements to the numerical approximation scheme 
o Evaluate the developed schemes in terms of stability and computational times 
1.3.2 Fractal advection-dispersion equation 
The objectives for this research for using a fractal advection-dispersion equation includes: 
o Investigate the use of the fractal derivative for fractured groundwater systems and previous 
applications 
o Develop an improved fractal advection-dispersion equation 
o Investigate numerical solution methods for the new equation 
o Evaluate the developed schemes in terms of stability 
o Perform simulations to test the validity of the equation to fractured systems 
1.3.3 Fractional advection-dispersion equation 
The objectives for application of a new fractional advection-dispersion equation includes: 
o Develop a fractional advection-dispersion equation specifically for fractured systems using the 
new derivatives 
o Investigate numerical solution methods for the new equation 
o Evaluate the developed schemes in terms of stability 
o Perform simulations to test the validity of the equation to fractured systems 
1.3.4 Fractional-fractal advection-dispersion equation 
The objectives for the development of a fractional-fractal advection-dispersion equation includes: 
o Consider the need and implications of developing a fractional-fractal advection-dispersion 
equation 
o Develop a fractional-fractal advection-dispersion equation 
o Perform simulations to test the validity of the equation to fractured systems 
8 
 
1.4 Structure of thesis 
The structure of the thesis is designed as follows: 
Chapter 1 Introduction 
The rationale for the current research is explained, each aspect is briefly introduced, and the aims and 
objectives are defined. 
Chapter 2 Classical advection-dispersion equation 
Augmented upwind numerical schemes are developed to better simulate groundwater transport in 
advection-dominated fractured systems. The numerical schemes are evaluated for stability and 
computational times for a simple one-dimensional problem.  
Chapter 3 Fractal advection-dispersion equation 
A fractal advection-dispersion equation for fractured groundwater systems is developed, along with 
numerical approximation methods. A generic transport problem is simulated to test the validity of the 
fractal advection-dispersion equation to fractured groundwater transport.  
Chapter 4 Fractional derivatives: singular and non-singular 
An overview of fractional calculus in the form of the functions required for the definition of fractional 
integrals and derivatives, fundamental fractional derivative definitions, and an analysis of the kernels 
associated with each fractional definition.   
Chapter 5 and 6 Fractional advection-dispersion equations 
New fractional advection-dispersion equations are developed with the Caputo and Atangana-Baleanu 
fractional derivative definitions. The advection-dominated numerical schemes are applied and tested 
for stability. Some simulations of fractional systems are investigated. 
Chapter 7 Fractional-Fractal advection-dispersion equation 
Considering the developed fractal in space advection-dispersion equation, and the fractional 
advection-dispersion equation, a fractional-fractal advection-dispersion equation is conceptualised. 
The developed equation is simulated in a simple transport problem to determine the meaning of 
fractional-fractal and evaluate the fractal dimension and fractional order.  
Chapter 8 Discussion 
A critical discussion of the research performed and outcomes are provided. 
Chapter 9 Conclusions 
The main findings and outcomes from the performed research are presented. 
References 
Appendices 
 
9 
 
 
2  CLASSICAL ADVECTION-DISPERSION   
  EQUATION 
 
The advection-dispersion equation is infamously challenging to solve, due to the simultaneous 
presence of two processes, namely advection and hydrodynamic dispersion. Additionally, in complex 
groundwater systems, applying an analytical solution of the advection-dispersion equation is often 
unsuccessful and thus numerical techniques are usually applied. There are numerous numerical 
techniques available, yet often the stability criteria are stringent; and oscillations and numerical 
dispersion are frequently found. Advection-dominated solute transport is often the cause for the 
numerical instabilities, where the advection-dispersion equation approximates to a hyperbolic-type 
partial differential equation, and numerical approximation methods become prone to these problems 
in this environment. To address these common numerical instabilities, an upwind numerical scheme 
can serve to correct these problems by damping responses to produce a more realistic solution in both 
heat transfer and fluid flow simulations (Hirt et al., 1975; Gray and Pinder, 1976; Patankar, 1980; 
Busnaina et al., 1991; Baptista et al., 1995; Ewing and Wang, 2001; Witek et al., 2008; Company et al., 
2009; Aswin et al., 2015; Appadu et al., 2016; Kajishima and Taira, 2016).  
Upwind numerical schemes have been applied to finite difference and finite element methods. For 
the finite difference method, it is common to make use of the upwind scheme to approximate the 
advective terms of the advection-dispersion equation by applying first-order one-sided (flow 
direction-biased) differences (Diersch, 2014). However, for finite element methods, the first order 
10 
 
upwinding creates strong smearing effects and is rarely used (Kuzmin, 2010). For this investigation, 
the traditional and augmented upwind schemes are applied for the finite difference method.  
Several authors have investigated the upwind scheme for the numerical solution of the advection-
dispersion equation, or the more general form the convection-dispersion equation (Busnaina et al., 
1991; Vested et al., 1992; Lazarov et al., 1996; Appadu et al., 2016). For instance, upstream weighting 
is often applied to finite element methods, where an element upstream of a node is weighted more 
heavily than the element located downstream, namely the Petrov-Galerkin finite element method 
(Sun and Yeh, 1983; Diersch, 2014). In Computational Fluid Dynamics (CFD), the flux-vector splitting 
upwind scheme splits the advective term into two fluxes to both represent both a positive and 
negative wave direction (Van Leer, 1982; Lomax at al., 2013). Applying aspects from the finite element 
upstream weighting and the CFD flux-splitting methods, a weighted upwind-downwind finite 
difference scheme is developed and investigated for groundwater transport simulations.  
To improve the solution of the classical advection-dispersion equation for fractured groundwater 
systems, the traditional finite difference upwind scheme is reviewed and then applied, along with 
augmented upwind schemes, to the advection-dispersion equation. The traditional explicit and 
implicit schemes, as well as the Crank-Nicolson scheme, are developed and analysed for numerical 
stability to form a base of comparison. Two new numerical approximation schemes are considered, 
namely a Crank-Nicolson scheme that is only applied for the advection term, and a weighted upwind-
downwind scheme. These newly developed schemes are analysed for numerical stability and 
compared to the traditional schemes.  
2.1 Upwind finite difference scheme for local operator 
Upwind schemes discretise hyperbolic partial differential equations with a finite differencing biased 
in the direction determined by the sign of the groundwater velocity, thus applying a solution-sensitive 
approach to simulate the direction of movement in a flow field (Courant et al., 1952; Ewing and Wang, 
2001).  
The movement of the particle or contaminant is towards the negative direction (left), when the vector 
term (𝑎) is negative; and when the vector term (𝑎) is positive, the movement of the particle or 
contaminant is towards the positive direction (right). Considering a one-dimensional derivative of a 
function 𝑓(𝑢) with a directional vector (𝑎): 
𝜕𝑓(𝑢) (2-1) 
𝑎  
𝜕𝑥
The first-order upwind scheme (explicit) numerical approximation is:  
11 
 
𝑢𝑛𝑖 − 𝑢
𝑛
𝑖−1
      𝑎                        𝑓𝑜𝑟 𝑎 > 0 (2-2) 
∆𝑥
𝑢𝑛𝑖+1 − 𝑢
𝑛
𝑖 (2-3) 
     𝑎                       𝑓𝑜𝑟 𝑎 < 0 
∆𝑥
Thus, for a positive direction (upwind side) vector term backward differences are applied, and for the 
negative direction (downwind side) vector term forward differences are applied.  
The first-order upwind finite difference scheme (implicit) numerical approximation is: 
𝑢𝑛+1𝑖 − 𝑢
𝑛+1
𝑖−1
𝑎                         𝑓𝑜𝑟 𝑎 > 0 (2-4) 
∆𝑥
𝑢𝑛+1 − 𝑢𝑛+1𝑖+1 𝑖 (2-5) 
𝑎                         𝑓𝑜𝑟 𝑎 < 0 
∆𝑥
Upwind methods eliminate undesirable oscillations, yet may have other related shortcomings, 
especially in terms of accuracy (Diersch, 2014). Thus, an upwind scheme is a compromise between 
accuracy and stability. Ewing and Wang (2001) found the upwind finite difference scheme to eliminate 
artificial oscillations in the classical finite difference method solutions, and to provide more stable 
solutions for complex multiphase and multicomponent flow systems. The upwind finite difference 
scheme can be expressed as a second-order approximation for the one-dimensional advection-
dispersion equation with a modified diffusion 𝐷(1 + (𝑃𝑒/2)(1 −  𝐶𝑟)), where 𝑃𝑒  is the Peclet 
number and 𝐶𝑟 is the Courant number (Ewing and Wang, 2001). Ewing and Wang (2001) further 
confirm the warning by Diersch (2014) where the upwind finite difference scheme introduces 
excessive numerical diffusion and the numerical solutions are dependent upon grid orientation.  
Summary of the key benefits and limitations for the upwind scheme (Diersch, 2014): 
Benefits: 
 Finite difference and finite element numerical approximation methods are prone to create 
artificial oscillations in advection-dominated problems, but upwinding can stabilise the 
solution to obtain more realistic solutions (albeit less accurate) (Diersch, 2014) 
 Upwind methods improve the efficiency of numerical solutions and reduce the need for 
extremely fine meshes (Diersch, 2014) 
Limitations: 
 The artificial damping measures by upwinding may suppress other issues in the model such as 
limitations in the spatial and temporal discretization (Diersch, 2014) 
12 
 
 Diffusion is artificially increased in dependence on the chosen mesh, and considered 
dependent on the mesh geometry, yet can still be considered accurate where the numerical 
diffusion is significantly less than the physical diffusion (Diersch, 2014) 
In summary, the upwind scheme is a powerful tool that has benefits for numerical stability. Yet, this 
tool should be applied with an understanding of the limitations, and the compromise on accuracy 
appropriately compensated for with numerical stability.  
2.2 Upwind finite difference schemes for the classical advection-dispersion equation 
The first-order upwind scheme is applied to the classical advection-dispersion equation for numerical 
approximation. Considering the one-dimensional advection-dispersion equation: 
𝜕𝑐 𝜕𝑐 𝜕2𝑐
+ 𝑣 − 𝐷 = 0 (2-6) 
𝜕𝑡 𝑥 𝜕𝑥 𝐿 𝜕𝑥2
The first-order upwind scheme for the classical advection-dispersion equation finite difference 
approximation influences the advection term, where backward or forward differences are considered 
depending on the direction of the transporting velocity. 
The traditional explicit and implicit schemes, as well as the Crank-Nicolson scheme, are developed and 
analysed for numerical stability to form a comparison base. Two new numerical approximation 
schemes are proposed, namely a Crank-Nicolson scheme where only for the advection term is applied, 
and a weighted upwind-downwind scheme. The numerical stability for the newly developed schemes 
are evaluated to validate their use in solving the advection-dispersion equation. 
2.2.1 Explicit upwind  
An explicit first-order upwind finite difference scheme for the one-dimensional advection-dispersion 
equation uses a one-sided finite difference in the upstream direction to approximate the advection 
term in the advection-dispersion equation and can be expressed as follows (assuming 𝑣 >  0) (Ewing 
and Wang, 2001): 
𝑐𝑛+1 − 𝑐𝑛 𝑐𝑛𝑖 𝑖 𝑖 − 𝑐
𝑛 𝑛 𝑛 𝑛
𝑖−1 𝑐𝑖+1 − 2𝑐𝑖 + 𝑐𝑖−1
+ 𝑣 − 𝐷 = 0 (2-7) 
∆𝑡 𝑥 ∆𝑥 𝐿 (∆𝑥)2
Rearranging, 
1 𝑛+1 1 𝑣𝑥 2𝐷𝐿 𝑛 𝑣𝑥 𝐷𝐿 𝐷𝐿( ) 𝑐𝑖 = ( − − ) 𝑐 + ( + ) 𝑐
𝑛 + ( )𝑐𝑛  
∆𝑡 ∆𝑡 ∆𝑥 (∆𝑥)2 𝑖 ∆𝑥 (∆𝑥)2 𝑖−1 (∆𝑥)2 𝑖+1  
Simplifying by using constants 𝑎1, 𝑏1, 𝑐1, and 𝑑1 
𝑎 𝑛+1 𝑛1𝑐𝑖 = 𝑏1𝑐𝑖 + 𝑐
𝑛 𝑛
1𝑐𝑖−1 + 𝑑1𝑐𝑖+1 (2-8) 
13 
 
where, 
1
 𝑎1 =  ∆𝑡
1 𝑣 2𝐷
 𝑏 𝑥 𝐿1 = − −  ∆𝑡 ∆𝑥 (∆𝑥)2
𝑣 𝐷
 𝑐 𝑥 𝐿1 = +  ∆𝑥 (∆𝑥)2
𝐷
 𝑑1 =
𝐿  
(∆𝑥)2
Equation (2-8) is the explicit upwind finite difference approximation of the one-dimensional 
advection-dispersion equation. 
2.2.2 Implicit upwind  
The implicit formulation of the first-order upwind finite difference scheme for the one-dimensional 
advection-dispersion equation is: 
𝑐𝑛+1 − 𝑐𝑛 𝑐𝑛+1 − 𝑐𝑛+1 𝑐𝑛+1 − 2𝑐𝑛+1 + 𝑐𝑛+1𝑖 𝑖 𝑖 𝑖−1 𝑖+1 𝑖 𝑖−1
+ 𝑣𝑥 − 𝐷𝐿 = 0 
(2-9) 
∆𝑡 ∆𝑥 (∆𝑥)2
Rearranging, 
1 𝑣𝑥 2𝐷𝐿
( + + )𝑐𝑛+1
1 𝑛 𝑣𝑥 𝐷𝐿 𝑛+1 𝐷𝐿= ( ) 𝑐 + ( + )𝑐 + ( )𝑐𝑛+1 
∆𝑡 ∆𝑥 (∆𝑥)2 𝑖 ∆𝑡 𝑖 ∆𝑥 (∆𝑥)2 𝑖−1 (∆𝑥)2 𝑖+1  
Simplifying by using constants 𝑎1, 𝑒1, 𝑐1, and 𝑑1 
𝑒 𝑐𝑛+1
(2-10) 
1 𝑖 = 𝑎
𝑛
1𝑐𝑖 + 𝑐 𝑐
𝑛+1
1 𝑖−1 + 𝑑
𝑛+1
1𝑐𝑖+1  
where, 
1 𝑣 2𝐷
 𝑒 𝑥 𝐿1 = + +  ∆𝑡 ∆𝑥 (∆𝑥)2
Equation (2-10) is the implicit upwind finite difference approximation of the one-dimensional 
advection-dispersion equation. 
2.2.3 Upwind Crank-Nicolson scheme 
Upwind and Crank-Nicolson schemes were compared in terms of accuracy and numerical dispersion 
by Karahan (2006). The Crank-Nicolson scheme was found to be more accurate and reduced numerical 
dispersion, and for this reason, a combination of these two schemes is investigated here. The Crank-
Nicolson scheme applied for the first-order upwind finite difference scheme approximation for the 
advection-dispersion equation 
𝑐𝑛 − 𝑐𝑛 𝑐𝑛 𝑛 𝑛
 𝑖 𝑖−1 𝑖+1
− 2𝑐𝑖 + 𝑐𝑖−1
0.5(𝑣 − 𝐷 )  
𝑐𝑛+1 𝑛 𝑥 𝐿𝑖 − 𝑐𝑖  ∆𝑥 (∆𝑥)
2  
+
∆𝑡  
= 0 
𝑐𝑛+1 − 𝑐𝑛+1 𝑐𝑛+1 − 2𝑐𝑛+1 + 𝑐𝑛+1  (2-11) 
 𝑖 𝑖−1 𝑖+1 𝑖 𝑖−1+0.5(𝑣  𝑥 − 𝐷 )
[ ∆𝑥
𝐿 (∆𝑥)2 ]
14 
 
Expanding and rearranging, 
1 𝑣𝑥 𝐷𝐿 1 𝑣𝑥 𝐷𝐿
( + 0.5 + ) 𝑐𝑛+1 = ( − 0.5 − ) 𝑐𝑛 
∆𝑡 ∆𝑥 (∆𝑥)2 𝑖 ∆𝑡 ∆𝑥 (∆𝑥)2 𝑖
(2-12) 
𝑣𝑥 𝐷𝐿 𝑛 𝑣𝑥 𝐷𝐿 𝐷𝐿 𝐷𝐿+0.5 ( + ) 𝑐𝑖−1 + 0.5 ( + )𝑐
𝑛+1 + 0.5 ( ) 𝑐𝑛 𝑛+1
∆𝑥 (∆𝑥)2 ∆𝑥 (∆𝑥)2 𝑖−1 (∆𝑥)2 𝑖+1
+ 0.5 ( ) 𝑐𝑖+1  (∆𝑥)2
Simplifying by using constants 𝑓1, 𝑔1, ℎ1, and 𝑗1 
𝑓 𝑐𝑛+1 = 𝑔 𝑐𝑛 + ℎ 𝑐𝑛 + ℎ 𝑐𝑛+11 𝑖 1 𝑖 1 𝑖−1 1 𝑖−1 + 𝑗1𝑐
𝑛
𝑖+1 + 𝑗 𝑐
𝑛+1 (2-13) 1 𝑖+1
where, 
1 𝑣𝑥 𝐷𝐿
𝑓1 = + 0.5 +  ∆𝑡 ∆𝑥 (∆𝑥)2
1 𝑣𝑥 𝐷𝐿
𝑔1 = − 0.5 −  ∆𝑡 ∆𝑥 (∆𝑥)2
𝑣𝑥 𝐷𝐿
ℎ1 = 0.5 ( + ) ∆𝑥 (∆𝑥)2
𝐷𝐿
𝑗1 = 0.5 ( ) (∆𝑥)2
Equation (2-13) is the upwind Crank-Nicolson finite difference approximation of the one-dimensional 
advection-dispersion equation. 
2.2.4 Upwind advection Crank-Nicolson scheme  
The Crank-Nicolson scheme applied only for advection term of the advection-dispersion equation for 
the first-order upwind finite difference scheme approximation 
𝑐𝑛+1𝑖 − 𝑐
𝑛 𝑐𝑛 − 𝑐𝑛 𝑐𝑛+1 − 𝑐𝑛+1 𝑐𝑛 − 2𝑐𝑛 𝑛𝑖 𝑖 𝑖−1 𝑖 𝑖−1 𝑖+1 𝑖 + 𝑐𝑖−1
+ [0.5(𝑣𝑥 ) + 0.5 (𝑣 )] − 𝐷 = 0 ∆𝑡 ∆𝑥 𝑥 ∆𝑥 𝐿 (∆𝑥)2 (2-14) 
Expanding and rearranging, 
𝑐𝑛+1 𝑐𝑛 𝑛 𝑛 𝑛+1 𝑛+1 𝑛 𝑛 𝑛𝑖 𝑖 𝑐𝑖 𝑐𝑖−1 𝑐𝑖 𝑐𝑖−1 𝑐𝑖+1 2𝑐𝑖 𝑐𝑖−1
− + 0.5𝑣 − 0.5𝑣 + 0.5𝑣 − 0.5𝑣 − 𝐷 + 𝐷 − 𝐷 = 0 
∆𝑡 ∆𝑡 𝑥 ∆𝑥 𝑥 ∆𝑥 𝑥 ∆𝑥 𝑥 ∆𝑥 𝐿 (∆𝑥)2 𝐿 (∆𝑥)2 𝐿 (∆𝑥)2
Simplifying,  
1 𝑣𝑥 1 𝑣𝑥 2𝐷𝐿 𝑣𝑥 𝐷𝐿 𝐷𝐿
( + 0.5 ) 𝑐𝑛+1𝑖 = ( − 0.5 + ) 𝑐
𝑛
𝑖 + (0.5 + ) 𝑐
𝑛
𝑖−1 + ( ) 𝑐
𝑛
𝑖+1 ∆𝑡 ∆𝑥 ∆𝑡 ∆𝑥 (∆𝑥)2 ∆𝑥 (∆𝑥)2 (∆𝑥)2
𝑣𝑥
+(0.5 ) 𝑐𝑛+1
∆𝑥 𝑖−1
 
Simplifying by using constants 𝑘1, 𝑙1, ℎ1, 𝑑1 and 𝑜1 
𝑘 𝑛+1 𝑛 𝑛 𝑛 𝑛+1 (2-15) 1𝑐𝑖 = 𝑙1𝑐𝑖 +𝑚1𝑐𝑖−1 + 𝑑1𝑐𝑖+1 + 𝑜1𝑐𝑖−1  
where, 
1 𝑣𝑥
𝑘1 = + 0.5  ∆𝑡 ∆𝑥
15 
 
1 𝑣𝑥 2𝐷𝐿
𝑙1 = − 0.5 +  ∆𝑡 ∆𝑥 (∆𝑥)2
𝑣𝑥 𝐷𝐿
𝑚1 = 0.5 +  ∆𝑥 (∆𝑥)2
𝑣𝑥
𝑜1 = 0.5  ∆𝑥
Equation (2-15) is the upwind advection Crank-Nicolson finite difference approximation of the one-
dimensional advection-dispersion equation. 
2.2.5 Explicit upwind-downwind weighted scheme  
Considering both the upwind and downwind direction for the advection term for the first-order 
upwind finite difference scheme approximation (explicit), where the ratio of upwind to downwind is 
controlled by 𝜃, where 0 ≤ 𝜃 ≤ 1: 
𝑐𝑛+1 − 𝑐𝑛 𝑐𝑛 − 𝑐𝑛 𝑛 𝑛 𝑛 𝑛 𝑛𝑖 𝑖 𝑖 𝑖−1 𝑐𝑖+1 − 𝑐𝑖 𝑐𝑖+1 − 2𝑐𝑖 + 𝑐( ) 𝑖−1+ [𝜃 (𝑣
∆𝑡 𝑥
) + 1 − 𝜃 (𝑣𝑥 )] − 𝐷𝐿 = 0 ∆𝑥 ∆𝑥 (∆𝑥)2 (2-16) 
Expanding and rearranging, 
𝑐𝑛+1 𝑛 𝑛 𝑛 𝑛 𝑛 𝑛 𝑛 𝑛𝑖 𝑐𝑖 𝑐𝑖 𝑐𝑖−1 𝑐𝑖+1 𝑐𝑖 𝑐𝑖+1 2𝑐𝑖 𝑐𝑖−1
− + 𝜃𝑣𝑥 − 𝜃𝑣𝑥 + (1 − 𝜃)𝑣𝑥 − (1 − 𝜃)𝑣𝑥 − 𝐷 + 𝐷 −𝐷 = 0 ∆𝑡 ∆𝑡 ∆𝑥 ∆𝑥 ∆𝑥 ∆𝑥 𝐿 (∆𝑥)2 𝐿 (∆𝑥)2 𝐿 (∆𝑥)2
Simplifying, 
1 1 𝑣𝑥 𝑣𝑥 2𝐷𝐿 𝑣𝑥 𝐷𝐿 𝐷𝐿 𝑣𝑥
( ) 𝑐𝑛+1𝑖 = ( − 𝜃 + (1 − 𝜃) − ) 𝑐
𝑛 + (𝜃 + )𝑐𝑛 + ( − (1 − 𝜃) ) 𝑐𝑛  
∆𝑡 ∆𝑡 ∆𝑥 ∆𝑥 (∆𝑥)2 𝑖 ∆𝑥 (∆𝑥)2 𝑖−1 (∆𝑥)2 ∆𝑥 𝑖+1
Simplifying by using constants 𝑎1, 𝑝1, 𝑞1, and 𝑟1 
𝑎 𝑐𝑛+1 = 𝑝 𝑐𝑛 + 𝑞 𝑐𝑛 + 𝑟 𝑐𝑛  (2-17) 1 𝑖 1 𝑖 1 𝑖−1 1 𝑖+1
where, 
1 𝑣𝑥 𝑣𝑥 2𝐷𝐿
𝑝1 = − 𝜃 + (1 − 𝜃) −  ∆𝑡 ∆𝑥 ∆𝑥 (∆𝑥)2
𝑣𝑥 𝐷𝐿
𝑞1 = 𝜃 +  ∆𝑥 (∆𝑥)2
𝐷𝐿 𝑣𝑥
𝑟1 = − (1 − 𝜃)  (∆𝑥)2 ∆𝑥
Equation (2-17) is the explicit upwind-downwind weighted finite difference approximation of the one-
dimensional advection-dispersion equation. 
2.2.6 Implicit upwind-downwind weighted scheme  
Now for the implicit formulation, the upwind and downwind direction, where the ratio of upwind to 
downwind is controlled by 𝜃, where 0 ≤ 𝜃 ≤ 1: 
16 
 
𝑐𝑛+1 − 𝑐𝑛 𝑐𝑛+1 − 𝑐𝑛+1 𝑐𝑛+1 𝑛+1 𝑛+1 𝑛+1 𝑛+1𝑖 𝑖 𝑖 𝑖−1 ( ) 𝑖+1
− 𝑐𝑖 𝑐𝑖+1 − 2𝑐𝑖 + 𝑐𝑖−1
+ [𝜃 (𝑣𝑥 ) + 1 − 𝜃 (𝑣𝑥 )] − 𝐷𝐿 = 0 (2-18) ∆𝑡 ∆𝑥 ∆𝑥 (∆𝑥)2
Expanding and rearranging, 
𝑐𝑛+1𝑖 𝑐
𝑛 𝑛+1 𝑛+1 𝑛+1 𝑛+1
𝑖 𝑐𝑖 𝑐𝑖−1 𝑐𝑖+1 𝑐𝑖
− + 𝜃𝑣 − 𝜃𝑣 + (1 − 𝜃)𝑣 − (1 − 𝜃)𝑣  
∆𝑡 ∆𝑡 𝑥 ∆𝑥 𝑥 ∆𝑥 𝑥 ∆𝑥 𝑥 ∆𝑥
𝑐𝑛+1 𝑛+1 𝑛+1𝑖+1 2𝑐𝑖 𝑐𝑖−1
−𝐷𝐿 + 𝐷 − 𝐷 = 0 (∆𝑥)2 𝐿 (∆𝑥)2 𝐿 (∆𝑥)2
Simplifying, 
1 𝑣𝑥 𝑣𝑥 2𝐷𝐿
( + 𝜃 − (1 − 𝜃) + ) 𝑐𝑛+1𝑖  ∆𝑡 ∆𝑥 ∆𝑥 (∆𝑥)2
1 𝑛 𝑣𝑥 𝐷𝐿 𝐷𝐿 𝑣𝑥= ( ) 𝑐𝑖 + (𝜃 + ) 𝑐
𝑛+1 𝑛+1
∆𝑡 ∆𝑥 (∆𝑥)2 𝑖−1
+ ( − (1 − 𝜃) ) 𝑐  
(∆𝑥)2 ∆𝑥 𝑖+1
Simplifying by using constants 𝑣1, 𝑎1, 𝑞1, and 𝑟1 
𝑣 𝑐𝑛+1 𝑛 𝑛+1 𝑛+1 (2-19) 1 𝑖 = 𝑎1𝑐𝑖 + 𝑞1𝑐𝑖−1 + 𝑟1𝑐𝑖+1  
where, 
1 𝑣𝑥 𝑣𝑥 2𝐷𝐿
𝑣1 = + 𝜃 − (1 − 𝜃) +  ∆𝑡 ∆𝑥 ∆𝑥 (∆𝑥)2
Equation (2-19) is the implicit upwind-downwind weighted finite difference approximation of the one-
dimensional advection-dispersion equation. 
2.3 Numerical stability analysis 
A finite difference scheme is considered stable if the errors incurred at a discrete time step are not 
propagated throughout the simulation. Assuming a Fourier expansion in space, 
𝑐(𝑥, 𝑡) =∑?̂? (𝑡)exp (𝑖𝑘𝑚𝑥) (2-20) 
𝑛
where, 
𝑐𝑛+1 = ?̂? 𝑒𝑖𝑘𝑚𝑥𝑖 𝑛+1   
𝑐𝑛 = ?̂? 𝑒𝑖𝑘𝑚𝑥𝑖 𝑛   
         𝑐𝑛+1𝑖−1 = ?̂? 𝑒
𝑖𝑘𝑚(𝑥−∆𝑥)
𝑛+1   
         𝑐𝑛+1 𝑖𝑘𝑚(𝑥+∆𝑥)𝑖+1 = ?̂?𝑛+1𝑒   
The recursive numerical stability analysis is performed in two parts, firstly it is proved for a set ∀ 𝑛 > 1,  
|?̂?𝑛| < |?̂?𝑜| (2-21) 
17 
 
Secondly, making the assumption that |?̂?𝑛| < |?̂?𝑜| is true for all time steps, the second part of the 
numerical stability analysis is to demonstrate for a set ∀ 𝑛 ≥ 1, that 
|?̂?𝑛+1| < |?̂?𝑜| (2-22) 
By examining these two conditions, where the error becomes less over each successive step, the 
stability criteria for the numerical scheme can be determined (Gnitchogna and Atangana, 2017; 
Atangana, 2015). This method is applied to investigate the stability of the numerical schemes for the 
advection-dispersion equation discussed in the previous section.  
2.3.1 Explicit upwind  
The first-order explicit upwind finite difference approximation of the one-dimensional advection-
dispersion equation was determined to be (Equation (2-8)), and substituting induction method terms 
gives, 
𝑎 ?̂? 𝑖𝑘𝑚𝑥 𝑖𝑘𝑚𝑥 𝑖𝑘𝑚(𝑥−∆𝑥) 𝑖𝑘𝑚(𝑥+∆𝑥)1 𝑛+1𝑒 = 𝑏1?̂?𝑛𝑒 + 𝑐1?̂?𝑛𝑒 + 𝑑1?̂?𝑛𝑒   
Multiple out, 
𝑎1?̂?
𝑖𝑘𝑚𝑥
𝑛+1𝑒 = 𝑏
𝑖𝑘𝑚𝑥 𝑖𝑘𝑚𝑥 −𝑖∆𝑥𝑘𝑚 𝑖𝑘𝑚𝑥 𝑖𝑘𝑚∆𝑥
1?̂?𝑛𝑒 + 𝑐1?̂?𝑛𝑒 𝑒 + 𝑑1?̂?𝑛𝑒 𝑒   
Divide by 𝑒𝑖𝑘𝑚𝑥, 
𝑎1?̂?𝑛+1 = 𝑏1?̂?𝑛 + 𝑐 ?̂? 𝑒
−𝑖𝑘𝑚∆𝑥
1 𝑛 + 𝑑1?̂?
𝑖𝑘𝑚∆𝑥
𝑛𝑒  (2-23) 
The induction numerical stability analysis is performed in two parts; firstly it is proved for a set ∀ 𝑛 > 1,  
|?̂?𝑛| < |?̂?𝑜|  
If 𝑛 = 0, then 
𝑎1?̂?1 = 𝑏1?̂?0 + 𝑐1?̂?
−𝑖𝑘𝑚∆𝑥
0𝑒 + 𝑑 ?̂? 𝑒
𝑖𝑘𝑚∆𝑥
1 0   
1
Simplifying, and remembering that 𝑎 =  
∆𝑡
1
?̂? = ?̂? (𝑏 −𝑖𝑘𝑚∆𝑥 𝑖𝑘𝑚∆𝑥  
∆𝑡 1 0 1
+ 𝑐1𝑒 + 𝑑1𝑒 ) 
Rearranging, 
?̂?1
= ∆𝑡(𝑏 + 𝑐 𝑒−𝑖𝑘𝑚∆𝑥 + 𝑑 𝑒𝑖𝑘𝑚∆𝑥1 1 1 )  ?̂?0
A norm is applied, 
|?̂?1|
< ∆𝑡(|𝑏 | + |𝑐 𝑒−𝑖𝑘𝑚∆𝑥1 1 | + |𝑑1𝑒
𝑖𝑘𝑚∆𝑥|)  
|?̂?0|
The first condition required, for Equation (2-23), becomes, 
|?̂?1|
< 1 
|?̂?  0|
18 
 
Remembering |𝑒𝑛| = 1, the condition becomes 
∆𝑡(|𝑏1| + |𝑐1| + |𝑑1|) < 1  
It can be concluded that |?̂?1| < |?̂?𝑜|, when 
∆𝑡(|𝑏1| + |𝑐1| + |𝑑1|) < 1 
The term is expanded using the simplification terms associated with Equation (2-8) 
1 𝑣𝑥 2𝐷𝐿 𝑣𝑥 𝐷𝐿 𝐷𝐿
∆𝑡 (| − + | + | + | + | |) < 1  (2-24) 
∆𝑡 ∆𝑥 (∆𝑥)2 ∆𝑥 (∆𝑥)2 (∆𝑥)2
If the assumption is made, where 
1 2𝐷𝐿 𝑣𝑥
+ >  
∆𝑡 (∆𝑥)2 ∆𝑥
Then, 
1 𝑣𝑥 2𝐷𝐿 𝑣𝑥 𝐷𝐿 𝐷𝐿
∆𝑡 ( − + + + + ) < 1    
∆𝑡 ∆𝑥 (∆𝑥)2 ∆𝑥 (∆𝑥)2 (∆𝑥)2
Simplifying, 
1 4𝐷𝐿
∆𝑡 ( + ) < 1    
∆𝑡 (∆𝑥)2 (2-25) 
For this numerical scheme, the solution is unstable under this assumption. Considering the nature of 
the explicit scheme to the use information from the current time step, or initial time step to calculate 
the next time step thus imposes these restrictions on the solution.  
However, considering the situation where the opposite assumption is made, 
1 2𝐷𝐿 𝑣𝑥
+ <  
∆𝑡 (∆𝑥)2 ∆𝑥
Then, 
1 𝑣𝑥 2𝐷𝐿 𝑣𝑥 𝐷𝐿 𝐷𝐿
∆𝑡 (− + − + + + ) < 1    (2-26) 
∆𝑡 ∆𝑥 (∆𝑥)2 ∆𝑥 (∆𝑥)2 (∆𝑥)2
Simplifying, 
∆𝑡
𝑣𝑥 < 1 ∆𝑥    (2-27) 
This condition corresponds to the Courant-Friedrichs-Lewy (CFL) condition (Cr) which is a well-known 
convergence criterion for the classical advection-dispersion equation when solved using an explicit 
finite difference numerical approximation method (Courant et al., 1967; Ewing and Wang, 2001). 
Therefore, the first-order explicit upwind finite difference approximation of the one-dimensional 
advection-dispersion equation is stable when: 
1 2𝐷𝐿 𝑣𝑥
+ <  
∆𝑡 (∆𝑥)2 ∆𝑥
19 
 
Thus, the use of information from the current time step, or initial time step to calculate the next time 
step requires that the ratio of groundwater velocity to the cell size be greater than the inverse of the 
time step and the ratio of dispersivity to the cell size.  
Furthermore for the stability analysis, the assumption that |?̂?𝑛| < |?̂?𝑜| is true for all time steps is made, 
and the second part of the numerical stability analysis is to demonstrate for a set ∀ 𝑛 ≥ 1, that 
|?̂?𝑛+1| < |?̂?𝑜|  
Rearranging Equation (2-23), 
(𝑏 + 𝑐 𝑒−𝑖𝑘𝑚∆𝑥1 1 + 𝑑 𝑒
𝑖𝑘𝑚∆𝑥
1 )
?̂?𝑛+1 = ?̂?𝑛 
 
𝑎1
Applying a norm on both sides, 
(|𝑏1| + |𝑐1𝑒
−𝑖𝑘𝑚∆𝑥| + |𝑑1𝑒
𝑖𝑘𝑚∆𝑥|)
|?̂?𝑛+1| < |?̂? | |𝑎 | 𝑛  1
1
Remembering |𝑒𝑛| = 1 and 𝑎 = , the condition becomes 
∆𝑡
|?̂?𝑛+1| < ∆𝑡(|𝑏1| + |𝑐1| + |𝑑1|) |?̂?𝑛|  
Remembering that it has been proven that for a set ∀ 𝑛 > 1, 
|?̂?𝑛| < |?̂?𝑜|  
Thus,  
|?̂?𝑛+1| < ∆𝑡(|𝑏1| + |𝑐1| + |𝑑1|) |?̂?𝑛| < ∆𝑡(|𝑏| + |𝑐1| + |𝑑1|) |?̂?𝑜|  
and, it can be inferred that 
|?̂?𝑛+1| < ∆𝑡(|𝑏1| + |𝑐1| + |𝑑1|)|?̂?𝑜|  
Thus, the solution will be stable when, 
∆𝑡(|𝑏1| + |𝑐1| + |𝑑1|) < 1 
The term is expanded using the simplification terms associated with Equation (2-8) 
1 𝑣𝑥 2𝐷𝐿 𝑣𝑥 𝐷𝐿 𝐷𝐿
∆𝑡 (| − + | + | + | + | |) < 1 (2-28) 
∆𝑡 ∆𝑥 (∆𝑥)2 ∆𝑥 (∆𝑥)2 (∆𝑥)2
If the assumption is made, where 
1 2𝐷𝐿 𝑣𝑥
+ >  
∆𝑡 (∆𝑥)2 ∆𝑥
Then, 
1 𝑣𝑥 2𝐷𝐿 𝑣𝑥 𝐷𝐿 𝐷𝐿
∆𝑡 ( − + + + + ) < 1    
∆𝑡 ∆𝑥 (∆𝑥)2 ∆𝑥 (∆𝑥)2 (∆𝑥)2
20 
 
Simplifying, the condition for |?̂?𝑛+1| < |?̂?𝑜| is similar to the condition determined for |?̂?𝑛| < |?̂?𝑜|, 
where the explicit scheme is unstable under this assumption. Conversely, consider the complementary 
assumption, 
1 2𝐷𝐿 𝑣𝑥
+ <  
∆𝑡 (∆𝑥)2 ∆𝑥
Then, 
1 𝑣𝑥 2𝐷𝐿 𝑣𝑥 𝐷𝐿 𝐷𝐿
∆𝑡 (− + − + + + ) > 1    
∆𝑡 ∆𝑥 (∆𝑥)2 ∆𝑥 (∆𝑥)2 (∆𝑥)2
Simplifying, 
∆𝑡
𝑣𝑥 < 1 ∆𝑥 (2-29) 
Thus, the first-order implicit upwind scheme is conditionally stable under this assumption, and similar 
to the condition for |?̂?𝑛| < |?̂?𝑜|, where the solution is stable under this assumption. The use 
information from the current time step or initial time step to calculate the next time step requires that 
the ratio of groundwater velocity to the cell size be greater than the inverse of the time step and the 
ratio of dispersivity to the cell size.  
In summary, the first-order explicit upwind finite difference approximation of the one-dimensional 
advection-dispersion equation has the following stability criterion (Courant condition) (Equation (2-
29)). Under this assumption and condition, the error of the approximation is not propagated 
throughout the solution, but rather decreases with each time step, as according to the induction 
method, where for all values of n, |?̂?𝑛+1| < |?̂?𝑜|. 
2.3.2 Implicit upwind  
The implicit formulation of the first-order upwind finite difference scheme for the one-dimensional 
advection-dispersion equation was determined to be (Equation (2-10)), and substituting induction 
method terms gives, 
𝑒 ?̂? 𝑒𝑖𝑘𝑚𝑥 = 𝑎 ?̂? 𝑒𝑖𝑘𝑚𝑥1 𝑛+1 1 𝑛 + 𝑐1?̂?𝑛+1𝑒
𝑖𝑘𝑚(𝑥−∆𝑥) + 𝑑 𝑖𝑘𝑚(𝑥+∆𝑥)1?̂?𝑛+1𝑒   
Multiple out, 
𝑒 𝑖𝑘𝑚𝑥 𝑖𝑘𝑚𝑥 𝑖𝑘𝑚𝑥1?̂?𝑛+1𝑒 = 𝑎1?̂?𝑛𝑒 + 𝑐1?̂?𝑛+1𝑒 𝑒
−𝑖𝑘𝑚∆𝑥 + 𝑑 ?̂? 𝑒𝑖𝑘𝑚𝑥𝑒𝑖𝑘𝑚∆𝑥1 𝑛+1   
Divide by 𝑒𝑖𝑥𝑘𝑚, 
𝑒1?̂?𝑛+1 = 𝑎 ?̂? + 𝑐 ?̂? 𝑒
−𝑖𝑘𝑚∆𝑥 + 𝑑 ?̂? 𝑒𝑖𝑘𝑚∆𝑥1 𝑛 1 𝑛+1 1 𝑛+1  (2-30) 
The induction numerical stability analysis is performed in two parts, firstly it is proved for a set ∀ 𝑛 ≥ 1,  
|?̂?𝑛| < |?̂?𝑜|  
If 𝑛 = 0, then 
21 
 
𝑒1?̂?
−𝑖𝑘𝑚∆𝑥
1 = 𝑎1?̂?0 + 𝑐1?̂?1𝑒 + 𝑑
𝑖𝑘𝑚∆𝑥
1?̂?1𝑒   
Rearrange, 
𝑒 ?̂? − 𝑐 ?̂? 𝑒−𝑖𝑘𝑚∆𝑥 − 𝑑 𝑖𝑘𝑚∆𝑥1 1 1 1 1?̂?1𝑒 = 𝑎1?̂?0  
1
Simplifying, and remembering that 𝑎1 =  ∆𝑡
1
?̂? (𝑒 − 𝑐 𝑒−𝑖𝑘𝑚∆𝑥1 1 1 − 𝑑
𝑖𝑘𝑚∆𝑥
1𝑒 ) = ?̂?  
 
∆𝑡 0
Rearranging, 
?̂?1 1
=  
?̂?0 ∆𝑡(𝑒1 − 𝑐 𝑒
−𝑖𝑘
1 𝑚
∆𝑥 − 𝑑1𝑒
𝑖𝑘𝑚∆𝑥)  
A norm is applied on both sides, 
|?̂?1| 1
<   
|?̂?0| ∆𝑡(|𝑒 | + |−𝑐 𝑒
−𝑖𝑘𝑚∆𝑥| + |−𝑑 𝑒𝑖𝑘1 1 1 𝑚
∆𝑥|)
The condition required |?̂?𝑛| < |?̂?𝑜|, thus becomes, 
|?̂?1|
< 1  
|?̂?0|
Remembering |𝑒𝑛| = 1, the condition becomes 
1
< 1 
∆𝑡(|𝑒  1| + |𝑐1| + |𝑑1|)
It can be concluded that |?̂?1| < |?̂?𝑜|, when 
∆𝑡(|𝑒1| + |𝑐1| + |𝑑1|) > 1 
The term is expanded using the simplification terms associated with Equation (2-10) 
1 𝑣𝑥 2𝐷𝐿 𝑣𝑥 𝐷𝐿 𝐷𝐿
∆𝑡 (| + + | + | + +2 | | |) > 1  (2-31) ∆𝑡 ∆𝑥 (∆𝑥) ∆𝑥 (∆𝑥)2 (∆𝑥)2
Simplifying, 
2𝑣𝑥 4𝐷𝐿
 + > 0 
∆𝑥 (∆𝑥)2 (2-32) 
The condition confirms that the first-order upwind implicit scheme is unconditionally stable. 
Secondly, making the assumption that |?̂?𝑛| < |?̂?𝑜| is true for all time steps, the second part of the 
numerical stability analysis is to demonstrate for a set ∀ 𝑛 ≥ 1, that 
|?̂?𝑛+1| < |?̂?𝑜|  
Rearranging Equation (2-30), 
𝑒1?̂?𝑛+1 − 𝑐
−𝑖𝑘𝑚∆𝑥
1?̂?𝑛+1𝑒 − 𝑑 ?̂? 𝑒
𝑖𝑘𝑚∆𝑥
1 𝑛+1 = 𝑎1?̂?𝑛  
22 
 
Simplifying, 
𝑎1
?̂?𝑛+1 = ?̂?  
 
(𝑒 − 𝑐 𝑒−𝑖𝑘𝑚∆𝑥1 1 − 𝑑1𝑒
𝑖𝑘 𝑛𝑚∆𝑥)
Applying the norm, 
|𝑎1|
|?̂?𝑛+1| < |?̂? | (|𝑒 | + |−𝑐 𝑒−𝑖𝑘𝑚∆𝑥| + |−𝑑 𝑒𝑖𝑘1 1 1 𝑚
∆𝑥|) 𝑛  
1
Remembering |𝑒𝑛| = 1 and 𝑎1 = , the condition becomes ∆𝑡
1
|?̂?𝑛+1| < |?̂? | ∆𝑡(|𝑒1| + |𝑐1| + |𝑑1|)
𝑛  
Remembering that it has been proven that for a set ∀ 𝑛 > 1, 
|?̂?𝑛| < |?̂?𝑜|  
Thus,  
1 1
|?̂?𝑛+1| < |?̂? | < |?̂? | ∆𝑡(|𝑒1| + |𝑐1| + |𝑑1|)
𝑛 ∆𝑡(|𝑒1| + |𝑐1| + |𝑑1|)
𝑜  
and, it can be inferred that 
1
|?̂?𝑛+1| < |?̂? | ∆𝑡(|𝑒1| + |𝑐1| + |𝑑1|)
𝑜  
Thus, the solution will be stable when, 
1
< 1 
∆𝑡(|𝑒1| + |𝑐1| + |𝑑1|)
It can be concluded that |?̂?𝑛+1| < |?̂?𝑜|, when 
∆𝑡(|𝑒1| + |𝑐1| + |𝑑1|) > 1 
The term is expanded using the simplification terms associated with Equation (2-10) 
1 𝑣𝑥 2𝐷𝐿 𝑣𝑥 𝐷𝐿 𝐷𝐿
∆𝑡 (| + + | + | + | + | ) > 1  (2-33) 
∆𝑡 ∆𝑥 (∆𝑥)2 ∆𝑥 (∆𝑥)2 (∆𝑥)2
|
Simplifying, 
2𝑣𝑥 4𝐷𝐿
 + > 0 
∆𝑥 (∆𝑥)2    (2-34) 
The condition for |?̂?𝑛+1| < |?̂?𝑜| is similar to the condition determined for |?̂?𝑛| < |?̂?𝑜|, where for the 
first-order implicit upwind scheme is unconditionally stable.  
In summary, the first-order implicit upwind finite difference approximation of the one-dimensional 
advection-dispersion equation is unconditionally stable since under these conditions, the error of the 
approximation is not propagated throughout the solution, but rather decreases with each time step. 
23 
 
2.3.3 Upwind Crank-Nicolson scheme 
Crank-Nicolson scheme applied for the first-order upwind finite difference scheme approximation for 
the advection-dispersion equation was determined to be (Equation (2-13)), and now induction 
method terms are substituted to facilitate the stability analysis, 
𝑓 ?̂? 𝑒𝑖𝑘𝑚𝑥 = 𝑔 ?̂? 𝑒𝑖𝑘𝑚𝑥 + ℎ ?̂? 𝑒𝑖𝑘𝑚(𝑥−∆𝑥) + ℎ ?̂? 𝑒𝑖𝑘𝑚(𝑥−∆𝑥) + 𝑗 ?̂? 𝑒𝑖𝑘𝑚(𝑥+∆𝑥)1 𝑛+1 1 𝑛 1 𝑛 1 𝑛+1 1 𝑛  
 
+𝑗 ?̂? 𝑒𝑖(𝑥+∆𝑥)𝑘𝑚1 𝑛+1  
Multiple out, 
𝑓1?̂?𝑛+1𝑒
𝑖𝑘𝑚𝑥 = 𝑔 𝑖𝑘𝑚𝑥 𝑖𝑘𝑚𝑥 −𝑖𝑘𝑚∆𝑥1?̂?𝑛𝑒 + ℎ1?̂?𝑛𝑒 𝑒 + ℎ1?̂? 𝑒
𝑖𝑘𝑚𝑥∆𝑒−𝑖𝑘𝑚∆𝑥𝑛+1 + 𝑗1?̂?𝑛𝑒
𝑖𝑘𝑚𝑥𝑒𝑖𝑘𝑚∆𝑥 
 
+𝑗 ?̂? 𝑒𝑖𝑘𝑚𝑥𝑒𝑖𝑘𝑚∆𝑥1 𝑛+1  
Divide by 𝑒𝑖𝑘𝑚𝑥, 
𝑓1?̂? = 𝑔 ?̂? + ℎ ?̂? 𝑒
−𝑖𝑘𝑚∆𝑥
𝑛+1 1 𝑛 1 𝑛 + ℎ1?̂?
−𝑖𝑘𝑚∆𝑥 𝑖𝑘𝑚∆𝑥 𝑖𝑘𝑚∆𝑥
𝑛+1𝑒 + 𝑗1?̂?𝑛𝑒 + 𝑗1?̂?𝑛+1𝑒  (2-35) 
The induction numerical stability analysis is performed in two parts, firstly it is proved for a set ∀ 𝑛 ≥ 1,  
|?̂?𝑛| < |?̂?𝑜|  
If 𝑛 = 0, then 
𝑓 −𝑖𝑘𝑚∆𝑥1?̂?1 = 𝑔1?̂?0 + ℎ1?̂?0𝑒 + ℎ ?̂? 𝑒
−𝑖𝑘𝑚∆𝑥 + 𝑗 ?̂? 𝑒𝑖𝑘𝑚∆𝑥 + 𝑗 ?̂? 𝑒𝑖𝑘𝑚∆𝑥1 1 1 0 1 1   
Rearrange, 
𝑓 ?̂? − ℎ ?̂? 𝑒−𝑖𝑘𝑚∆𝑥 − 𝑗 ?̂? 𝑒𝑖𝑘𝑚∆𝑥1 1 1 1 1 1 = 𝑔1?̂?0 + ℎ ?̂? 𝑒
−𝑖𝑘𝑚∆𝑥
1 0 + 𝑗
𝑖𝑘𝑚∆𝑥
1?̂?0𝑒   
Simplifying, 
?̂? (𝑓 − ℎ 𝑒−𝑖𝑘𝑚∆𝑥 − 𝑗 𝑒𝑖𝑘𝑚∆𝑥) = ?̂? (𝑔 + ℎ 𝑒−𝑖𝑘𝑚∆𝑥 + 𝑗 𝑒𝑖𝑘𝑚∆𝑥  1 1 1 1 0 1 1 1 ) 
Rearranging, 
?̂? (𝑔 + ℎ 𝑒−𝑖𝑘𝑚∆𝑥1 1 1 + 𝑗1𝑒
𝑖𝑘𝑚∆𝑥)
=   
?̂?0 (𝑓1 − ℎ
−𝑖𝑘
1𝑒 𝑚
∆𝑥 − 𝑗 𝑖𝑘 ∆𝑥1𝑒 𝑚 )
A norm is applied, 
|?̂?1| |𝑔 | + |ℎ 𝑒
−𝑖𝑘𝑚∆𝑥
1 1 | + |𝑗 𝑒
𝑖𝑘𝑚∆𝑥
1 |
<  
|?̂?0| |𝑓1| + |−ℎ1𝑒
−𝑖𝑘𝑚∆𝑥| + |−𝑗 𝑒𝑖𝑘𝑚∆𝑥  1 |
The condition required|?̂?𝑛| < |?̂?𝑜|, thus becomes: 
|?̂?1|
< 1  
|?̂?0|
Remembering |𝑒𝑛| = 1, the condition becomes 
|𝑔1| + |ℎ1| + |𝑗1|
< 1 
|𝑓1| + |ℎ1| + |𝑗
 
1|
24 
 
The term is expanded using the simplification terms associated with Equation (2-13) 
1 𝑣 𝐷
| − 0.5 𝑥 − 𝐿
𝑣 𝐷 𝐷
2 + 0.5 (
𝑥 + 𝐿 ) + 0.5 𝐿
∆𝑡 ∆𝑥 ∆𝑥 |(∆𝑥) (∆𝑥)2 (∆𝑥)2
< 1  
1 𝑣 𝐷 𝑣 𝐷 𝐷
|∆𝑡 + 0.5
𝑥 + 𝐿 + 0.5 ( 𝑥 + 𝐿 ) + 0.5 𝐿∆𝑥 (∆𝑥)2 ∆𝑥 2
|
(∆𝑥) (∆𝑥)2
Simplifying, 
𝑣𝑥 2𝐷𝐿
+ > 0 
∆𝑥 (∆𝑥)2
(2-36) 
The first-order upwind Crank-Nicolson scheme for the advection-dispersion equation is 
unconditionally stable.  
Secondly, making the assumption that |?̂?𝑛| < |?̂?𝑜| is true for all time steps, the second part of the 
numerical stability analysis is to demonstrate for a set ∀ 𝑛 ≥ 1, that 
|?̂?𝑛+1| < |?̂?𝑜|  
Rearranging Equation (2-35), 
𝑓1?̂?𝑛+1 − ℎ
−𝑖𝑘𝑚∆𝑥 𝑖𝑘𝑚∆𝑥 −𝑖𝑘𝑚∆𝑥 𝑖𝑘𝑚∆𝑥
1?̂?𝑛+1𝑒 − 𝑗1?̂?𝑛+1𝑒 = 𝑔1?̂?𝑛 + ℎ1?̂?𝑛𝑒 + 𝑗1?̂?𝑛𝑒   
Simplifying, 
𝑔 + ℎ 𝑒−𝑖𝑘𝑚∆𝑥 + 𝑗 𝑒𝑖𝑘𝑚∆𝑥1 1 1  
?̂?𝑛+1 = ?̂?  𝑓 − ℎ 𝑒−𝑖𝑘𝑚∆𝑥 − 𝑗 𝑒𝑖𝑘𝑚∆𝑥 𝑛1 1 1
Applying a norm on both sides, 
|𝑔 | + |ℎ 𝑒−𝑖𝑘𝑚∆𝑥| + |𝑗 𝑒𝑖𝑘𝑚∆𝑥1 1 1 |
|?̂?𝑛+1| < |?̂? | |𝑓1| + |−ℎ 𝑒
−𝑖𝑘
1 𝑚
∆𝑥| + |−𝑗 𝑒𝑖𝑘 𝑛  1 𝑚
∆𝑥|
Remembering |𝑒𝑛| = 1, the condition becomes 
|𝑔1| + |ℎ1| + |𝑗1|
|?̂?𝑛+1| < |?̂? | |𝑓1| + |ℎ1| + |𝑗1|
𝑛  
Remembering that it has been proven that for a set ∀ 𝑛 > 1, 
|?̂?𝑛| < |?̂?𝑜|  
Thus,  
|𝑔1| + |ℎ1| + |𝑗1| |𝑔1| + |ℎ1| + |𝑗1|
|?̂?𝑛+1| < |?̂?𝑛| < |?̂? | |𝑓1| + |ℎ1| + |𝑗1| |𝑓1| + |ℎ1| + |𝑗1|
𝑜  
and, it can be inferred that 
|𝑔1| + |ℎ1| + |𝑗1|
|?̂?𝑛+1| < |?̂?𝑜| |𝑓1| + |ℎ1| + |𝑗1|
 
Thus, the solution will be stable when, 
25 
 
|𝑔1| + |ℎ1| + |𝑗1|
< 1 
|𝑓1| + |ℎ1| + |𝑗1|
The term is expanded using the simplification terms associated with Equation (2-13), 
1 𝑣𝑥 𝐷𝐿 𝑣𝑥 𝐷𝐿 𝐷|∆𝑡 − 0.5∆𝑥 − 2 + 0.5 (∆𝑥 + ) + 0.5
𝐿 |
(∆𝑥) (∆𝑥)2 (∆𝑥)2
< 1  
1 𝑣𝑥 𝐷| + 0.5 + 𝐿
𝑣𝑥 𝐷𝐿 𝐷+ 0.5 ( + ) + 0.5 𝐿
∆𝑡 ∆𝑥 |(∆𝑥)2 ∆𝑥 (∆𝑥)2 (∆𝑥)2
Simplifying,  
𝑣𝑥 2𝐷𝐿
+ > 0 
∆𝑥 (∆𝑥)2 (2-37) 
Equation (2-37) is similar to Equation (2-36), and thus by following a similar procedure the condition 
for |?̂?𝑛+1| < |?̂?𝑜| is, unconditionally stable for both perspectives.  
2.3.4 Upwind advection Crank-Nicolson scheme 
Remembering that the advection-dominated system is the main culprit for the numerical instabilities, 
the Crank-Nicolson scheme is applied to the advection term only for the first-order. The upwind 
advection Crank-Nicolson scheme is determined to be (Equation (2-15)), and now substituting 
induction method terms, 
𝑘 ?̂? 𝑒𝑖𝑘𝑚𝑥 = 𝑙 ?̂? 𝑒𝑖𝑘𝑚𝑥 +𝑚 ?̂? 𝑒𝑖𝑘𝑚(𝑥−∆𝑥) + 𝑑 ?̂? 𝑒𝑖𝑘𝑚(𝑥+∆𝑥) 𝑖(𝑥−∆𝑥)𝑘𝑚1 𝑛+1 1 𝑛 1 𝑛 1 𝑛 + 𝑜1?̂?𝑛+1𝑒   
Multiple out, 
𝑘 ?̂? 𝑒𝑖𝑘𝑚𝑥 = 𝑙 ?̂? 𝑒𝑖𝑘𝑚𝑥 +𝑚 ?̂? 𝑖𝑘𝑚𝑥 −𝑖𝑘𝑚∆𝑥 𝑖𝑘𝑚𝑥 𝑖𝑘𝑚∆𝑥 𝑖𝑘𝑚𝑥 𝑖𝑘𝑚∆𝑥1 𝑛+1 1 𝑛 1 𝑛𝑒 𝑒 + 𝑑1?̂?𝑛𝑒 𝑒 + 𝑜1?̂?𝑛+1𝑒 𝑒   
Divide by 𝑒𝑖𝑘𝑚𝑥, 
𝑘 −𝑖𝑘𝑚∆𝑥1?̂?𝑛+1 = 𝑙1?̂?𝑛 +𝑚1?̂?𝑛𝑒 + 𝑑1?̂? 𝑒
𝑖𝑘𝑚∆𝑥
𝑛 + 𝑜
𝑖𝑘𝑚∆𝑥
1?̂?𝑛+1𝑒  (2-38) 
The induction numerical stability analysis is performed in two parts, firstly it is proved for a set ∀ 𝑛 ≥ 1,  
|?̂?𝑛| < |?̂?𝑜|  
If 𝑛 = 0, then 
𝑘1?̂? = 𝑙 ?̂? + 𝑚 ?̂? 𝑒
−𝑖𝑘𝑚∆𝑥
1 1 0 1 0 + 𝑑1?̂?0𝑒
𝑖𝑘𝑚∆𝑥 + 𝑜1?̂? 𝑒
𝑖𝑘𝑚∆𝑥
1   
Rearrange, 
𝑘 ?̂? − 𝑜 ?̂? 𝑒𝑖𝑘𝑚∆𝑥 = 𝑙 ?̂? + 𝑚 ?̂? 𝑒−𝑖𝑘𝑚∆𝑥 𝑖𝑘𝑚∆𝑥1 1 1 1 1 0 1 0 + 𝑑1?̂?0𝑒   
Simplifying, 
?̂?1(𝑘 − 𝑜
𝑖𝑘𝑚∆𝑥 −𝑖𝑘𝑚∆𝑥 𝑖𝑘𝑚∆𝑥
1 1𝑒 ) = ?̂?0(𝑙1 +𝑚1𝑒 + 𝑑1𝑒 )  
Rearranging, 
26 
 
?̂? 𝑙 + 𝑚 𝑒−𝑖𝑘𝑚∆𝑥1 1 1 + 𝑑 𝑒
𝑖𝑘𝑚∆𝑥
1
=   
?̂?0 𝑘1 − 𝑜 𝑒
𝑖𝑘𝑚∆𝑥1
The norm is taken on both sides, 
|?̂? | |𝑙 | + |𝑚 𝑒−𝑖𝑘𝑚∆𝑥| + |𝑑 𝑒𝑖𝑘𝑚∆𝑥1 1 1 1 |
<  
|?̂?0| |𝑘1| + |−𝑜1𝑒
𝑖𝑘  𝑚∆𝑥|
The stability condition required |?̂?𝑛| < |?̂?𝑜|, develops to, 
|?̂?1|
< 1  
|?̂?0|
Remembering |𝑒𝑛| = 1, the condition becomes 
|𝑙1| + |𝑚1| + |𝑑1|
< 1  
|𝑘1| + |𝑜1|
The term is expanded using the simplification terms associated with Equation (2-15) 
1 𝑣𝑥 2𝐷| − 0.5 + 𝐿
𝑣 𝐷 𝐷
∆𝑡 ∆𝑥 | + |0.5
𝑥 𝐿 𝐿
(∆𝑥)2 ∆𝑥
+ | + | |
(∆𝑥)2 (∆𝑥)2
< 1 (2-39) 
1 𝑣
| + 0.5 𝑥
𝑣
∆𝑡 ∆𝑥|
+ |0.5 𝑥∆𝑥|
If the assumption is made, where 
1 2𝐷𝐿 𝑣𝑥
+ > 0.5  
∆𝑡 (∆𝑥)2 ∆𝑥
Then, 
1 𝑣𝑥 2𝐷𝐿 𝑣𝑥 𝐷 𝐷
∆𝑡 − 0.5∆𝑥 + 2 + 0.5∆𝑥 +
𝐿 + 𝐿
(∆𝑥) (∆𝑥)2 (∆𝑥)2
< 1  
1 𝑣 𝑣
+ 0.5 𝑥 + 0.5 𝑥
∆𝑡 ∆𝑥 ∆𝑥
Simplifying, 
4𝐷𝐿 𝑣𝑥
<  
(∆𝑥)2 ∆𝑥 (2-40) 
Under this assumption, the upwind advection Crank-Nicolson scheme for the advection-dispersion 
equation is conditionally stable. 
However, if the complementary assumption is made, where 
1 2𝐷𝐿 𝑣𝑥
+ < 0.5  
∆𝑡 (∆𝑥)2 ∆𝑥
Then, 
1 𝑣 2𝐷 𝑣 𝐷 𝐷
−∆𝑡 + 0.5
𝑥
∆𝑥 −
𝐿
2 + 0.5
𝑥 𝐿
∆𝑥 + +
𝐿
(∆𝑥) (∆𝑥)2 (∆𝑥)2
< 1  
1 𝑣 𝑣
∆𝑡 + 0.5
𝑥
∆𝑥 + 0.5
𝑥
∆𝑥
Simplifying, 
27 
 
2
> 0 
∆𝑡 (2-41) 
Under this assumption, the upwind advection Crank-Nicolson scheme for the advection-dispersion 
equation is unconditionally stable. 
Secondly, making the assumption that |?̂?𝑛| < |?̂?𝑜| is true for all time steps, the second part of the 
numerical stability analysis is to demonstrate for a set ∀ 𝑛 ≥ 1, that 
|?̂?𝑛+1| < |?̂?𝑜|  
Rearranging Equation (2-38), 
𝑘1?̂?𝑛+1 − 𝑜1?̂?𝑛+1𝑒
𝑖𝑘𝑚∆𝑥 = 𝑙 −𝑖𝑘𝑚∆𝑥1?̂?𝑛 +𝑚1?̂?𝑛𝑒 + 𝑑 ?̂? 𝑒
𝑖𝑘𝑚∆𝑥
1 𝑛   
Simplifying, 
𝑙 + 𝑚 −𝑖𝑘𝑚∆𝑥 𝑖𝑘𝑚∆𝑥1 1𝑒 + 𝑑1𝑒
?̂?𝑛+1 = ?̂?𝑛 
 
𝑘 − 𝑜 𝑒𝑖𝑘𝑚∆𝑥1 1
A norm is applied to, 
|𝑙1| + |𝑚
−𝑖𝑘𝑚∆𝑥 𝑖𝑘𝑚∆𝑥
1𝑒 | + |𝑑1𝑒 |
|?̂?𝑛+1| < |?̂? |  |𝑘 | + |−𝑜 𝑒𝑖𝑘𝑚∆𝑥 𝑛1 1 |
Remembering |𝑒𝑛| = 1, the condition becomes 
|𝑙1| + |𝑚1| + |𝑑1|
|?̂?𝑛+1| < |?̂? | |𝑘 𝑛  1| + |𝑜1|
Remembering that it has been proven that for a set ∀ 𝑛 > 1, 
|?̂?𝑛| < |?̂?𝑜| 
 
Thus,  
|𝑙1| + |𝑚1| + |𝑑1| |𝑙1| + |𝑚1| + |𝑑1|
|?̂?𝑛+1| < |?̂?𝑛| < |?̂?𝑜|  |𝑘1| + |𝑜1| |𝑘1| + |𝑜1|
and, it can be inferred that 
|𝑙1| + |𝑚1| + |𝑑1|
|?̂?𝑛+1| < |?̂?|𝑘 𝑜
|  
1| + |𝑜1|
Thus, the solution will be stable when, 
|𝑙1| + |𝑚1| + |𝑑1|
< 1 
|𝑘1| + |𝑜1|
The term is expanded using the simplification terms associated with Equation (2-15) 
1 𝑣𝑥 2𝐷𝐿 𝑣𝑥 𝐷| 𝐿
𝐷𝐿
∆𝑡 − 0.5∆𝑥 + | + |0.5 + | + | |(∆𝑥)2 ∆𝑥 (∆𝑥)2 (∆𝑥)2
< 1 (2-42) 
1 𝑣 𝑣
|∆𝑡 + 0.5
𝑥
∆𝑥| + |0.5
𝑥
∆𝑥|
28 
 
Equation (2-42) is similar to Equation (2-39), and thus by following a similar procedure the condition 
for |?̂?𝑛+1| < |?̂?𝑜| becomes,  
4𝐷𝐿 𝑣𝑥
<  
(∆𝑥)2 ∆𝑥
under the assumption 
1 2𝐷𝐿 𝑣𝑥
+ > 0.5 . 
∆𝑡 (∆𝑥)2 ∆𝑥
And, if the opposite assumption applies 
1 2𝐷𝐿 𝑣𝑥
+ < 0.5  
∆𝑡 (∆𝑥)2 ∆𝑥
then, the first-order upwind advection Crank-Nicolson scheme for the advection-dispersion equation 
is unconditionally stable.  
In summary, the first-order upwind advection Crank-Nicolson finite difference approximation of the 
one-dimensional advection-dispersion equation has the following stability criterion,  
4𝐷𝐿 𝑣𝑥
<  
(∆𝑥)2 ∆𝑥
Under these conditions, the error of the approximation is not proliferated throughout the solution. 
2.3.5 Explicit upwind-downwind weighted scheme 
The explicit first-order upwind-downwind weighted finite difference approximation of the one-
dimensional advection-dispersion equation was determined to be (Equation (2-17)). Substituting 
induction stability analysis method terms, 
𝑎 ?̂? 𝑒𝑖𝑘𝑚𝑥 = 𝑝 ?̂? 𝑒𝑖𝑘𝑚𝑥 + 𝑞 ?̂? 𝑒𝑖𝑘𝑚(𝑥−∆𝑥)1 𝑛+1 1 𝑛 1 𝑛 + 𝑟1?̂?𝑛𝑒
𝑖𝑘𝑚(𝑥+∆𝑥)  
Multiple out, 
𝑎 ?̂? 𝑒𝑖𝑘𝑚𝑥 = 𝑝 ?̂? 𝑒𝑖𝑘𝑚𝑥 + 𝑞 ?̂? 𝑒𝑖𝑘𝑚𝑥𝑒−𝑖𝑘𝑚∆𝑥 + 𝑟 ?̂? 𝑒𝑖𝑘𝑚𝑥1 𝑛+1 1 𝑛 1 𝑛 1 𝑛 𝑒
𝑖𝑘𝑚∆𝑥  
Divide by 𝑒𝑖𝑘𝑚𝑥, 
𝑎 ?̂? = 𝑝 ?̂? + 𝑞 ?̂? 𝑒−𝑖𝑘𝑚∆𝑥1 𝑛+1 1 𝑛 1 𝑛 + 𝑟1?̂? 𝑒
𝑖𝑘𝑚∆𝑥
𝑛  (2-43) 
The induction numerical stability analysis is performed in two parts, firstly it is proved for a set ∀ 𝑛 ≥ 1,  
|?̂?𝑛| < |?̂?𝑜|  
If 𝑛 = 0, then 
𝑎1?̂?1 = 𝑝1?̂?0 + 𝑞1?̂?
−𝑖𝑘𝑚∆𝑥
0𝑒 + 𝑟 ?̂? 𝑒
𝑖𝑘𝑚∆𝑥
1 0   
Rearrange, 
29 
 
𝑎 ?̂? = 𝑝 ?̂? + 𝑞 ?̂? 𝑒−𝑖𝑘𝑚∆𝑥 + 𝑟 ?̂? 𝑒𝑖𝑘𝑚∆𝑥1 1 1 0 1 0 1 0   
Simplifying, 
𝑎1?̂? = ?̂? (𝑝 + 𝑞 𝑒
−𝑖𝑘𝑚∆𝑥 + 𝑟 𝑒𝑖𝑘𝑚∆𝑥)  1 0 1 1 1
Rearranging, 
?̂? 𝑝 + 𝑞 𝑒−𝑖𝑘𝑚∆𝑥1 1 1 + 𝑟 𝑒
𝑖𝑘𝑚∆𝑥
1
=   
?̂?0 𝑎1
Applying the norm on both sides, 
|?̂? −𝑖𝑘𝑚∆𝑥1| |𝑝1| + |𝑞1𝑒 | + |𝑟1𝑒
𝑖𝑘𝑚∆𝑥|
<   
|?̂?0| |𝑎1|
The condition required can be expressed as, 
|?̂?1|
< 1  
|?̂?0|
1
Remembering |𝑒𝑛| = 1 and 𝑎1 = , the condition becomes ∆𝑡
∆𝑡(|𝑝1| + |𝑞1| + |𝑟1|) < 1  
The term is expanded using the simplification terms associated with Equation (2-17) 
1 𝑣𝑥 𝑣𝑥 2𝐷𝐿 𝑣𝑥 𝐷𝐿 𝐷𝐿 𝑣𝑥
∆𝑡 (| − 𝜃 + (1 − 𝜃) − | + |𝜃 + | + | − (1 − 𝜃) |) < 1 (2-44) 
∆𝑡 ∆𝑥 ∆𝑥 (∆𝑥)2 ∆𝑥 (∆𝑥)2 (∆𝑥)2 ∆𝑥
If the assumption is made, where 
1 𝑣𝑥 𝑣𝑥 2𝐷𝐿
+ (1 − 𝜃) > 𝜃 + , 
∆𝑡 ∆𝑥 ∆𝑥 (∆𝑥)2
and 
𝐷𝐿 𝑣𝑥
> (1 − 𝜃)  
(∆𝑥)2 ∆𝑥
Then, 
1 𝑣𝑥 𝑣𝑥 2𝐷𝐿 𝑣𝑥 𝐷𝐿 𝐷𝐿 𝑣𝑥
∆𝑡 (( − 𝜃 + (1 − 𝜃) − ) + (𝜃 + ) + ( − (1 − 𝜃) )) < 1  
∆𝑡 ∆𝑥 ∆𝑥 (∆𝑥)2 ∆𝑥 (∆𝑥)2 (∆𝑥)2 ∆𝑥
Simplifying, 
1 𝑣𝑥 𝑣𝑥 𝑣𝑥 𝑣𝑥 2𝐷𝐿 𝐷𝐿 𝐷𝐿
∆𝑡 ( − 𝜃 + 𝜃 + (1 − 𝜃) − (1 − 𝜃) − + + ) < 1 (2-45) 
∆𝑡 ∆𝑥 ∆𝑥 ∆𝑥 ∆𝑥 (∆𝑥)2 (∆𝑥)2 (∆𝑥)2
Under these assumptions, the first-order explicit upwind-downwind weighted scheme for the one-
dimensional advection-dispersion equation is highly unstable. Conversely, if the assumptions are 
made, where 
1 𝑣𝑥 𝑣𝑥 2𝐷𝐿
+ (1 − 𝜃) < 𝜃 + , 
∆𝑡 ∆𝑥 ∆𝑥 (∆𝑥)2
and 
30 
 
𝐷𝐿 𝑣𝑥
< (1 − 𝜃)  
(∆𝑥)2 ∆𝑥
Then, 
1 𝑣𝑥 𝑣𝑥 2𝐷𝐿 𝑣𝑥 𝐷𝐿 𝐷𝐿 𝑣𝑥
∆𝑡 ((− + 𝜃 − (1 − 𝜃) + ) + (𝜃 + ) + (− + (1 − 𝜃) )2 2 2 ) < 1  ∆𝑡 ∆𝑥 ∆𝑥 (∆𝑥) ∆𝑥 (∆𝑥) (∆𝑥) ∆𝑥
Simplifying, 
1 𝑣𝑥 𝑣𝑥 𝑣𝑥 𝑣𝑥 2𝐷𝐿 𝐷𝐿 𝐷𝐿
∆𝑡 (− + 𝜃 + 𝜃 − (1 − 𝜃) + (1 − 𝜃) + + − ) < 1 
∆𝑡 ∆𝑥 ∆𝑥 ∆𝑥 ∆𝑥 (∆𝑥)2 (∆𝑥)2 (∆𝑥)2
(2-46) 
𝑣𝑥 2𝐷𝐿
∆𝑡 (𝜃 + ) < 1 
∆𝑥 (∆𝑥)2
Under these assumptions, the first-order explicit upwind-downwind weighted scheme for the one-
dimensional advection-dispersion equation is conditionally stable, when 𝜃 < 1. Thus, under these 
assumptions, the weighted explicit upwind-downwind scheme (0 < 𝜃 < 1) and the full downwind 
scheme (𝜃 = 0) are stable. Yet, the full upwind scheme (𝜃 = 1) is invalid under these assumptions.   
Secondly, making the assumption that |𝑐?̂?| < |𝑐?̂?| is true for all time steps, the second part of the 
numerical stability analysis is to demonstrate for a set ∀ 𝑛 ≥ 1, that 
|?̂?𝑛+1| < |?̂?𝑜|  
Remembering Equation (2-43), and simplifying, 
𝑎 ?̂? = ?̂? (𝑝 + 𝑞 𝑒−𝑖𝑘𝑚∆𝑥 + 𝑟 𝑒𝑖𝑘𝑚∆𝑥1 𝑛+1 𝑛 1 1 1 ) 
 
𝑝1 + 𝑞1𝑒
−𝑖𝑘𝑚∆𝑥 + 𝑟 𝑖𝑘𝑚∆𝑥1𝑒
?̂?𝑛+1 = ?̂?  𝑎 𝑛1
Applying a norm, 
|𝑝 −𝑖𝑘𝑚∆𝑥1| + |𝑞1𝑒 | + |𝑟 𝑒
𝑖𝑘𝑚∆𝑥
1 |
|?̂?𝑛+1| < |?̂? | |𝑎1|
𝑛  
1
Remembering |𝑒𝑛| = 1 and 𝑎1 = , the condition becomes ∆𝑡
|?̂?𝑛+1| < ∆𝑡(|𝑝1| + |𝑞1| + |𝑟1|)|?̂?𝑛|  
Remembering that it has been proven that for a set ∀ 𝑛 > 1, 
|?̂?𝑛| < |?̂?𝑜|  
Thus,  
|?̂?𝑛+1| < ∆𝑡(|𝑝 | + |𝑞 | + |𝑟1|)|?̂?𝑛| < ∆𝑡(|𝑝 | + |𝑞 | + |𝑟1 1 1 1 1|)|?̂?𝑜|  
and, it can be inferred that 
31 
 
|?̂?𝑛+1| < ∆𝑡(|𝑝 | + |𝑞 | + |𝑟1|)|?̂?1 1 𝑜|  
Thus, the solution will be stable when, 
∆𝑡(|𝑝1| + |𝑞1| + |𝑟1|) < 1 
The term is expanded using the simplification terms associated with Equation (2-17) 
1 𝑣𝑥 𝑣𝑥 2𝐷𝐿 𝑣𝑥 𝐷𝐿 𝐷𝐿 𝑣𝑥
∆𝑡 (| − 𝜃 + (1 − 𝜃) − | + |𝜃 + | + | − (1 − 𝜃) |) < 1 
∆𝑡 ∆𝑥 ∆𝑥 (∆𝑥)2 ∆𝑥 (∆𝑥)2 (∆𝑥)2 ∆𝑥 (2-47) 
Equation (2-47) is similar to Equation (2-44), and thus by following a similar procedure the explicit 
weighted upwind-downwind numerical scheme is unstable under the assumptions,  
1 𝑣𝑥 2𝐷𝐿 𝑣𝑥
+ (1 − 𝜃) + > 𝜃 , 
∆𝑡 ∆𝑥 (∆𝑥)2 ∆𝑥
and 
𝐷𝐿 𝑣𝑥
> (1 − 𝜃)  
(∆𝑥)2 ∆𝑥
Yet, is conditionally stable, when 𝜃 < 1, under the assumptions 
1 𝑣𝑥 2𝐷𝐿 𝑣𝑥
+ (1 − 𝜃) + < 𝜃 , 
∆𝑡 ∆𝑥 (∆𝑥)2 ∆𝑥
and 
𝐷𝐿 𝑣𝑥
< (1 − 𝜃)  
(∆𝑥)2 ∆𝑥
In summary, the weighted explicit upwind-downwind numerical scheme for the advection-dispersion 
equation for 0 < 𝜃 < 1 is conditionally stable. Yet, under certain assumptions the extreme cases of 
the full upwind scheme (𝜃 = 0) are invalid.  
2.3.6 Implicit upwind-downwind weighted scheme  
The first-order upwind-downwind (implicit) weighted finite difference approximation of the one-
dimensional advection-dispersion equation was determined to be (Equation (2-19)). Now, substituting 
induction method terms to assess the stability of the scheme, 
𝑣 ?̂? 𝑒𝑖𝑘𝑚𝑥 = 𝑎 ?̂? 𝑒𝑖𝑘𝑚𝑥 + 𝑞 ?̂? 𝑒𝑖𝑘𝑚(𝑥−∆𝑥) 𝑖𝑘𝑚(𝑥+∆𝑥)1 𝑛+1 1 𝑛 1 𝑛+1 + 𝑟1?̂?𝑛+1𝑒   
Multiple out, 
𝑣 ?̂? 𝑒𝑖𝑘𝑚𝑥 = 𝑎 ?̂? 𝑒𝑖𝑘𝑚𝑥 + 𝑞 ?̂? 𝑒𝑖𝑘𝑚𝑥𝑒−𝑖𝑘𝑚∆𝑥 + 𝑟 ?̂? 𝑒𝑖𝑘𝑚𝑥𝑒𝑖𝑘𝑚∆𝑥1 𝑛+1 1 𝑛 1 𝑛+1 1 𝑛+1   
Divide by 𝑒𝑖𝑘𝑚𝑥, 
𝑣1?̂? = 𝑎 ?̂? + 𝑞 ?̂? 𝑒
−𝑖𝑘𝑚∆𝑥 + 𝑟 ?̂? 𝑖𝑘𝑚∆𝑥𝑛+1 1 𝑛 1 𝑛+1 1 𝑛+1𝑒  (2-48) 
The induction numerical stability analysis is performed in two parts, firstly it is proved for a set ∀ 𝑛 ≥ 1,  
32 
 
|?̂?𝑛| < |?̂?𝑜|  
If 𝑛 = 0, then 
𝑣 ?̂? = 𝑎 ?̂? + 𝑞 ?̂? 𝑒−𝑖𝑘𝑚∆𝑥 + 𝑟 𝑖𝑘𝑚∆𝑥1 1 1 0 1 1 1?̂?1𝑒   
Rearrange, 
𝑣 −𝑖𝑘𝑚∆𝑥 𝑖𝑘𝑚∆𝑥1?̂?1 − 𝑞1?̂?1𝑒 − 𝑟1?̂?1𝑒 = 𝑎1?̂?0  
Simplifying, 
?̂?1(𝑣 − 𝑞 𝑒
−𝑖𝑘𝑚∆𝑥
1 1 − 𝑟 𝑒
𝑖𝑘𝑚∆𝑥
1 ) = ?̂?0(𝑎1)  
Rearranging, 
?̂?1 𝑎1
=   
?̂?0 𝑣1 − 𝑞1𝑒
−𝑖𝑘𝑚∆𝑥 − 𝑟 𝑒𝑖𝑘𝑚∆𝑥1
Applying a norm, 
|?̂?1| |𝑎1|
<  
|?̂? | |𝑣 | + |−𝑞 𝑒−𝑖𝑘𝑚∆𝑥| + |−𝑟 𝑒𝑖𝑘𝑚∆𝑥
 
0 1 1 1 |
The stability condition necessary |?̂?𝑛| < |?̂?𝑜|, becomes: 
|?̂?1|
< 1  
|?̂?0|
1
Remembering |𝑒𝑛| = 1 and 𝑎1 = , the condition becomes ∆𝑡
1
< 1 
∆𝑡(|𝑣1| + |𝑞1| + |𝑟1|)
 
It can be concluded that |?̂?1| < |?̂?𝑜|, when 
∆𝑡(|𝑣1| + |𝑞1| + |𝑟1|) > 1 
The term is expanded using the simplification terms associated with Equation (2-19) 
1 𝑣𝑥 𝑣𝑥 2𝐷𝐿 𝑣𝑥 𝐷𝐿 𝐷𝐿 𝑣𝑥
∆𝑡 (| + 𝜃 − (1 − 𝜃) + | + |𝜃 + | + | − (1 − 𝜃) |) > 1 (2-49) 
∆𝑡 ∆𝑥 ∆𝑥 (∆𝑥)2 ∆𝑥 (∆𝑥)2 (∆𝑥)2 ∆𝑥
If the assumption is made, where 
1 𝑣𝑥 2𝐷𝐿 𝑣𝑥
+ 𝜃 + > (1 − 𝜃) , 
∆𝑡 ∆𝑥 (∆𝑥)2 ∆𝑥
and 
𝐷𝐿 𝑣𝑥
> (1 − 𝜃)  
(∆𝑥)2 ∆𝑥
Then, 
1 𝑣𝑥 𝑣𝑥 2𝐷𝐿 𝑣𝑥 𝐷𝐿 𝐷𝐿 𝑣𝑥
∆𝑡 (( + 𝜃 − (1 − 𝜃) + ) + (𝜃 + ) + ( − (1 − 𝜃) )) > 1  
∆𝑡 ∆𝑥 ∆𝑥 (∆𝑥)2 ∆𝑥 (∆𝑥)2 (∆𝑥)2 ∆𝑥
Simplifying, 
𝑣𝑥 2𝐷𝐿
(2𝜃 − 1) + > 1 (2-50) 
∆𝑥 (∆𝑥)2
33 
 
Under these assumptions, the first-order implicit upwind-downwind weighted scheme for the one-
dimensional advection-dispersion equation is conditionally stable, where 𝜃 > 0. When 𝜃 > 0, the 
solution becomes the weighted upwind-downwind solution, while when 𝜃 = 1, the solution becomes 
the upwind formation. When 𝜃 = 0, the solution becomes the downward formulation of the 
advection, and this is shown to be numerically unstable under these assumptions.  
On the other hand, if the assumptions are made, where 
1 𝑣𝑥 𝑣𝑥 2𝐷𝐿
+ 𝜃 < (1 − 𝜃) + , 
∆𝑡 ∆𝑥 ∆𝑥 (∆𝑥)2
and 
𝐷𝐿 𝑣𝑥
< (1 − 𝜃)  
(∆𝑥)2 ∆𝑥
Then, 
1 𝑣𝑥 𝑣𝑥 2𝐷𝐿 𝑣𝑥 𝐷𝐿 𝐷𝐿 𝑣𝑥
∆𝑡 ((− − 𝜃 + (1 − 𝜃) − ) + (𝜃 + ) + (− + (1 − 𝜃) )) > 1  
∆𝑡 ∆𝑥 ∆𝑥 (∆𝑥)2 ∆𝑥 (∆𝑥)2 (∆𝑥)2 ∆𝑥
Simplifying, 
𝑣𝑥∆𝑡
(1 − 𝜃) > 1 (2-51) 
∆𝑥
Under these assumptions, the first-order implicit upwind-downwind weighted scheme for the one-
dimensional advection-dispersion equation is conditionally stable, where 𝜃 < 1. When 𝜃 = 0, the 
solution becomes the downward formulation of the advection, while when 0 < 𝜃 < 1, the solution 
becomes the weighted upwind-downwind solution. When 𝜃 = 1, the solution becomes the upwind 
formation and is numerically unstable under these assumptions.  
Secondly, making the assumption that |?̂?𝑛| < |?̂?𝑜| is true for all time steps, the second part of the 
numerical stability analysis is to demonstrate for a set ∀ 𝑛 ≥ 1, that 
|?̂?𝑛+1| < |?̂?𝑜|  
Remembering Equation (2-48), 
𝑣 ?̂? = 𝑎 ?̂? + 𝑞 ?̂? 𝑒−𝑖𝑘𝑚∆𝑥1 𝑛+1 1 𝑛 1 𝑛+1 + 𝑟 ?̂? 𝑒
𝑖𝑘𝑚∆𝑥
1 𝑛+1   
Simplifying, 
?̂? −𝑖𝑘𝑚∆𝑥𝑛+1(𝑣1 − 𝑞1𝑒 − 𝑟
𝑖𝑘𝑚∆𝑥
1𝑒 ) = 𝑎1?̂?𝑛 
𝑎1  
?̂?𝑛+1 = ?̂?  𝑣1 − 𝑞 𝑒
−𝑖𝑘
1 𝑚
∆𝑥 − 𝑟 𝑒𝑖𝑘𝑚∆𝑥 𝑛1
Taking the norm on both sides, 
|𝑎1|
|?̂?𝑛+1| < |?̂? | |𝑣 | + |−𝑞 −𝑖𝑘 ∆𝑥 𝑖𝑘 ∆𝑥 𝑛  1 1𝑒 𝑚 | + |−𝑟1𝑒 𝑚 |
1
Remembering |𝑒𝑛| = 1 and 𝑎1 = , the condition becomes ∆𝑡
34 
 
1
|?̂?𝑛+1| < |?̂?𝑛| ∆𝑡(|𝑣1| + |𝑞1| + |𝑟1|)
 
Remembering that it has been proven that for a set ∀ 𝑛 > 1, 
|?̂?  𝑛| < |?̂?𝑜| 
Thus,  
1 1
|?̂?𝑛+1| < |?̂? | < |?̂? | ∆𝑡(|𝑣1| + |𝑞1| + |𝑟1|)
𝑛 ∆𝑡(|𝑣1| + |𝑞1| + |𝑟 |)
𝑜  
1
and, it can be inferred that 
1
|?̂?𝑛+1| < |?̂? | ∆𝑡(|𝑣1| + |𝑞1| + |𝑟1|)
𝑜  
Thus, the solution will be stable when, 
1
< 1 
∆𝑡(|𝑣1| + |𝑞1| + |𝑟1|)
It can be concluded that |?̂?𝑛+1| < |?̂?𝑜|, when 
∆𝑡(|𝑣1| + |𝑞1| + |𝑟1|) > 1 
The term is expanded using the simplification terms associated with Equation (2-19) 
1 𝑣𝑥 𝑣𝑥 2𝐷𝐿 𝑣𝑥 𝐷𝐿 𝐷𝐿 𝑣𝑥
∆𝑡 (| + 𝜃 − (1 − 𝜃) + | + |𝜃 + | + | − (1 − 𝜃) |) > 1 (2-52) 
∆𝑡 ∆𝑥 ∆𝑥 (∆𝑥)2 ∆𝑥 (∆𝑥)2 (∆𝑥)2 ∆𝑥
Equation (2-52) is similar to Equation (2-49), and thus by following a similar procedure the condition 
for |?̂?𝑛+1| < |?̂?𝑜| becomes,  
𝑣𝑥 2𝐷𝐿
(2𝜃 − 1) + > 1 
∆𝑥 (∆𝑥)2
when the first assumptions are made.  
And, similarly, the condition for |?̂?𝑛+1| < |?̂?𝑜| becomes,  
𝑣𝑥∆𝑡
(1 − 𝜃) > 1 
∆𝑥
when the second set of assumptions are made. 
In summary, the weighted implicit upwind-downwind numerical scheme for the advection-dispersion 
equation for 0 < 𝜃 < 1 is conditionally stable. Yet, under certain assumptions, the extreme cases of 
the full upwind scheme (𝜃 = 0) and full downwind (𝜃 = 1) are unstable under certain assumptions.  
35 
 
2.4 Comparison of numerical schemes stability conditions 
Traditional upwind numerical schemes for the one-dimensional advection-dispersion equation, as well 
as new upwind numerical schemes, are developed and subjected to numerical stability analysis using 
the recursive method. The stability conditions and related assumptions are tabulated for the 
traditional approaches in Table 2-1, and for the new schemes in Table 2-2.  
The first-order explicit upwind finite difference scheme is found to be unstable for the first assumption 
made, and conditionally stable for the second assumption, where the condition found is the Courant 
condition traditionally applied for explicit finite difference schemes (Table 2-1). The correspondence 
to the Courant condition for this scheme validates the stability analysis method used.  
The first-order upwind implicit finite difference scheme is unconditionally stable for all conditions. The 
first-order upwind implicit scheme is more stable than the explicit formulation, which is typically the 
case when comparing explicit and implicit formulations. However, implicit schemes tend to 
compromise on accuracy to achieve the improved stability.  
The first-order upwind Crank-Nicolson finite difference scheme is found to be unconditionally stable, 
and thus the combination of the first-order upwind numerical scheme with the Crank-Nicolson 
numerical scheme applied to the entire advection-dispersion equation is appropriate. This leads on to 
the next numerical scheme to be analysed, where now the Crank-Nicolson scheme is only considered 
for the advection term of the advection-dispersion equation (Table 2-1).  
The newly proposed upwind advection Crank-Nicolson finite difference scheme is conditionally stable 
for the first assumption made, and unconditionally stable for the second assumption (Table 2-2). Thus, 
it can be concluded that an upwind and advection Crank-Nicolson scheme only applied to the 
advection term of the advection-dispersion equation is appropriate under certain conditions. 
Furthermore, a weighted upwind-downwind finite difference numerical scheme is proposed for the 
one-dimensional advection-dispersion equation. The explicit weighted upwind-downwind scheme is 
found to be unstable for the first assumption made, yet unconditionally stable for the second 
assumption (Table 2-2). 
The weighted explicit upwind-downwind numerical scheme for the advection-dispersion equation for 
0 < 𝜃 < 1 is conditionally stable. Yet, under certain assumptions the extreme cases of the full upwind 
scheme (𝜃 = 0) are unstable. The weighted implicit upwind-downwind numerical scheme for the 
advection-dispersion equation for 0 < 𝜃 < 1 is conditionally stable. Yet, under certain assumptions, 
the extreme cases of the full upwind scheme (𝜃 = 0) and full downwind (𝜃 = 1) are unstable under 
certain assumptions (Table 2-2). 
36 
 
When comparing the explicit first-order upwind scheme with the explicit weighted first-order upwind-
downwind scheme, both solutions have an instability under one assumption, However, the explicit 
weighted first-order upwind-downwind scheme is unconditionally stable for the second assumption 
for certain weighting, while the explicit first-order upwind scheme is conditionally stable. From a 
comparison on stability, the new explicit weighted first-order upwind-downwind scheme is an 
improvement on the explicit first-order upwind scheme for the finite difference method of numerical 
approximation. Considering that generally explicit formulations are less stable, the weighted approach 
could improve the stability of the scheme while retaining the accuracy advantages of the explicit 
formulation.  
Table 2-1 Traditional upwind numerical schemes for the one dimensional advection-dispersion equation 
with associated assumptions in the numerical stability analysis and the resulting stability conditions 
Stability 
Scheme Numerical approximation Assumptions 
condition 
1 2𝐷𝐿 𝑣𝑥
1 1 𝑣𝑥 2𝐷𝑛+1 𝐿 + >  Unstable ( ) 𝑐 = ( − − ) 𝑐𝑛 ∆𝑡 (∆𝑥)2 ∆𝑥
Explicit ∆𝑡 𝑖 ∆𝑡 ∆𝑥 (∆𝑥)2 𝑖
Upwind  𝑣𝑥 𝐷𝐿 𝐷𝐿
+ ( + ) 𝑐𝑛 𝑛
2 𝑖−1
+ ( ) 𝑐
2 𝑖+1
 1 2𝐷 𝑣 ∆𝑡
∆𝑥 (∆𝑥) (∆𝑥) 𝐿 𝑥+ <  𝑣𝑥 < 1 
∆𝑡 (∆𝑥)2 ∆𝑥 ∆𝑥Conditionally stable 
1 𝑣𝑥 2𝐷𝐿
( + + ) 𝑐𝑛+1
∆𝑡 ∆𝑥 (∆𝑥)2 𝑖 2𝑣𝑥 4𝐷𝐿+ > 0 
Implicit 1 𝑣𝑥 𝐷𝐿 No assumptions ∆𝑥 (∆𝑥)2
= ( ) 𝑐𝑛 + ( + ) 𝑐𝑛+1
Upwind  ∆𝑡 𝑖 ∆𝑥 (∆𝑥)2 𝑖−1
 
required Unconditionally 
𝐷𝐿
+( ) 𝑐𝑛+1 stable 
(∆𝑥)2 𝑖+1
1 𝑣𝑥 𝐷𝐿
( + 0.5 + ) 𝑐𝑛+1
∆𝑡 ∆𝑥 (∆𝑥)2 𝑖
1 𝑣𝑥 𝐷 𝑣𝐿 𝑥 2𝐷𝐿
Upwind = ( − 0.5 − ) 𝑐𝑛 + > 0 
∆𝑡 ∆𝑥 (∆𝑥)2 𝑖 No assumptions ∆𝑥 (∆𝑥)2
Crank- 𝑣𝑥 𝐷𝐿 𝑣𝑥 𝐷𝐿 required 
Nicolson +0.5 ( + ) 𝑐𝑛 𝑛+1 Unconditionally 
∆𝑥 (∆𝑥)2 𝑖−1
+ 0.5 ( + ) 𝑐  
∆𝑥 (∆𝑥)2 𝑖−1 stable 
𝐷𝐿 𝐷𝐿
+0.5 ( ) 𝑐𝑛 𝑛+1
(∆𝑥)2 𝑖+1
+ 0.5 ( ) 𝑐  
(∆𝑥)2 𝑖+1
 
Interestingly, the implicit weighted first-order upwind-downwind scheme is not more stable than the 
implicit first-order upwind scheme. This is contrasting to the usual trend where implicit formulations 
are more stable than explicit formulations. It would thus appear that the explicit weighted first-order 
upwind-downwind scheme is dependent on dispersivity, cell size and time step; while the implicit 
weighted first-order upwind-downwind scheme is dependent on chosen weighting factor (𝜃). 
However, if an appropriate weighting factor is selected the scheme is stable under all assumptions.  
37 
 
In summary, based on the stability analysis solely, an upwind and Crank-Nicolson scheme is 
appropriate when the Crank-Nicolson scheme is applied to the advection term of the advection-
dispersion equation. Furthermore, the proposed explicit weighted first-order upwind-downwind finite 
difference numerical scheme is an improvement on the traditional explicit first-order upwind scheme, 
while the implicit weighted first-order upwind-downwind finite difference numerical scheme is stable 
under all assumption when the appropriate weighting factor (𝜃) is assigned. 
Table 2-2 New upwind schemes for the one dimensional advection-dispersion equation with associated 
assumptions in the numerical stability analysis and the resulting stability conditions 
Scheme Numerical approximation Assumptions Stability condition 
1 2𝐷 𝑣 4𝐷𝐿𝐿 𝑥
1 𝑣 < 𝑣𝑥  𝑥 1 𝑣𝑥 2𝐷𝐿 + > 0.5  ∆𝑥
( + 0.5 ) 𝑐𝑛+1𝑖 = ( − 0.5 + ) 𝑐
𝑛 ∆𝑡 (∆𝑥)2 ∆𝑥
Upwind ∆𝑡 ∆𝑥 ∆𝑡 ∆𝑥 (∆𝑥)2 𝑖 Conditionally stable 
advection 𝑣𝑥 𝐷𝐿 𝐷𝐿+(0.5 + ) 𝑐𝑛 + ( ) 𝑐𝑛  
Crank- ∆𝑥 (∆𝑥)2 𝑖−1 (∆𝑥)2 𝑖+1
Nicolson 𝑣𝑥+(0.5 ) 𝑐𝑛+1𝑖−1  2∆𝑥 1 2𝐷𝐿 𝑣𝑥 > 0 
 + < 0.5  ∆𝑡∆𝑡 (∆𝑥)2 ∆𝑥 Unconditionally stable 
1 𝑣𝑥 𝑣𝑥 2𝐷𝐿
1 + (1 − 𝜃) > 𝜃 +  
𝑛+1 ∆𝑡 ∆𝑥 ∆𝑥 (∆𝑥)2( ) 𝑐  Unstable 
∆𝑡 𝑖 𝐷𝐿 𝑣𝑥
1 𝑣 𝑣 2𝐷 > (2 1 − 𝜃)  Explicit 𝑥 𝑥 𝐿 (∆𝑥) ∆𝑥
= ( − 𝜃 + (1 − 𝜃) − ) 𝑐𝑛 
upwind- ∆𝑡 ∆𝑥 ∆𝑥 (∆𝑥)2 𝑖
downwind 𝑣𝑥 𝐷𝐿
+(𝜃 + ) 𝑐𝑛  
weighted ∆𝑥 (∆𝑥)2 𝑖−1
scheme  𝐷𝐿 𝑣𝑥 𝑛 1 𝑣 𝑣 2𝐷+( − (1 − 𝜃) ) 𝑐  𝑥 𝑥 𝐿
2 𝑖+1 + (1 − 𝜃) < 𝜃 +  𝑣𝑥 2𝐷( 𝐿∆𝑥) ∆𝑥 ∆𝑡 ∆𝑥 ∆𝑥 (∆𝑥)2 ∆𝑡 (𝜃 + 2) < 1 ∆𝑥 (∆𝑥)
 𝐷𝐿 𝑣𝑥(
2 < 1 − 𝜃)  
Conditionally stable 
(∆𝑥) ∆𝑥
1 𝑣𝑥 2𝐷𝐿 𝑣𝑥
+ 𝜃 + > (1 − 𝜃) , 
∆𝑡 ∆𝑥 (∆𝑥)2 ∆𝑥 𝑣𝑥 2𝐷(2𝜃 − 1) + 𝐿 > 1 
𝐷 2𝐿 𝑣𝑥 ∆𝑥 (∆𝑥)
1 𝑣 𝑣 2𝐷 > (2 1 − 𝜃)  𝑥 𝑥 𝐿 (∆𝑥) ∆𝑥 Conditionally stable 
Implicit ( + 𝜃 − (1 − 𝜃) + ) 𝑐𝑛+1
∆𝑡 ∆𝑥 ∆𝑥 (∆𝑥)2 𝑖
  
upwind- 1 𝑣𝑥 𝐷𝐿
downwind = ( ) 𝑐𝑛 𝑛+1
∆𝑡 𝑖
+ (𝜃 + ) 𝑐
∆𝑥 (∆𝑥)2 𝑖−1
weighted 𝐷𝐿 𝑣𝑥
( ) 𝑛+1
1 𝑣𝑥 𝑣𝑥 2𝐷𝐿
scheme  + ( − 1 − 𝜃 ) 𝑐  + 𝜃 < (1 − 𝜃) + , 
(∆𝑥)2 ∆𝑥 𝑖+1 ∆𝑡 ∆𝑥 ∆𝑥 (∆𝑥)2 𝑣 ∆𝑡(1 − 𝜃) 𝑥 > 1 
𝐷𝐿 𝑣𝑥 ∆𝑥
(
2 < 1 − 𝜃)  (∆𝑥) ∆𝑥 Conditionally stable 
 
 
38 
 
2.5 Numerical simulation 
The developed numerical schemes have been assessed based on the stability analysis performed. To 
supplement the analysis a simulation using these schemes is performed to additionally assess the 
computational times and practicality. A generic one-dimensional transport problem is made use of for 
the simulation (Figure 2-1). The velocity within the aquifer is constant at 0.5 m/d, the hydrodynamic 
dispersivity in the x-direction is 0.3 m2/d, and the initial contaminant concentration is 10 mg/l. The 
initial condition and boundary conditions for the defined problem are: 
 
 
Figure 2-1 Generic one-dimensional transport problem, where an initial contaminant concentration in an 
aquifer is simulated along a single line in the x-direction. 
The numerical schemes described in Section 2.2 are applied to this problem, and coding the numerical 
approximations in the software program Scilab, the resulting breakthrough curves are displayed for in 
Figure 2-2 to Figure 2-8. The codes for each numerical scheme are provided in Appendix A. The 
traditional first-order upwind (implicit and explicit) schemes are included as a base with which to 
compare the newly developed schemes. The traditional upwind explicit formulation is unstable at 
39 
 
early times (Figure 2-2), and corresponds to the stability analysis where no unconditionally stable 
condition was found (Section 2.3.1). The traditional upwind implicit formulation is stable (Figure 2-3), 
and confirms the stability analysis where the scheme is found to be unconditionally stable under 
certain assumptions (2.3.2).  
A combination of the traditional upwind and Crank-Nicolson schemes are considered as a foundation 
with which to compare the newly developed upwind advection Crank-Nicolson schemes (explicit and 
implicit). The combination of the traditional upwind and Crank-Nicolson scheme is found to produce 
the expected breakthrough curve, but with instabilities at early times (Figure 2-4). However, the 
stability analysis did not indicate instabilities for this formulation (Section 2.3.3 and 2.4). The explicit 
formulation of the advection-specific upwind Crank-Nicolson combination is also found to be unstable 
(Figure 2-5), yet the implicit formulation improves the instabilities where the instabilities smooth out 
quicker (Figure 2-6).  
The developed weighted upwind-downwind formulation is simulated for this generic transport 
problem in Figure 2-7 (explicit) and Figure 2-8 (implicit). The weighting factor (𝜃) provides a means to 
adapt the scheme, and the simulated breakthrough curve for variable weightings is displayed. 
Incorporating the weighted component of the downward scheme for the explicit formulation creates 
instabilities in the solution where increasing the weighting artificially reduces the plume extent (Figure 
2-7). Thus, for this specific transport problem, the weighted explicit formulation is not an 
improvement on the traditional explicit upwind formulation. However, the stability analysis found that 
for some cases/assumptions the weighted upwind-downwind formulation could improve the stability 
of the scheme while retaining the accuracy advantages of the explicit formulation. The weighted 
upwind-downwind implicit formulation is stable for all theta values (Figure 2-8), yet the solution has 
some variability with the theta value. Thus, this formulation could provide a more flexible solution to 
calibrate and match the simulated break-through curve with the measured concentrations.  
The computational times for each of these schemes is considered for this simple 1D transport 
problem. The computational times recorded in Scilab for each scheme are reported in Table 2-3. For 
all the schemes, the implicit formulations had a slightly higher computational time, which can be 
expected due to the additional inverse simulation required for implicit formulations. The advection 
Crank-Nicolson schemes had the highest computational times, and the weighted implicit is slightly 
quicker than the traditional upwind implicit scheme. The difference in computational times on this 
scale are small, but for a larger complex 3D system, these differences could become substantial. 
 
 
 
40 
 
 
Distance (x-axis) (m) 
 
Figure 2-2 Simulated breakthrough curve of concentration (concentration against distance) for the First-
order upwind explicit numerical scheme 
Distance (x-axis) (m) 
 
Figure 2-3 Simulated breakthrough curve of concentration (concentration against distance) for the First-
order upwind implicit numerical scheme 
 
 
 
41 
 
Concentration (mg/l) Concentration (mg/l) 
 
Distance (x-axis) (m)  
Figure 2-4 Simulated breakthrough curve of concentration (concentration against distance) for the First-
order upwind Crank-Nicolson numerical scheme 
Distance (x-axis) (m) 
 
Figure 2-5 Simulated breakthrough curve of concentration (concentration against distance) for the First-
order upwind advection Crank-Nicolson (explicit) numerical scheme 
 
 
42 
 
Concentration (mg/l) Concentration (mg/l) 
 
Distance (x-axis) (m) 
 
Figure 2-6 Simulated breakthrough curve of concentration (concentration against distance) for the First-
order upwind advection Crank-Nicolson (implicit) numerical scheme 
𝜽 = 𝟏 
𝜽 = 𝟎. 𝟗 
𝜽 = 𝟎. 𝟔 
𝜽 = 𝟎. 𝟏 
Distance (x-axis) (m) 
 
Figure 2-7 Simulated breakthrough curve of concentration (concentration against distance) for the First-
order upwind-downwind weighted (explicit) numerical scheme with variable theta (𝜽) values from 1 (black), 
0.9 (blue), 0.6 (dark blue), and 0.1 (green) 
 
 
 
43 
 
Concentration (mg/l) Concentration (mg/l) 
 
Distance (x-axis) (m)  
Figure 2-8 Simulated breakthrough curve of concentration (concentration against distance) for the First-
order upwind-downwind weighted (implicit) numerical scheme with variable theta (𝜽) values from 1 
(green), 0.5 (dark blue), and 0.1 (black) 
Table 2-3 Computational times (in nanoseconds) for each numerical scheme investigated 
Computational time (ns – Nano seconds) 
Numerical approximation scheme 
1 2 3 Average 
Upwind (explicit) 1.093 1.015 1.328 1.15 
Upwind (implicit) 1.531 1.578 1.421 1.51 
Upwind-Crank-Nicolson (full) 1.531 1.390 1.406 1.44 
Upwind-Crank-Nicolson (advection-explicit) 2.250 1.656 1.328 1.74 
Upwind-Crank-Nicolson (advection-implicit) 2.265 1.859 1.703 1.94 
Weighted upwind-downwind (explicit) 1.375 1.453 1.390 1.41 
Weighted upwind-downwind (implicit) 1.421 1.437 1.593 1.48 
 
In general, the implicit formulations are more numerically stable, yet have slightly higher 
computational times. The first-order upwind advection Crank-Nicolson (implicit) scheme is an 
improvement on the combination of the traditional upwind and Crank-Nicolson schemes, yet runs 
slightly longer. The weighted implicit upwind-downwind scheme is an improvement on the traditional 
upwind scheme for the considered system as theta provides a more flexible solution and the 
computational times are slightly faster.  
44 
 
Concentration (mg/l) 
2.6 Chapter summary 
In this chapter, the limitations identified with the advection-dispersion equation, especially with 
respect to fractured systems, are improved within the local or traditional space by applying 
augmented upwind numerical approximation schemes that are better suited for advection-dominated 
fractured systems. A numerical scheme combining the traditional upwind and Crank-Nicolson 
schemes is developed, along with new schemes including the advection-specific upwind Crank-
Nicolson combination, and weighted upwind-downwind schemes. The developed numerical schemes, 
along with the traditional approaches for comparison, are analysed for stability using the recursive 
stability analysis method, applied to a simple one-dimensional transport problem, and computational 
time assessment. From these analyses, the implicit formulations are found to be more stable and 
practically viable, and the new advection Crank-Nicolson and weighted upwind-downwind schemes 
are found to be improvements from the traditional methods. Thus, the augmented upwind schemes 
have the potential to improve the simulation of the local advection-dispersion equation for fractured 
systems. However, these improves do not address the problem of anomalous diffusion and therefore 
in the following chapters the reformulation of the advection-dispersion equation will be investigated.  
  
45 
 
 
3  FRACTAL ADVECTION-DISPERSION  
     EQUATION 
 
The typical application of a Fickian advection-dispersion transport equation to model groundwater 
transport depends on the availability of information to characterise the physical system in its entirety. 
Yet, where preferential flow pathways (fractures, faults or dykes) are present there is often 
inadequate data to accurately characterise the heterogeneity of these systems. Thus, when there is 
an incomplete characterisation of heterogeneity, the classical Fickian advection-dispersion transport 
equation is unable to simulate the observed plume movement accurately. The dissimilarity between 
the modelled and measured plume is commonly referred to as anomalous diffusion, because the 
plume diverges from the Fickian model of groundwater transport (Zheng and Bennett, 2002; Pickens 
and Grisak, 1981).  
Describing the heterogeneity of a system is complicated by dispersivity increasing with the scale of 
measurement (Freeze and Cherry, 1979; Pickens and Grisak, 1981; Molz et al., 1983; Neuman, 1990; 
Zheng and Bennett, 2002). Solutions to the dispersivity scale-dependency problem include 
establishing scaling relationships (Pickens and Grisak, 1981), or models incorporating fractures (or 
preferential flow pathways). Modelling methods to physically include preferential pathways include 
the dual porosity model; random array of fractures; multiple continuum; and others (Acuna and 
Yortsos, 1995; Cello et al., 2009). Yet, these models cannot account for the fractal nature of fractured 
systems. Fractal geometry and fractured systems were originally associated in 1985 for a nuclear 
46 
 
waste disposal site investigation (Barton and Hsieh, 1989), and since several authors have validated 
the relationship (Cello et al., 2009; Rodrigues and Pandolfelli, 1998; Roy et al., 2007; Borodich, 1999).  
An alternative approach to overcome the difficulty of defining dispersion is to reformulate the 
traditional advection-dispersion equation into fractional advection-dispersion equations, where 
fractional-order derivatives are used rather than integer-order derivatives (Benson, 1998). 
Formulating the transport equation in terms of fractional derivatives can account for non-Fickian long-
tailed breakthrough curves and improves the simulation of anomalous diffusion (Chen et al., 2010), 
yet has also proven to be susceptible to scale-dependency problems in three-dimensional natural 
porous media applications (Lu et al., 2002).  
Considering the current limitations of transport modelling, especially in fractured groundwater 
systems, a fractal groundwater advection-dispersion transport equation is developed to provide a tool 
to simulate anomalous diffusion in these systems.  
Fractal differentiation is discussed in terms of the fractal derivative and the fractal integral. The fractal 
derivative is commonly used and known, yet the fractal integral is developed in this paper along with 
the appropriate theorem and proof. The numerical approximation of the fractal derivative and integral 
are given, where Simpson’s 3/8 Rule and Boole’s Rule for numerical integration are applied for the 
fractal integral. Upon the given foundation, the fractal advection-dispersion transport equation is 
formulated to develop a new groundwater transport model. The qualitative properties of the new 
groundwater transport model are investigated to determine boundedness, existence and uniqueness 
of the solution. To validate the developed fractal advection-dispersion equation, a numerical 
simulation is performed with different fractal dimensions to demonstrate the applicability of the 
model to fractal groundwater systems. 
3.1 Fractal differentiation 
3.1.1 Fractal derivative 
A fractal derivative and conventional integer-order derivative are different in that the integer-order 
derivative represents the change of a function (dependent variable) with the change of another 
quantity (independent variable) in ordinary space, while the fractal derivative represents the ratio of 
change of two quantities in a fractal space. The fractal derivative can be defined by transforming 
standard integer dimensional space-time (𝑢, 𝑡) into fractal time (𝑢, 𝑡𝛼) (Chen et al., 2010a; Chen et 
al., 2010b; Cheng, 2016): 
 𝜕𝑢 𝑢(𝑡) − 𝑢(𝑡1) (3-1) 
                                  = lim 𝛼                     𝛼 > 0     𝜕𝑡𝛼 𝑡→ 𝑡 𝛼1 𝑡 − 𝑡1
where, 
 𝛼 denotes fractal dimension of time. 
47 
 
Furthermore, the fractal derivative can be defined by transforming the standard dimensional space-
time (𝑢, 𝑡) into fractal space-time (𝑢𝛽 , 𝑡𝛼) (Chen et al., 2010a): 
 𝜕𝑢𝛽 𝑢𝛽(𝑡) − 𝑢𝛽(𝑡1) (3-2) 
                                       = lim 𝛼              𝛼 > 0, 𝛽 > 0 𝜕𝑡𝛼 𝑡 → 𝑡 𝛼1 𝑡 − 𝑡1
where, 
 𝛽 denotes fractal dimension of space. 
3.1.2 Fractal integral 
The fractal integral or antiderivative can be determined by considering the fractal time derivative of a 
function, and assuming that the derivative is known and denoted by a function 𝑢(𝑡): 
 𝑑
𝑓(𝑡) = 𝑢(𝑡) (3-3) 
𝑑𝑡𝛼
Expanding and approximating the fractal derivative contributes, 
 𝑓(𝑡) − 𝑓(𝑡1)
= 𝑢(𝑡) (3-4) 
𝑡𝛼 − 𝑡 𝛼1
Multiply the fractal derivative by a unity, 
 𝑓(𝑡) − 𝑓(𝑡1) 𝑡 − 𝑡1
× ( ) = 𝑢(𝑡) (3-5) 
𝑡𝛼 − 𝑡 𝛼1 𝑡 − 𝑡1
Rearrange, 
 𝑓(𝑡) − 𝑓(𝑡1) 𝑡 − 𝑡1
× ( ) = 𝑢(𝑡) (3-6) 
𝑡 − 𝑡 𝑡𝛼 − 𝑡 𝛼1 1
Recognising the left component is the integer-order derivative (𝑓′(𝑡)), and taking the inverse of the 
inverse, 
 
1
𝑓′(𝑡) × ( ) = 𝑢(𝑡) (3-7) 
𝑡𝛼 − 𝑡 𝛼1
𝑡 − 𝑡1
Simplifying, 
 1
𝑓′(𝑡) × ( ) = 𝑢(𝑡) (3-8) 
𝛼𝑡𝛼−1
Rearranging, 
 𝑓′(𝑡) = 𝛼𝑡𝛼−1 𝑢(𝑡) (3-9) 
To further solve Equation (3-9) the Laplace transform (ℒ) is applied, 
 ℒ{𝑓′(𝑡)} = ℒ{𝛼𝑡𝛼−1 𝑢(𝑡)} (3-10) 
48 
 
Remembering the Laplace transform of an integer-order derivative: 
 𝓈ℒ{𝑓(𝑡)} − 𝑓(0) = ℒ{𝛼𝑡𝛼−1 𝑢(𝑡)} (3-11) 
Rearranging, simplifying and dividing by the Laplace space (𝓈): 
 𝛼 𝑓(0)
ℒ{𝑓(𝑡)} = ℒ{𝑡𝛼−1 𝑢(𝑡)} +  (3-12) 
𝓈 𝓈
The inverse Laplace transform is applied to convert back from the Laplace domain: 
 𝛼 𝑓(0)
ℒ−1[ℒ{𝑓(𝑡)}] = ℒ−1 [ ℒ{𝑡𝛼−1 𝑢(𝑡)} + ] (3-13) 
𝓈 𝓈
The convolution theorem provides the inverse transform of the product of two transforms. 
Considering two functions 𝑓 and 𝑔, which are piecewise continuous on 𝑡 ≥ 0, the Laplace transform 
is (Logan, 2006): 
 ℒ(𝑓 ∗ 𝑔)(𝓈) = 𝐹(𝓈)𝐺(𝓈) (3-14) 
The convolution of 𝑓 and 𝑔 is: 
 𝑡
(𝑓 ∗ 𝑔)(𝑡) ≡ ∫ 𝑓(𝜏) 𝑔(𝑡 − 𝜏) 𝑑𝜏 (3-15) 
0
and, the inverse Laplace transform is thus: 
 ℒ−1(𝐹𝐺)(𝓈) = (𝑓 ∗ 𝑔)(𝑡) (3-16) 
The Laplace transform is additive, but not multiplicative. This means that the Laplace transform of a 
product is not the product of the Laplace transforms. The convolution theorem gives the transform 
required to get a product of Laplace transforms, i.e. the convolution (Logan, 2006). Applying the 
convolution theorem to Equation (3-13): 
 𝑡
𝑓(𝑡) = 𝛼∫ 𝜏𝛼−1 𝑓(𝜏) 𝑑𝜏  +  𝑓(0) (3-17) 
0
Thus, the fractal integral can be defined as: 
 𝑡
 𝐹𝐼𝛼  𝑓(𝑡) = 𝛼∫ 𝜏𝛼−10 𝑡  𝑓(𝜏) 𝑑𝜏 (3-18) 
0
Theorem: Let the function (𝑓) be differentiable in an open interval 𝐼. Then the fractal integral of the 
function (𝑓) is given as: 
 𝑡
 𝐹 𝛼 𝛼−10𝐼𝑡  𝑓(𝑡) = 𝛼∫ 𝜏  𝑓(𝜏) 𝑑𝜏 (3-19) 
0
𝑑
Proof:  𝐹0𝐼
𝛼
𝑡  ( 𝛼 𝑓(𝑡)) = 𝑓(𝑡) − 𝑓(0) 𝑑𝑡
49 
 
Consider the definition of the fractal integral (Equation (3-19) for a fractal-order derivative function: 
 𝑑 𝑡 𝑑
 𝐹 𝛼 𝛼−10𝐼𝑡  ( 𝛼 𝑓(𝑡)) = 𝛼∫ 𝜏  𝑓(𝛼 𝜏) 𝑑𝜏 (3-20) 𝑑𝑡 0 𝑑𝜏
𝑑 1
Remembering that the fractal derivative can be expressed as 𝑓(𝑡)𝛼 = 𝑓
′(𝑡) (
𝑑𝑡 𝛼𝑡𝛼−1
) (Equations 
((3-4) to (3-8)): 
 𝑑 𝑡 1
 𝐹0𝐼
𝛼
𝑡  ( 𝛼 𝑓(𝑡)) = 𝛼∫  𝜏
𝛼−1𝑓′(𝜏) ( )  𝑑𝜏 (3-21) 
𝑑𝑡 0 𝛼𝜏
𝛼−1
Simplifying, 
 𝑑 𝑡
 𝐹𝐼𝛼  ( 𝑓(𝑡) ′) = ∫  𝑓 (𝜏) 𝑑𝜏 (3-22) 0 𝑡 𝑑𝑡𝛼 0
Applying the integral from 𝑡 to 0: 
 𝑑
 𝐹 𝛼0𝐼𝑡  ( 𝑓(𝑡)) = 𝑓(𝑡)− 𝑓(0) 𝑑𝑡𝛼 (3-23) 
𝑑
Secondly, the following is considered for the proof (𝛼  
𝐹
0𝐼
𝛼
𝑡 𝑓(𝑡)) = 𝑓(𝑡) 𝑑𝑡
Let the new function 𝐹 be defined as 𝐹(𝑡) = 𝐹0𝐼
𝛼
𝑡 𝑓(𝑡): 
 𝑑 𝑑
 𝐹𝐼𝛼0 𝑡  𝑓(𝑡) = 𝐹(𝑡) (3-24) 𝑑𝑡𝛼 𝑑𝑡𝛼
Applying the definition of a fractal derivative (Equation (3-1)): 
 𝑑 𝐹(𝑡) − 𝐹(𝑡1)
𝐹(𝑡) = lim  (3-25) 
𝑑𝑡𝛼 𝑡 → 𝑡1 𝑡𝛼 𝛼 − 𝑡1
𝑑 1
Remembering that the fractal derivative can be expressed as 𝑓(𝑡)𝛼 = 𝑓
′(𝑡) ( 𝛼−1) (Equations 𝑑𝑡 𝛼𝑡
(3-4) to (3-8)): 
 𝑑 1
𝐹(𝑡) = 𝐹′(𝑡) ( ) (3-26) 
𝑑𝑡𝛼 𝛼𝑡𝛼−1
𝑑
Substituting back the new function 𝐹, where 𝐹(𝑡) = 𝐹 𝛼0𝐼𝑡 𝑓(𝑡), and and 𝐹
′ =  
𝑑𝑡
 𝑑 𝑑 1
𝐹(𝑡) = ( 𝐹 𝛼0𝐼𝑡 𝑓(𝑡))
 ( ) (3-27) 
𝑑𝑡𝛼 𝑑𝑡 𝛼𝑡𝛼−1
Consider the definition of the fractal integral (Equation (3-19)): 
 
 𝑑 𝑑 𝑡 1
 𝐹 𝛼0𝐼𝑡  𝑓(𝑡) = (𝛼∫ 𝜏
𝛼−1 𝑓(𝜏) 𝑑𝜏) ( ) 
𝛼 (3-28) 𝑑𝑡 𝑑𝑡 0 𝛼𝑡
𝛼−1
50 
 
𝑑 𝑡
Simplifying using the fact that  (∫ 𝑓(𝜏) 𝑑𝜏) = 𝑓(𝑡): 
𝑑𝑡 0
 𝑑 1
𝐹 𝛼 𝛼−1 ( ) (3-29)  0𝐼𝑡  𝑓(𝑡) = 𝛼𝑡  𝑓 𝑡 ( ) 𝑑𝑡𝛼 𝛼𝑡𝛼−1
                                                            = 𝑓(𝑡)  
This completes the proof. 
3.2 New groundwater transport model in a fractured aquifer with self-similarities 
Groundwater transport within a fractured aquifer with a fractal nature exhibiting self-similarity cannot 
be accurately simulated by the classical Fickian advection-dispersion transport equation due to: 1) lack 
of a detailed characterisation of the fracture network and heterogeneity of the system, and 2) the 
occurrence of subdiffusion and superdiffusion (limitation of the mathematical formulation). Because 
the information to characterise such a system to an appropriate level of detail is often not available, 
and the advection-dispersion equation cannot account for anomalous diffusion, most applications fail 
to accurately simulate the observed contaminant transport.  
The limitations of the classical advection-dispersion equation have motivated alternative non-local 
conceptualisations of flow and transport, and various methods for addressing scale and space-time 
dependencies. Unconventional methods include stochastic averaging of the classical advection-
dispersion equation, multiple-rate mass transfer method, continuous time random walk method, time 
fractional advection-dispersion equation method, space fractional advection-dispersion equation 
method, and others (Zhang et al., 2009). In these alternative methods, the dispersive state is allowed 
to vary between superdiffusion, subdiffusion and normal diffusion, termed transient dispersion (Sun 
et al., 2014). Yet, each method is formulated for a specific transition and thus might not be appropriate 
for all types of transient dispersion. The fractional advection-dispersion equation formulations have 
proven successful in describing non-Fickian transport, but three-dimensional applications have been 
found to also show scale-dependent dispersivity problems (Lu et al., 2002; Huang et al., 2006).  
Anomalous behaviour has been recorded in unsaturated flow systems, defined by Richard’s equation, 
in heterogeneous systems of preferential pathways in soil, where the development of a horizontal 
wetting front deviates from the Boltzmann scaling (anomalous Boltzmann scaling) in a similar manner 
to how groundwater transport deviates from the Fickian model for diffusion (anomalous diffusion) 
(Gerolymatou et al., 2006; Hall, 2007). Sun et al. (2013) developed a fractal Richard’s equation to 
model unsaturated flow in heterogeneous soils that exhibit anomalous Boltzmann scaling. The fractal 
Richard’s equation was able to model the full range of observed non-Boltzmann behaviour, from 
subdiffusion to superdiffusion, which is related to the well-established fractal model for soils (Rieu 
and Sposito, 1991; Tyler and Wheatcraft, 1990).  
51 
 
In response to the current limitations of groundwater transport modelling in fractured systems and 
the success of the fractal Richard’s equation, a fractal groundwater advection-dispersion transport 
equation is deemed a suitable and meaningful investigation. A fractal advection-dispersion equation 
has the potential to provide the same advantages as the proven fractal Richard’s equation, where a 
fractal model could simulate the full range of observed non-Fickian behaviour, from subdiffusion to 
superdiffusion, which is related to the well-established fractal model for fractured systems without 
the detailed characterisation of the heterogeneity of the system. This formulation of a fractal 
advection-dispersion equation has not been developed previously because there was no definition of 
a fractal integral. Now however, with the developed definition of the fractal integral (proven with a 
theorem and proofs in Section 3.1.2, it is possible and described in detail in the following section. 
3.2.1 Fractal formulation of the advection-dispersion transport equation 
The effects of self-similarity inherent in groundwater transport with preferential pathways formed by 
fractures are included by incorporating a fractal space component into the mathematical formulation 
of the one dimensional advection-dispersion equation: 
 𝜕 𝜕 𝜕 𝜕
𝑐(𝑥, 𝑡) = −𝑣 𝑐(𝑥, 𝑡) + 𝐷 [ 𝑐(𝑥, 𝑡)] (3-30) 
𝜕𝑡 𝑥 𝜕𝑥𝛼 𝐿 𝜕𝑥𝛼 𝜕𝑥𝛼
To express the fractal ADE in terms of integer-order dimensions, the defined property of the fractal 
𝑑 1
derivative ( ) ′( )𝛼 𝑓 𝑡 = 𝑓 𝑡 ( 𝛼−1) (Equations (3-4) to (3-8)) is considered: 𝑑𝑡 𝛼𝑡
 𝜕 𝜕 1
𝑐(𝑥, 𝑡) = 𝑐(𝑥, 𝑡) ∙ ( ) (3-31) 
𝜕𝑥𝛼 𝜕𝑥 𝛼𝑥𝛼−1
Rearranging,  
 𝜕 𝜕 𝑥1−𝛼
𝑐(𝑥, 𝑡) = 𝑐(𝑥, 𝑡) ∙ ( ) (3-32) 
𝜕𝑥𝛼 𝜕𝑥 𝛼
Without the loss of generality, the advection transport term of the fractal advection-dispersion 
(Equation (3-30)) is considered: 
 𝜕 𝜕 𝑥1−𝛼
−𝑣𝑥 𝑐(𝑥, 𝑡) = −𝑣𝑥 𝑐(𝑥, 𝑡) ∙ ( ) 𝜕𝑥𝛼 𝜕𝑥 𝛼 (3-33) 
Now, the dispersion transport term of the fractal advection-dispersion (Equation (3-30)) is considered, 
𝑑 1
and the defined property of the fractal derivative 𝑓(𝑡) = 𝑓′( )𝛼 𝑡 ( 𝛼−1) (Equations (3-4) to (3-8) is 𝑑𝑡 𝛼𝑡
applied: 
 𝜕 𝜕 𝜕 𝜕 1
𝐷𝐿 [ 𝑐(𝑥, 𝑡)] = 𝐷𝐿 [ 𝑐(𝑥, 𝑡) ( )] 𝛼 𝛼 𝛼 (3-34) 𝜕𝑥 𝜕𝑥 𝜕𝑥 𝜕𝑥 𝛼𝑥𝛼−1
𝜕 1
Let [ 𝑐(𝑥, 𝑡) ( 𝛼−1)] be equal to a function 𝐹(𝑥): 𝜕𝑥 𝛼𝑡
52 
 
 𝜕 𝜕 𝜕
𝐷𝐿 [ 𝑐(𝑥, 𝑡)] = 𝐷𝐿 [𝐹(𝑥)] 𝛼 𝛼 𝛼 (3-35) 𝜕𝑥 𝜕𝑥 𝜕𝑥
𝑑 1
Applying the defined property of the fractal derivative 𝛼 𝑓(𝑡) = 𝑓
′(𝑡) (
𝑑𝑡 𝛼𝑡𝛼−1
) (Equations (3-4) to (3-
8)) again to remove the second fractal derivative, 
 𝜕 𝜕 1
𝐷𝐿 [ 𝑐(𝑥, 𝑡)] = 𝐷𝐿  𝐹′(𝑥) ( ) 𝜕𝑥𝛼 𝜕𝑥𝛼
(3-36) 
𝛼𝑥𝛼−1
Applying the product rule (ℎ(𝑥) = 𝑓(𝑥)𝑔(𝑥); 𝑡ℎ𝑒𝑛 ℎ′(𝑥) = 𝑓′(𝑥)𝑔(𝑥) + 𝑓(𝑥)𝑔′(𝑥)) to determine 
𝜕
 𝐹(𝑥): 
𝜕𝑥
 𝜕 𝜕 1 𝜕 𝜕 1
𝐹′(𝑥) = ( 𝑐(𝑥, 𝑡)) ( ) + 𝑐(𝑥, 𝑡) ( ( )) 
𝜕𝑥 𝜕𝑥 𝛼𝑥𝛼−1 𝜕𝑥 𝜕𝑥 𝛼𝑥𝛼−1
(3-37) 
𝜕2 𝑥1−𝛼 𝜕 1 − 𝛼
                                 = 𝑐(𝑥, 𝑡) ( ) + 𝑐(𝑥, 𝑡) ( 𝑥−𝛼) 
𝜕𝑥2 𝛼 𝜕𝑥 𝛼
Substituting back into the function 𝐹(𝑥) (Equation (3-36)): 
 
𝜕 𝜕 𝜕2 𝑥1−𝛼 𝜕 1 − 𝛼 𝑥1−𝛼
𝐷𝐿 [ 𝑐(𝑥, 𝑡)] = 𝐷𝐿 ( 𝑐(𝑥, 𝑡) ( ) + 𝑐(𝑥, 𝑡) ( 𝑥
−𝛼))( ) 
𝜕𝑥𝛼 𝜕𝑥𝛼 𝜕𝑥2 𝛼 𝜕𝑥 𝛼 𝛼
(3-38) 
𝜕2 𝑥2−2𝛼 (1 − 𝛼) 𝜕
                                      = 𝐷 ( 𝑐(𝑥, 𝑡) ∙  +  𝑥1−2𝛼𝐿 ∙ 𝑐(𝑥, 𝑡)) 𝜕𝑥2 𝛼2 𝛼 𝜕𝑥
Substituting the advection (Equation (3-32)) and dispersion (Equation (3-38)) terms back into the one-
dimensional fractal (space) advection-dispersion equation (Equation (3-30)): 
𝜕 𝜕 𝑥1−𝛼 𝜕2 𝑥2−2𝛼 (1 − 𝛼) 𝜕
𝑐(𝑥, 𝑡) = −𝑣 𝑐(𝑥, 𝑡) ∙ ( ) +𝐷 𝑐(𝑥, 𝑡) ∙   +  𝐷 𝑥1−2𝛼𝑥 𝐿 𝐿 ∙ 𝑐(𝑥, 𝑡) (3-39) 𝜕𝑡 𝜕𝑥 𝛼 𝜕𝑥2 𝛼2 𝛼 𝜕𝑥
Factorising, 
𝜕 𝑥1−𝛼 (1 − 𝛼) 𝜕 𝑥2−2𝛼 𝜕2
𝑐(𝑥, 𝑡) = (−𝑣 ∙ ( ) + 𝐷 𝑥1−2𝛼) 𝑐(𝑥, 𝑡) + (𝐷 ∙ ) 𝑐(𝑥, 𝑡)  
𝜕𝑡 𝑥 𝛼 𝐿 𝛼 𝜕𝑥 𝐿 𝛼2 𝜕𝑥2 (3-40) 
𝑥1−𝛼 (1−𝛼) 𝑥2−2𝛼
Let a function 𝑉𝛼𝐹 (𝑥) = −𝑣𝑥 ∙ ( ) + 𝐷𝐿 𝑥
1−2𝛼, and function 𝐷𝛼𝐹 (𝑥) = 𝐷𝐿 ∙𝛼 𝛼 𝛼2 : 
𝜕 𝜕 𝜕2
𝑐(𝑥, 𝑡) = 𝑉𝛼𝐹 (𝑥) 𝑐(𝑥, 𝑡) + 𝐷
𝛼
𝐹 (𝑥) 𝑐(𝑥, 𝑡)  (3-41) 𝜕𝑡 𝜕𝑥 𝜕𝑥2
where, 
 𝑉𝛼𝐹 (𝑥) is the velocity with fractal dimension with respect to 𝑥, and 
 𝐷𝛼𝐹 (𝑥) is the dispersion coefficient with fractal dimension with respect to 𝑥. 
53 
 
Equation (3-41) is the developed new transport model called the fractal (space) advection dispersion 
equation (FADE). The FADE equation resembles the traditional advection-dispersion equation, yet the 
newly defined velocity and dispersion coefficients (𝑉𝛼𝐹 (𝑥) and 𝐷
𝛼
𝐹 (𝑥)) have fractal dimensions with 
respect to 𝑥. 
Remark: The fractal advection-dispersion equation should return to the classical formulation of the 
advection-dispersion equation when 𝛼 = 1, to be valid.  
Considering the fractal dimension velocity term (𝛼 = 1): 
(1 − 𝛼) 𝑥1−𝛼
𝑉𝛼(𝑥) = 𝐷 𝑥2−𝛼𝐹 𝐿 − 𝑣𝑥 ∙2 ( )  𝛼 𝛼 (3-42) 
                                                                 = −𝑣𝑥 
Considering the fractal dimension dispersion term (𝛼 = 1): 
𝑥1−𝛼
𝐷𝛼𝐹 (𝑥) = 𝐷𝐿 ∙   𝛼2 (3-43) 
                                                                                    = 𝐷𝐿 
When 𝛼 = 1, the fractal advection-dispersion equation reverts to the classical formulation of the 
advection-dispersion equation, and thus can be considered valid. To further validate the developed 
fractal advection-dispersion equation, the qualitative properties of the formulation are investigated 
in the following section.4 
3.2.2 Qualitative properties 
The qualitative properties of boundedness, existence and uniqueness for the developed fractal 
advection-dispersion equation are presented, using the Picard-Lindelöf theorem.   
Lipschitz condition boundedness for partial differential 
The Lipschitz condition is used for a bound on the modulus of continuity for a function. A function 𝑓 
satisfies a Lipschitz condition on a set 𝐵, if there is a constant 𝑀 ≥  0, such that: 
‖𝑓(𝑡, 𝑢) − 𝑓(𝑡, 𝑣)‖ ≤ 𝑀‖𝑢 − 𝑣‖ (3-44) 
for all (𝑡, 𝑢), (𝑡, 𝑣) ∈ 𝐵. 
A function that has a bounded first derivative is considered Lipschitz, and this has been assumed true 
for partial differential operators. A theorem and proof are proposed here for the partial differential 
equations. 
𝜕
Theorem: Let 𝐵 be a set of non-zero differentiable functions, such that 𝑓, 𝑔 ∈ 𝐵 and ‖ 𝑔‖ < 𝑀 <
𝜕𝑥
∞. For 𝑓, 𝑔 ∈ 𝐵: 
54 
 
𝜕 𝜕
‖ 𝑓 − 𝑔‖ ≤ 𝑘‖𝑓 − 𝑔‖ 
𝜕𝑥 𝜕𝑥 (3-45) 
Proof: Let 𝑓, 𝑔 ∈ 𝐵, such that that 𝑓 ≠ 𝑔, then 
𝜕 𝜕 𝜕
‖ 𝑓 − 𝑔‖ = ‖ (𝑓 − 𝑔)‖ 
𝜕𝑥 𝜕𝑥 𝜕𝑥 (3-46) 
Since 𝑓, 𝑔 ∈ 𝐵, and similarly 𝑓 − 𝑔 ∈ 𝐵, then by assumption a positive constant 𝑀 can be found, such 
that 
𝜕
‖ (𝑓 − 𝑔)‖ < 𝑀 (3-47) 
𝜕𝑥
‖𝑓−𝑔‖
And, multiplying by  
‖𝑓−𝑔‖
𝜕 ‖𝑓 − 𝑔‖
‖ (𝑓 − 𝑔)‖ < 𝑀  
𝜕𝑥 ‖𝑓 − 𝑔‖
(3-48) 
𝜕 𝑀
‖ (𝑓 − 𝑔)‖ < ∙ ‖𝑓 − 𝑔‖ 
𝜕𝑥 ‖𝑓 − 𝑔‖
𝑀
Let  = 𝑘, 
‖𝑓−𝑔‖
𝜕
‖ (𝑓 − 𝑔)‖ < 𝑘‖𝑓 − 𝑔‖ (3-49) 
𝜕𝑥
This completes the proof. 
Fixed-point theorem for existence and uniqueness 
An integral is applied to both sides of the fractal transport equation (Equation (3-41)), to obtain: 
𝑡 𝜕 𝜕2
𝑐(𝑥, 𝑡) − 𝑐(𝑥, 0) = ∫ {𝑉𝛼𝐹 (𝑥) 𝑐(𝑥, 𝜏) + 𝐷
𝛼
𝜕𝑥 𝐹
(𝑥) 𝑐(𝑥, 𝜏)} 𝑑𝜏 
𝜕𝑥2 (3-50) 0
Let a new function 𝐹𝐶(𝑥, 𝑡) be expressed as: 
𝑡 𝜕 𝜕2
𝐹𝐶(𝑥, 𝑡) = ∫ {𝑉𝛼𝐹 (𝑥) 𝑐(𝑥, 𝜏) + 𝐷
𝛼
𝜕𝑥 𝐹
(𝑥) 𝑐(𝑥, 𝜏)
2 } 𝑑𝜏 𝜕𝑥 (3-51) 0
and, state the Lipschitz condition (Equation (3-44)): 
‖𝐹𝐶1(𝑥, 𝑡) − 𝐹𝐶2(𝑥, 𝑡)‖ < 𝑘‖𝑐1(𝑥, 𝑡) − 𝑐2(𝑥, 𝑡)‖ (3-52) 
Without the loss of generality, consider the term left of the inequality sign, and substitute the 
function 𝐹𝐶(𝑥, 𝑡): 
55 
 
𝑡 𝜕 𝜕2
∫ {𝑉𝛼𝐹 (𝑥) 𝑐1(𝑥, 𝜏) + 𝐷
𝛼
𝐹 (𝑥) 𝑐 (𝑥, 𝜏) 𝑑𝜏𝜕𝑥 𝜕𝑥2 1
}
‖𝐹𝐶1(𝑥, 𝑡) − 𝐹𝐶2(𝑥, 𝑡)‖ =
‖ 0 ‖
‖ 𝑡 2 ‖ (3-53) 𝜕 𝜕
−∫ {𝑉𝛼𝐹 (𝑥) 𝑐2(𝑥, 𝜏) + 𝐷
𝛼
𝐹 (𝑥) 𝑐2(𝑥, 𝜏)}𝑑𝜏
0 𝜕𝑥 𝜕𝑥
2
Simplifying and factorising, 
𝑡 𝜕
∫ {𝑉𝛼𝐹 (𝑥) (𝑐1(𝑥, 𝜏) − 𝑐2(𝑥, 𝜏)} 𝑑𝜏
‖𝐹𝐶 (𝑥, 𝑡) − 𝐹𝐶 (𝑥, 𝑡)‖ = ‖ 0
𝜕𝑥 ‖
1 2 ‖ 𝑡 𝜕2 ‖
 (3-54) 
+∫ {𝐷𝛼𝐹 (𝑥) (𝑐1(𝑥, 𝜏) − 𝑐𝜕𝑥2 2
(𝑥, 𝜏)} 𝑑𝜏
0
Applying the triangular inequality, 
𝑡 𝜕
‖𝐹𝐶1(𝑥, 𝑡) − 𝐹𝐶2(𝑥, 𝑡)‖ ≤ ‖∫ {𝑉
𝛼
𝐹 (𝑥) (𝑐1(𝑥, 𝜏) − 𝑐2(𝑥, 𝜏)} 𝑑𝜏‖ 
0 𝜕𝑥
(3-55) 
𝑡 𝜕2
+‖∫ {𝐷𝛼𝐹 (𝑥) (𝑐1(𝑥, 𝜏) − 𝑐 (𝑥, 𝜏)} 𝑑𝜏‖ 
0 𝜕𝑥
2 2
𝑡 𝑡
And, remembering ‖∫ 𝑓(𝑥)‖ ≤ ∫ ‖𝑓(𝑥)‖𝑑𝑥, 0 0
𝑡 𝜕
‖𝐹𝐶1(𝑥, 𝑡) − 𝐹𝐶2(𝑥, 𝑡)‖ ≤ ∫ ‖𝑉
𝛼
𝐹 (𝑥)‖ ‖ (𝑐1(𝑥, 𝜏) − 𝑐2(𝑥, 𝜏)‖ 𝑑𝜏 
0 𝜕𝑥
(3-56) 
𝑡 𝜕2
+∫ ‖𝐷𝛼𝐹 (𝑥)‖‖ (𝑐1(𝑥, 𝜏) − 𝑐2(𝑥, 𝜏)‖𝑑𝜏 2
0 𝜕𝑥
Applying the presented theorem for Lipschitz condition boundedness for partial differential,  
𝑡
‖𝐹𝐶1(𝑥, 𝑡) − 𝐹𝐶2(𝑥, 𝑡)‖ ≤ ‖𝑉
𝛼
𝐹 (𝑥)‖ 𝜃1‖(𝑐1(𝑥, 𝜏) − 𝑐2(𝑥, 𝜏)‖∫ 𝑑𝜏 
0
𝑡
+‖𝐷𝛼𝐹 (𝑥)‖ 𝜃
2
2‖(𝑐1(𝑥, 𝜏) − 𝑐2(𝑥, 𝜏)‖∫ 𝑑𝜏 
0 (3-57) 
                                              ≤ {‖𝑉𝛼𝐹 (𝑥)‖ 𝜃1 𝑇
𝛼
𝑚𝑎𝑥 + ‖𝐷𝐹 (𝑥)‖ 𝜃
2
2  𝑇𝑚𝑎𝑥} ‖(𝑐1(𝑥, 𝜏) − 𝑐2(𝑥, 𝜏)‖ 
Let ‖𝑉𝛼𝐹 (𝑥)‖ 𝜃1 𝑇𝑚𝑎𝑥 + ‖𝐷
𝛼
𝐹 (𝑥)‖ 𝜃
2
2  𝑇𝑚𝑎𝑥 = 𝐾, 
‖𝐹𝐶1(𝑥, 𝑡) − 𝐹𝐶2(𝑥, 𝑡)‖ ≤ 𝐾‖(𝑐1(𝑥, 𝜏) − 𝑐2(𝑥, 𝜏)‖ (3-58) 
Thus, the fractal transport equation does uphold the Lipschitz condition.  
To facilitate the evaluation of the fractal transport equation, a new function is introduced 
𝜕 𝜕2
ℱ(𝑥, 𝑡, 𝑐(𝑥, 𝑡)) = 𝑉𝛼𝐹 (𝑥) 𝑐(𝑥, 𝑡) + 𝐷
𝛼
𝐹 (𝑥) 𝑐(𝑥, 𝑡) 
(3-59) 
𝜕𝑥 𝜕𝑥2
such that, 
𝜕
𝑐(𝑥, 𝑡) = ℱ(𝑥, 𝑡, 𝑐(𝑥, 𝑡) (3-60) 
𝜕𝑡
56 
 
Integrating on both sides of Equation (3-60), 
𝑡
𝑐(𝑥, 𝑡) = 𝑐(𝑥, 0) +∫ ℱ(𝑥, 𝜏, 𝑐(𝑥, 𝜏)) 𝑑𝜏 (3-61) 
0
A set is created in which to evaluate the fractal transport equation, 
𝐴 = 𝑇̅̅ ̅̅(̅̅𝑡 ̅̅) × ̅̅𝑌̅̅(̅̅𝑐 ̅̅) (3-62) 𝑎,𝑏 𝑎 0 𝑏 0
where, 
̅̅𝑇̅̅𝑎(̅̅𝑡 ̅̅0) = (𝑡0 − 𝑎, 𝑡0 + 𝑎), and  
𝑌̅̅ ̅̅(̅̅𝑐 ̅̅𝑏 0) = (𝑐0 − 𝑏, 𝑐0 + 𝑏). 
The Banach fixed-point theorem is applied by introducing the norm of the supremum (statistical limit 
of a set) for 𝑐(𝑌𝑏(𝑐0), 𝑇𝑎(𝑡0)) denoted as 𝜑,  
  𝑠𝑢𝑝|𝜑(𝑏)|
‖𝜑‖∞ =  (3-63) 𝐴𝑎,𝑏
Let 𝐴𝑎,𝑏 be a set where, 𝐹: 𝐴𝑎,𝑏  → 𝐴𝑎,𝑏, such that   
𝑡
𝐹𝜑 = 𝑐0 +∫ ℱ(𝑥, 𝜏, 𝑐(𝑥, 𝜏)) 𝑑𝜏 (3-64) 
0
Remembering the physical meaning of the fractal advection dispersion equation and the nature of 
groundwater transport, the spatial component (𝑏) of an aquifer domain is such that the simulated 
plume when the initial concentration source is removed cannot be greater than the aquifer total 
extent: 
‖𝐹𝜑 − 𝑐0‖ ≤ 𝑏 (3-65) 
Now that the domain of the equation is defined, let us consider the following sequence: 
𝑐𝑛+1(𝑥, 𝑡) = 𝐹𝑐𝑛(𝑥, 𝑡) 
(3-66) 
‖𝑐𝑛+1 − 𝑐𝑛‖∞ = ‖𝐹𝑐𝑛(𝑥, 𝑡) − 𝐹𝑐𝑛−1(𝑥, 𝑡)‖∞ 
Equation (3-50) to Equation (3-58) proves that 
‖𝐹𝑐𝑛(𝑥, 𝑡) − 𝐹𝑐𝑛−1(𝑥, 𝑡)‖∞ ≤ 𝐾‖(𝑐𝑛(𝑥, 𝑡) − 𝑐𝑛−1(𝑥, 𝑡)‖∞ (3-67) 
and it follows that 
‖𝐹𝑐𝑛−1(𝑥, 𝑡) − 𝐹𝑐𝑛−2(𝑥, 𝑡)‖
2
∞ ≤ 𝐾 ‖(𝑐𝑛−1(𝑥, 𝑡) − 𝑐𝑛−2(𝑥, 𝑡)‖∞ 
                                                      ≤ 𝐾3‖(𝑐𝑛−3(𝑥, 𝑡) − 𝑐𝑛−4(𝑥, 𝑡)‖∞ (3-68) 
… 
                                             ≤ 𝐾𝑛‖(𝑐1(𝑥, 𝑡) − 𝑐0(𝑥, 𝑡)‖∞ 
with 𝐾 < 1, then 
lim ‖𝑐 𝑛𝑛+1 − 𝑐𝑛‖∞ < lim 𝐾 ‖(𝑐1(𝑥, 𝑡) − 𝑐0(𝑥, 𝑡)‖∞ 
𝑛→∞ 𝑛→∞ (3-69) 
Thus, (𝑐𝑛+1)𝑛∈ℕ is Cauchy in complex space. Then 𝐹𝑐𝑛 has a fixed point using the Banach fixed point 
theorem and the fractal transport advection dispersion equation has a unique solution.  
57 
 
3.3 Numerical approximation methods (fractal derivative) 
The formulation of the fractal advection-dispersion equation has been proven to be bounded, exist 
and unique. Accordingly, numerical approximation methods for the developed fractal (space) 
advection-dispersion equation are provided for the solution of the new model. The fractal derivative 
is a local operator that allows for the application of traditional finite difference approximation 
methods for time and space. The numerical approximation makes use of the traditional explicit 
forward finite difference method, as well as the Crank-Nicolson finite difference method for the fractal 
derivative formulation. The numerical stability of these numerical solutions are investigated. In the 
following section, the numerical approximation for the fractal integral formulation makes use of the 
Simpson’s 3/8 and Boole’s numerical integration rules. 
3.3.1 Forward finite difference scheme 
Explicit forward difference in time and space, and the second-order approximation are applied to 
Equation (3-41): 
𝑐𝑛+1 − 𝑐𝑛𝑖 𝑖 𝑐
𝑛 − 𝑐𝑛 𝑛𝑖+1 𝑖 𝑐𝑖+1 − 2𝑐
𝑛
𝑖 + 𝑐
𝑛
𝛼 𝛼 𝑖−1= 𝑉 + 𝐷  (3-70) 
∆𝑡 𝐹 ∆𝑥 𝐹 (∆𝑥)2
where, 
 𝑖 denotes one-dimensional grid-centered framework with conventional unit vector (𝒊),  
 𝑉𝛼𝐹  is the fractal dimension velocity term, and 
 𝐷𝛼𝐹  is the fractal dimension dispersion term. 
Rearranging, 
𝑐𝑛+1 𝛼 𝛼 𝛼𝑖 1 𝑉𝐹 2𝐷𝐹 𝑛 𝑉𝐹 𝐷
𝛼 𝐷𝛼𝐹 𝐹
= ( − − )𝑐 𝑛 𝑛
( )2 𝑖
+ ( + )𝑐 + ( ) 𝑐  
( (3-71) ∆𝑡 ∆𝑡 ∆𝑥 ∆𝑥 ∆𝑥 ∆𝑥)2 𝑖+1 (∆𝑥)2 𝑖−1
Simplifying by using constants 𝑎2, 𝑏2, 𝑐2, and 𝑑2 
𝑎 𝑐𝑛+12 𝑖 = 𝑏
𝑛 𝑛
2𝑐𝑖 + 𝑐2𝑐𝑖+1 + 𝑑 𝑐
𝑛
2 𝑖−1 (3-72) 
where, 
1
𝑎2 =  ∆𝑡
1 𝑉𝛼𝐹 2𝐷
𝛼
𝐹
𝑏2 = − −  ∆𝑡 ∆𝑥 (∆𝑥)2
𝑉𝛼 𝛼𝐹 𝐷𝐹
𝑐2 = +  ∆𝑥 (∆𝑥)2
𝐷𝛼𝐹
𝑑2 =  (∆𝑥)2
Equation (3-72) is the traditional explicit finite difference approximation of the fractal (space) 
advection-dispersion equation. 
58 
 
A finite difference scheme is considered stable if the errors incurred at a discrete time step are not 
propagated throughout the simulation. A Von Neumann stability analysis is performed to evaluate the 
stability of the finite difference schemes applied to the fractal advection-dispersion equation. The 
error incurred in the numerical approximation can be defined as: 
(3-73) 
𝜆𝑘𝑖 = 𝑁
𝑘
𝑖 − 𝑐
𝑘
𝑖  
where, 
 𝜆𝑘𝑖  is the approximation or round-off error, 
 𝑁𝑘𝑖  is the numerical solution, and 
 𝑐𝑘𝑖  is the exact solution.  
Considering Equation (3-72) in terms of the recurrence relation for the error: 
𝑘+1 𝑘 𝑘 𝑘
𝑎2𝜆𝑖 = 𝑏2𝜆𝑖 + 𝑐2𝜆𝑖+1 + 𝑑2𝜆𝑖−1 (3-74) 
The error for each discrete point in time and space has been determined as: 
𝜆𝑘+1 = 𝑒𝑎(𝑡+Δ𝑡)𝑒𝑗𝑘𝑥𝑖  (3-75) 
𝜆𝑘 = 𝑒𝑎𝑡𝑒𝑗𝑘𝑥𝑖  (3-76) 
𝜆𝑘 𝑎𝑡 𝑗𝑘(𝑥+Δ𝑥) (3-77) 𝑖+1 = 𝑒 𝑒  
𝜆𝑘 = 𝑒𝑎𝑡𝑖−1 𝑒
𝑗𝑘(𝑥−Δ𝑥) 
(3-78) 
Substituting back into Equation (3-74): 
𝑎 𝑒𝑎(𝑡+Δ𝑡)𝑒𝑗𝑘𝑥 = 𝑏 𝑒𝑎𝑡𝑒𝑗𝑘𝑥 + 𝑐 𝑒𝑎𝑡𝑒𝑗𝑘(𝑥+Δ𝑥) + 𝑑 𝑒𝑎𝑡𝑒𝑗𝑘(𝑥−Δ𝑥)2 2 2 2  (3-79) 
Expanding, 
𝑎 𝑒𝑎𝑡𝑒𝑎Δ𝑡𝑒𝑗𝑘𝑥 = 𝑏 𝑒𝑎𝑡𝑒𝑗𝑘𝑥 + 𝑐 𝑒𝑎𝑡𝑒𝑗𝑘𝑥𝑒𝑗𝑘Δ𝑥2 2 2 + 𝑑2𝑒
𝑎𝑡𝑒𝑗𝑘𝑥𝑒−𝑗𝑘Δ𝑥 (3-80) 
Simplifying and rearranging, 
𝑏2 𝑐2 𝑑2
𝑒𝑎Δ𝑡 = + 𝑒𝑗𝑘Δ𝑥 + 𝑒−𝑗𝑘Δ𝑥 (3-81) 
𝑎2 𝑎2 𝑎2
The amplification factor (𝐺) can be defined as: 
𝜆𝑘+1 𝑎(𝑡+Δ𝑡)𝑖 𝑒 𝑒
𝑗𝑘𝑥
𝐺 = = = 𝑒𝑎Δ𝑡𝑘  𝑎𝑡 𝑗𝑘𝑥 (3-82) 𝜆𝑖 𝑒 𝑒
where, for the solution to remain bounded the condition |𝐺| ≤ 1. 
Thus, Equation (3-81) expresses the stability criteria for the forward finite difference scheme applied 
1
to the fractal advection-dispersion equation, where |𝑒𝑎Δ𝑡| ≤ 1, and remembering that 𝑎2 = : ∆𝑡
𝑒𝑎Δ𝑡 = ∆𝑡𝑏2 + ∆𝑡𝑐
𝑗𝑘Δ𝑥 −𝑗𝑘Δ𝑥
2𝑒 + ∆𝑡𝑑2𝑒  (3-83) 
59 
 
Applying known identities and the absolute value of a complex number: 
|𝑒𝑎Δ𝑡| = √[∆𝑡𝑏1 + ∆𝑡𝑐1{cos(𝑘∆𝑥) + 𝑗 sin(𝑘∆𝑥) }]
2 + [∆𝑡𝑑1{cos(𝑘∆𝑥) − 𝑗 sin(𝑘∆𝑥) }]
2 ≤ 1 (3-84) 
Equation (3-84) is an expression of the stability criteria for Equation (3-70).  
3.3.2 Crank-Nicolson finite difference scheme 
The Crank-Nicolson finite difference scheme is applied to Equation (3-30): 
𝑐𝑛+1   −   𝑐𝑛 𝑐𝑛 − 𝑐𝑛 𝑐𝑛 − 2𝑐𝑛 𝑛𝑖 𝑖 𝛼 𝑖+1 𝑖 𝛼 𝑖+1 𝑖 + 𝑐𝑖−1= 0.5(𝑉𝐹 + 𝐷𝐹 ) ∆𝑡 ∆𝑥 (∆𝑥)2
(3-85) 
𝑐𝑘+1 𝑘+1 𝑘+1 𝑘+1 𝑘+1
𝛼 𝑖+1 − 𝑐𝑖 𝑐𝛼 𝑖+1 − 2𝑐𝑖 + 𝑐𝑖−1+0.5(𝑉𝐹 + 𝐷𝐹 ) ∆𝑥 (∆𝑥)2
Rearranging, 
1 𝑉𝛼 2𝐷𝛼 1 𝑉𝛼 2𝐷𝛼
𝑐𝑛+1
𝐹 𝐹 𝐹 𝐹
( − 0.5 (− − )) = 0.5 ( − − )𝑐𝑛𝑖 + ∆𝑡 ∆𝑥 (∆𝑥)2 ∆𝑡 ∆𝑥 (∆𝑥)2 𝑖
𝑉𝛼 𝐷𝛼 𝛼 𝛼 𝛼𝐹 𝐹 𝑛 𝐷𝐹 𝑉𝐹 𝐷𝐹 (3-86) 0.5 ( + ) 𝑐 + 0.5 ( ) 𝑐𝑛 + 0.5 ( + ) 𝑐𝑛+1 
∆𝑥 (∆𝑥)2 𝑖+1 (∆𝑥)2 𝑖−1 ∆𝑥 (∆𝑥)2 𝑖+1
𝐷𝛼𝐹
+0.5 ( ) 𝑐𝑛+1
(∆𝑥)2 𝑖−1
 
Simplifying by using constants 𝑒2, 𝑏2, 𝑐2, and 𝑑2 
𝑒2𝑐
𝑛+1 = 𝑏 𝑐𝑛𝑖 2 𝑖 + 𝑐2𝑐
𝑛
𝑖+1 + 𝑑
𝑛 𝑛+1 𝑛+1 (3-87) 
2𝑐𝑖−1 + 𝑐2𝑐𝑖+1 + 𝑑2𝑐𝑖−1  
where, 
1 𝑉𝛼𝐹 2𝐷
𝛼
𝐹
𝑒2 = − 0.5 (− − ) ∆𝑡 ∆𝑥 (∆𝑥)2
Equation (3-87) is the Crank-Nicolson finite difference approximation of the fractal (space) advection-
dispersion equation. 
A Von Neumann stability analysis is performed to evaluate the stability of the Crank-Nicolson finite 
difference approximation applied to Equation (3-87), remembering Equations (3-75) to (3-78), and 
additionally: 
𝜆𝑘+1 = 𝑒𝑎(𝑡+Δ𝑡)𝑒𝑗𝑘(𝑥+∆𝑥) (3-88) 𝑖+1
𝜆𝑘+1 𝑎(𝑡+∆𝑡) 𝑗𝑘(𝑥−∆𝑥) (3-89) 𝑖−1 = 𝑒 𝑒  
To obtain: 
𝑒 𝑒𝑎(𝑡+Δ𝑡)𝑒𝑗𝑘𝑥 = 𝑏 𝑎𝑡 𝑗𝑘𝑥 𝑎𝑡 𝑗𝑘(𝑥+Δ𝑥) 𝑎𝑡 𝑗𝑘(𝑥−Δ𝑥) 𝑎(𝑡+∆𝑡) 𝑗𝑘(𝑥+Δ𝑥)2 2𝑒 𝑒 + 𝑐2𝑒 𝑒 + 𝑑2𝑒 𝑒 + 𝑐2𝑒 𝑒  
(3-90) 
+𝑑2𝑒
𝑎(𝑡+∆𝑡)𝑒𝑗𝑘(𝑥−Δ𝑥) 
60 
 
Expanding, 
𝑒 𝑒𝑎𝑡𝑒𝑎Δ𝑡𝑒𝑗𝑘𝑥 = 𝑏 𝑒𝑎𝑡𝑒𝑗𝑘𝑥 + 𝑐 𝑒𝑎𝑡𝑒𝑗𝑘𝑥𝑒𝑗𝑘Δ𝑥 + 𝑑 𝑒𝑎𝑡𝑒𝑗𝑘𝑥𝑒−𝑗𝑘Δ𝑥 + 𝑐 𝑒𝑎𝑡𝑒𝑎Δ𝑡𝑒𝑗𝑘𝑥𝑒𝑗𝑘Δ𝑥2 2 2 2 2  
(3-91) 
+𝑑 𝑒𝑎𝑡𝑒𝑎Δ𝑡2 𝑒
𝑗𝑘𝑥𝑒−𝑗𝑘Δ𝑥 
Simplifying and rearranging, 
𝑏 + 𝑐 𝑒𝑗𝑘Δ𝑥 + 𝑑 𝑒−𝑗𝑘Δ𝑥
𝑒𝑎Δ𝑡
2 2 2
=  
(𝑒 − 𝑐 𝑒𝑗𝑘Δ𝑥 −𝑗𝑘Δ𝑥
(3-92) 
2 2 − 𝑑2𝑒 )
Applying known identities, simplifying and applying the absolute value of a complex number, 
where 𝑧 = 𝑎 + 𝑖𝑏 and |𝑧| = √𝑎2 + 𝑏2: 
√[𝑏 + (𝑐 + 𝑑 )cos (𝑘∆𝑥)]2+[(𝑐 − 𝑑 ) sin(𝑘∆𝑥)]2 
|𝑒𝑎Δ𝑡
2 2 2 1 2
| = ≤ 1 (3-93) 
√[𝑒2 − (𝑐2 + 𝑑 )cos (𝑘∆𝑥)]22 +[(𝑐 21 − 𝑑2) sin(𝑘∆𝑥)]  
where, 
√[𝑏2 + (𝑐2 + 𝑑2)cos( 𝑘∆𝑥)]2+[(𝑐2 − 𝑑2) sin(𝑘∆𝑥)]2  (3-94) 
≤ √[𝑒 − (𝑐 + 𝑑 2 22 2 2)cos (𝑘∆𝑥)] +[(𝑐2 − 𝑑2) sin(𝑘∆𝑥)]   
Simplifying, 
𝑒2 − 𝑏2
cos (𝑘∆𝑥) ≤   
2(𝑐2 + 𝑑2)
(3-95) 
Equation (3-95) is an expression of the stability criteria for Equation (3-85).  
3.4 Numerical integration methods (fractal integral) 
The fractal derivative is a local operator, and thus the numerical solution of fractal derivative 
equations can be approximated using standard numerical techniques for the integer-order derivative 
equations (Chen et al., 2010a).  
The fractal advection-dispersion equation (Equation (3-41)) is converted to an integral formulation by 
applying the integral to both sides: 
𝑡 𝜕 𝑡 𝜕 𝜕2
∫ 𝑐(𝑥, 𝜏) = ∫ (𝑉𝛼𝐹 𝑐(𝑥, 𝜏) + 𝐷
𝛼
𝐹 𝑐(𝑥, 𝜏))  
0 𝜕𝜏 0 𝜕𝑥 𝜕𝑥
2 (3-96) 
Applying the integral to the time derivative, 
𝑡 𝜕 𝜕2
𝑐(𝑥, 𝑡) − 𝑐(𝑥, 0) = ∫ (𝑉𝛼𝐹 𝑐(𝑥, 𝜏) + 𝐷
𝛼
𝐹 𝑐(𝑥, 𝜏))  
(3-97) 
0 𝜕𝑥 𝜕𝑥
2
Rearranging, 
𝑡 𝜕 𝜕2
𝑐(𝑥, 𝑡) = 𝑐(𝑥, 0) + ∫ (𝑉𝛼 𝑐(𝑥, 𝜏) + 𝐷𝛼𝐹 𝐹 𝑐(𝑥, 𝜏))  
0 𝜕𝑥 𝜕𝑥
2 (3-98) 
61 
 
To facilitate the numerical integration of the fractal advection and dispersion terms, let 
𝐹(𝑥, 𝜏, 𝐶(𝑥, 𝜏)) be equal to the integral of the advection and dispersion terms, 
𝜕 𝜕2
where  𝐹(𝑥, 𝜏, 𝑐(𝑥, 𝜏)) = 𝑉𝛼𝐹 𝑐(𝑥, 𝜏) + 𝐷
𝛼
𝐹 𝑐(𝑥, 𝜏): 𝜕𝑥 𝜕𝑥2
𝑡
𝑐(𝑥, 𝑡) = 𝑐(𝑥, 0) + ∫ 𝐹(𝑥, 𝜏, 𝑐(𝑥, 𝜏))  (3-99) 
0
Considering Equation (3-99) for a specific time (𝑡𝑛): 
𝑡𝑛
𝑐(𝑥, 𝑡𝑛) = 𝑐(𝑥, 0) + ∫ 𝐹(𝑥, 𝜏, 𝑐(𝑥, 𝜏))  (3-100) 
0
The integral can be approximated applying numerical integration techniques, including the traditional 
Trapezoid Rule, or the alternative methods of Simpson’s 3/8 Rule and Boole’s Rule, which are more 
rigorous. For this reason, the Simpson’s 3/8 Rule and Boole’s Rule are applied. 
3.4.1 Simpson’s 3/8 Rule of numerical integration 
Consider the defined fractal integral over a specific time interval (𝑡𝑛), where, 0 = 𝑡0 < 𝑡1 < 𝑡2 <
𝑡3… < 𝑡𝑛 = 𝑡 
 𝑡 𝑡𝑛
𝛼∫ 𝜏𝛼−1 𝑓(𝜏) 𝑑𝜏 = 𝛼∫ 𝜏𝛼−1 𝑓(𝜏) 𝑑𝜏 (3-101) 
0 𝑡0
Applying Simpson’s 3/8 Rule for the defined fractal integral (Abramowitz and Stegun, 1966): 
𝑡𝑛 𝑡3 𝑡6 𝑡𝑛−1
∫ 𝛼𝜏𝛼−1 𝑓(𝜏) 𝑑𝜏 = ∫ 𝛼𝜏𝛼−1 𝑓(𝜏) 𝑑𝜏 + ∫ 𝛼𝜏𝛼−1 𝑓(𝜏) 𝑑𝜏 + ⋯+∫ 𝛼𝜏𝛼−1 𝑓(𝜏) 𝑑𝜏 (3-102) 
𝑡0 𝑡0 𝑡3 𝑡𝑛−4
According to Simpson’s 3/8 Rule (Abramowitz and Stegun, 1966): 
 𝑏 𝑏 − 𝑎 2𝑎 + 𝑏 𝑎 + 2𝑏
∫ 𝑓(𝑥)𝑑𝑥 = [𝑓(𝑎) + 3𝑓 ( ) + 3𝑓 ( ) + 𝑓(𝑏)] 
8 3 6  (3-103) 𝑎
Without losing generality, the integral is considered: 
𝑡3 𝑡 − 𝑡
𝛼−1 3 0∫ 𝛼𝜏  𝑓(𝜏) 𝑑𝜏 = [𝛼𝑡𝛼−10  𝑓(𝑡0) + 3𝛼𝑡
𝛼−1 𝛼−1 𝛼−1
8 1
𝑓(𝑡1) + 3𝛼𝑡2 𝑓(𝑡2) + 𝛼𝑡3 𝑓(𝑡3)] (3-104) 
𝑡0
Simplifying, 
𝑡3 𝑡3 − 𝑡0
∫ 𝛼𝜏𝛼−1 𝑓(𝜏) 𝑑𝜏 =𝛼 ( ) [𝑡𝛼−10  𝑓(𝑡0) + 3𝑡
𝛼−1
1 𝑓(𝑡1) + 3𝑡
𝛼−1
2 𝑓(𝑡
𝛼−1
2) + 𝑡3 𝑓(𝑡3)] (3-105) 
𝑡 80
𝑡
Similarly, the integral ∫ 6  is considered: 𝑡3
𝑡6 𝑡6 − 𝑡3
∫ 𝛼𝜏𝛼−1 𝑓(𝜏) 𝑑𝜏 =𝛼 ( ) [𝑡𝛼−1 𝛼−1 𝛼−1 𝛼−1 (3-106) 
8 3
 𝑓(𝑡3) + 3𝑡4 𝑓(𝑡4) + 3𝑡5 𝑓(𝑡5) + 𝑡6 𝑓(𝑡6)] 
𝑡3
62 
 
𝑡
and, the generalized integral 𝑛−1∫  is considered: 𝑡𝑛−4
𝑡𝑛−1
∫ 𝛼𝜏𝛼−1 𝑓(𝜏) 𝑑𝜏 = 
𝑡𝑛−4 (3-107) 
𝑡𝑛−1 − 𝑡𝑛−4
𝛼 ( ) [𝑡𝛼−1𝑛−4 𝑓(𝑡
𝛼−1 𝛼−1 𝛼−1
8 𝑛−4
) + 3𝑡𝑛−1𝑓(𝑡𝑛−3) + 3𝑡𝑛−2𝑓(𝑡𝑛−2) + 𝑡𝑛−1𝑓(𝑡𝑛−1)] 
Incorporating the integrals back into Equation (3-102), and simplifying: 
𝑡𝑛
∫ 𝛼𝜏𝛼−1 𝑓(𝜏) 𝑑𝜏 = 
𝑡0
𝑡𝛼−1 𝑓(𝑡 ) + 3𝑡𝛼−1𝑓(𝑡 ) + 3𝑡𝛼−1𝑓(𝑡 ) + 𝑡𝛼−1𝑓(𝑡 ) +
3 ∆𝑡 𝛼 0 0 1 1 2 2 3 3 (3-108) 
( ) { 𝑡𝛼−1 𝑓(𝑡 ) + 3𝑡𝛼−1𝑓(𝑡 ) + 3𝑡𝛼−1𝑓(𝑡 ) + 𝑡𝛼−13 3 4 4 5 5 6 𝑓(𝑡6) +⋯+ } 8
𝑡𝛼−1 𝑓(𝑡 𝛼−1𝑛−4 𝑛−4) + 3𝑡𝑛−1𝑓(𝑡𝑛−3) + 3𝑡
𝛼−1𝑓(𝑡 ) + 𝑡𝛼−1𝑛−2 𝑛−2 𝑛−1𝑓(𝑡𝑛−1)
Equation (3-108) is the numerical approximation of the fractal integral using Simpson’s 3/8 Rule for 
numerical integration.  
Applying the Simpson’s 3/8 Rule to the integral formulation of the fractal advection-dispersion 
equation, 
𝑡𝑛
∫ 𝐹(𝑥, 𝑡𝑛, 𝑐(𝑥, 𝑡𝑛))𝑑𝜏 = 
0
𝑡3 𝑡6 𝑡𝑛−1 (3-109) 
∫ 𝐹(𝑥, 𝑡𝑛, 𝑐(𝑥, 𝑡𝑛))𝑑𝜏+∫ 𝐹(𝑥, 𝑡𝑛, 𝑐(𝑥, 𝑡𝑛))𝑑𝜏+⋯+∫ 𝐹(𝑥, 𝑡𝑛, 𝑐(𝑥, 𝑡𝑛))𝑑𝜏 
𝑡0 𝑡3 𝑡𝑛−4
𝑡
Without losing generality, the integral, 3∫ 𝐹(𝑥, 𝑡𝑛, 𝑐(𝑥, 𝑡𝑛))𝑑𝜏, is considered: 𝑡0
𝑡3 𝑡3 − 𝑡0
∫ 𝐹(𝑥, 𝑡𝑛, 𝑐(𝑥, 𝑡𝑛))𝑑𝜏 = ∙ 
𝑡 80 (3-110) 
[𝐹(𝑥, 𝑡0, 𝑐(𝑥, 𝑡0)) + 3𝐹(𝑥, 𝑡1, 𝑣(𝑥, 𝑡1)) + 3𝐹(𝑥, 𝑡2, 𝑐(𝑥, 𝑡2)) + 𝐹(𝑥, 𝑡3, 𝑐(𝑥, 𝑡3))] 
𝑡
Similarly, the integral, ∫ 6 𝐹(𝑥, 𝑡𝑛, 𝑐(𝑥, 𝑡𝑛))𝑑𝜏, is considered: 𝑡3
𝑡6 𝑡6 − 𝑡3
∫ 𝐹(𝑥, 𝑡𝑛, 𝑐(𝑥, 𝑡𝑛))𝑑𝜏 = ∙ 
𝑡 83 (3-111) 
[𝐹(𝑥, 𝑡3, 𝑐(𝑥, 𝑡3)) + 3𝐹(𝑥, 𝑡4, 𝑐(𝑥, 𝑡4)) + 3𝐹(𝑥, 𝑡5, 𝑐(𝑥, 𝑡5)) + 𝐹(𝑥, 𝑡6, 𝑐(𝑥, 𝑡6))] 
𝑡𝑛−1
and, the generalized integral, ∫ 𝐹(𝑥, 𝑡𝑛, 𝑐(𝑥, 𝑡𝑡 𝑛))𝑑𝜏, is considered: 𝑛−4
𝑡𝑛−1 𝑡𝑛−1 − 𝑡𝑛−4 𝐹(𝑥, 𝑡𝑛−4, 𝑐(𝑥, 𝑡𝑛−4)) + 3𝐹(𝑥, 𝑡𝑛−3, 𝑐(𝑥, 𝑡𝑛−3)) +
∫ 𝐹(𝑥, 𝑡𝑛, 𝑐(𝑥, 𝑡𝑛))𝑑𝜏 = ∙ [ ] 
𝑡 8 3𝐹(𝑥, 𝑡𝑛−2, 𝑐(𝑥, 𝑡𝑛−2)) + 𝐹(𝑥, 𝑡 , 𝑐(𝑥, 𝑡 )) (3-112) 𝑛−4 𝑛−1 𝑛−1
Incorporating the integrals back into Equation (3-100), and simplifying: 
 
63 
 
𝐹(𝑥, 𝑡0, 𝑐(𝑥, 𝑡 0
)) + 3𝐹(𝑥, 𝑡1, 𝑐(𝑥, 𝑡1))  
 +3𝐹(𝑥, 𝑡2, 𝑐(𝑥, 𝑡2)) + 𝐹(𝑥, 𝑡3, 𝑐(𝑥, 𝑡3)) +  
  
3 ∆𝑡 𝐹(𝑥, 𝑡3, 𝑐(𝑥, 𝑡3)) + 3𝐹(𝑥, 𝑡4, 𝑐(𝑥, 𝑡4))
𝑐(𝑥, 𝑡𝑛) = 𝑐(𝑥, 0) + ( )  8  +3𝐹(𝑥, 𝑡5, 𝑐(𝑥, 𝑡5)) + 𝐹(𝑥, 𝑡6, 𝑐(𝑥, 𝑡6)) + ⋯+  (3-113) 
 
 𝐹(𝑥, 𝑡𝑛−4, 𝑐
(𝑥, 𝑡𝑛−4)) + 3𝐹(𝑥, 𝑡
 
𝑛−3, 𝑐(𝑥, 𝑡𝑛−3))  
{+3𝐹(𝑥, 𝑡𝑛−2, 𝑐(𝑥, 𝑡𝑛−2)) + 𝐹(𝑥, 𝑡𝑛−1, 𝑐(𝑥, 𝑡𝑛−1))}
Substituting back the function 𝐹(𝑥, 𝜏, 𝑐(𝑥, 𝜏)), and expanding 𝑐(𝑥, 𝑡𝑛): 
𝑐(𝑥, 𝑡𝑛) = 𝑐(𝑥, 0) + 
2
 𝜕 𝜕 𝜕 𝜕
2
(𝑉𝛼𝐹 𝑐(𝑥, 𝑡0) + 𝐷
𝛼
𝐹 𝑐(𝑥, 𝑡0)) + 3(𝑉
𝛼 𝛼  
𝜕𝑥 𝜕𝑥2 𝐹
𝑐(𝑥, 𝑡 ) + 𝐷 𝑐(𝑥, 𝑡 ))
 𝜕𝑥
1 𝐹 𝜕𝑥2 1  
  
 𝜕 𝜕2 𝜕 𝜕2  
 +3(𝑉𝛼𝐹 𝑐(𝑥, 𝑡2) + 𝐷
𝛼
𝐹 𝑐(𝑥, 𝑡2)) + (𝑉
𝛼
𝐹 𝑐(𝑥, 𝑡
𝛼
𝜕𝑥 𝜕𝑥2 𝜕𝑥 3
) + 𝐷𝐹 𝑐(𝑥, 𝑡3)) +2  
 𝜕𝑥  
3 ∆𝑡 (3-114) 
( ) +⋯+  
8  𝜕 𝜕2 𝜕 𝜕2  
 (𝑉𝛼𝐹 𝑐(𝑥, 𝑡𝑛−4) + 𝐷
𝛼
𝐹 𝑐(𝑥, 𝑡
𝛼
𝑛−4)) + 3(𝑉𝐹 𝑐(𝑥, 𝑡𝑛−3) + 𝐷
𝛼
𝐹 𝑐(𝑥, 𝑡𝑛−3))𝜕𝑥 𝜕𝑥2
 
 𝜕𝑥 𝜕𝑥
2
 
  
 𝜕 𝜕2 𝜕 𝜕2  
 +3(𝑉
𝛼
𝐹 𝑐(𝑥, 𝑡𝑛−2) + 𝐷
𝛼 𝛼
𝐹 𝑐(𝑥, 𝑡 )) + (𝑉 𝑐(𝑥, 𝑡 ) + 𝐷
𝛼 𝑐(𝑥, 𝑡 ))
𝜕𝑥 𝜕𝑥2 𝑛−2 𝐹 𝜕𝑥 𝑛−1 𝐹 𝜕𝑥2 𝑛−1
 
{  }
Equation (3-114) is the numerical approximation of the integral formulation of the fractal advection-
dispersion equation, where the numerical integration of the fractal integral is performed using the 
Simpson’s 3/8 Rule. 
3.4.2 Boole’s Rule of numerical integration 
Alternatively, the Boole’s Rule for numerical integration can be applied to the approximation of the 
fractal integral. Applying Boole’s Rule for the defined fractal integral (Abramowitz and Stegun, 1966): 
𝑡𝑛 𝑡4 𝑡8 𝑡𝑛−1
∫ 𝛼𝜏𝛼−1 𝑓(𝜏) 𝑑𝜏 = ∫ 𝛼𝜏𝛼−1 𝑓(𝜏) 𝑑𝜏 + ∫ 𝛼𝜏𝛼−1 𝑓(𝜏) 𝑑𝜏 + ⋯+∫ 𝛼𝜏𝛼−1 𝑓(𝜏) 𝑑𝜏 (3-115) 
𝑡0 𝑡0 𝑡4 𝑡𝑛−5
According to Boole’s Rule (Abramowitz and Stegun, 1966): 
 𝑏 𝑏 − 𝑎
∫ 𝑓(𝑥)𝑑𝑥 = [7𝑓0 + 32𝑓1 + 127𝑓2 + 32𝑓90 3
+ 7𝑓4] (3-116) 
𝑎
𝑡
Without losing generality, the integral ∫ 4  is considered: 𝑡0
𝑡4 𝑡4 − 𝑡𝛼−1 0 7𝛼𝑡
𝛼−1
0  𝑓(𝑡0) + 32𝛼𝑡
𝛼−1
1 𝑓(𝑡
𝛼−1
∫ 𝛼𝜏  𝑓(𝜏) 𝑑𝜏 = [ 1
) + 12𝛼𝑡2 𝑓(𝑡2)] (3-117) 
𝑡 90 +32𝛼𝑡
𝛼−1𝑓(𝑡 ) + 7𝛼𝑡𝛼−1
0 3 3 4
𝑓(𝑡4)
Simplifying, 
𝑡4 𝑡 − 𝑡 𝛼−1
𝛼−1 4 0 7𝑡0  𝑓(𝑡0) + 32𝑡
𝛼−1𝑓(𝑡 ) + 12𝑡𝛼−11 1 2 𝑓(𝑡2) + 32𝑡
𝛼−1
3 𝑓(𝑡 )∫ 𝛼𝜏  𝑓(𝜏) 𝑑𝜏 =𝛼 ( ) [ 3
90 +7𝑡𝛼−1
] 
𝑡 4 𝑓(𝑡4)
(3-118) 
0
64 
 
𝑡
Similarly, the integral 8∫  is considered: 𝑡4
𝑡8 𝑡 𝛼−1
𝛼−1 8
− 𝑡4 7𝑡  𝑓(𝑡
∫ 𝛼𝜏  𝑓(𝜏) 𝑑𝜏 =𝛼 ( ) [ 4 4
)
] 
𝑡 90 +32𝑡
𝛼−1
5 𝑓(𝑡5) + 12𝑡
𝛼−1
6 𝑓(𝑡 ) + 32𝑡
𝛼−1
6 7 𝑓(𝑡7) + 7𝑡
𝛼−1
8 𝑓(𝑡 )
(3-119) 
4 8
𝑡
and, the generalized integral 𝑛−1∫  is considered: 𝑡𝑛−5
𝑡𝑛−1 𝑡𝑛−5 − 𝑡𝑛−1
∫ 𝛼𝜏𝛼−1 𝑓(𝜏) 𝑑𝜏 =𝛼 ( ) 
𝑡 90𝑛−5
(3-120) 
7𝑡𝛼−1𝑛−5  𝑓(𝑡× [ 𝑛−5
)
+32𝑡𝛼−1 𝛼−1 𝛼−1 𝛼−1
] 
𝑛−4𝑓(𝑡𝑛−4) + 12𝑡𝑛−3𝑓(𝑡𝑛−3) + 32𝑡𝑛−2𝑓(𝑡𝑛−2) + 7𝑡𝑛−1𝑓(𝑡𝑛−1)
Incorporating the integrals back into Equation (3-115), and simplifying: 
𝑡𝑛
∫ 𝛼𝜏𝛼−1 𝑓(𝜏) 𝑑𝜏 = 
𝑡0
7𝑡𝛼−1 𝑓(𝑡 ) + 32𝑡𝛼−1𝑓(𝑡 ) + 12𝑡𝛼−1𝑓(𝑡 ) + 32𝑡𝛼−1𝑓(𝑡 ) + 7𝑡𝛼−1
4 ∆𝑡 𝛼 0 0 1 1 2 2 3 3 4
𝑓(𝑡4) + (3-121) 
( ){ 7𝑡𝛼−14  𝑓(𝑡4) + 32𝑡
𝛼−1
5 𝑓(𝑡 ) + 12𝑡
𝛼−1
5 6 𝑓(𝑡6) + 32𝑡
𝛼−1
7 𝑓(𝑡7) + 7𝑡
𝛼−1
8 𝑓(𝑡8) + ⋯+ } 90
7𝑡𝛼−1𝑛−5  𝑓(𝑡 ) + 32𝑡
𝛼−1𝑓(𝑡 ) + 12𝑡𝛼−1𝑛−5 𝑛−4 𝑛−4 𝑛−3𝑓(𝑡𝑛−3) + 32𝑡
𝛼−1
𝑛−2𝑓(𝑡
𝛼−1
𝑛−2) + 7𝑡𝑛−1𝑓(𝑡𝑛−1)
Equation (3-121) is the numerical approximation of the fractal integral using Boole’s Rule for numerical 
integration.  
Boole’s Rule of numerical integration is now considered for the solution of integral of the advection 
and dispersion terms (represented as 𝐹(𝑥, 𝑡, 𝑐(𝑥, 𝑡))) in Equation (3-100), where (Abramowitz and 
Stegun, 1966): 
𝑡𝑛
∫ 𝐹(𝑥, 𝑡𝑛, 𝑐(𝑥, 𝑡𝑛))𝑑𝜏 = 
0
𝑡4 𝑡8 𝑡𝑛−1 (3-122) 
∫ 𝐹(𝑥, 𝜏, 𝑐(𝑥, 𝜏)) 𝑑𝜏 + ∫ 𝐹(𝑥, 𝜏, 𝑐(𝑥, 𝜏)) 𝑑𝜏 + ⋯+∫ 𝐹(𝑥, 𝜏, 𝑐(𝑥, 𝜏)) 𝑑𝜏 
𝑡0 𝑡4 𝑡𝑛−5
𝑡
Without losing generality, the integral, ∫ 4 𝐹(𝑥, 𝜏, 𝑐(𝑥, 𝜏)) 𝑑𝜏, is considered: 𝑡0
𝑡4 𝑡4 − 𝑡0 7𝐹(𝑥, 𝑡0, 𝑐(𝑥, 𝑡0)) + 32𝐹(𝑥, 𝑡1, 𝑐(𝑥, 𝑡1))
∫ 𝐹(𝑥, 𝜏, 𝑐(𝑥, 𝜏))𝑑𝜏 = [ ] 
90 (3-123) 𝑡 +12𝐹(𝑥, 𝑡2, 𝑐(𝑥, 𝑡2)) + 32𝐹(𝑥, 𝑡3, 𝑐(𝑥, 𝑡3)) + 7𝐹(𝑥, 𝑡4, 𝑐(𝑥, 𝑡 ))0 4
𝑡8
Similarly, the integral, ∫ 𝐹(𝑥, 𝜏, 𝑐(𝑥, 𝜏)) 𝑑𝜏, is considered: 
𝑡4
𝑡8 𝑡8 − 𝑡4 7𝐹(𝑥, 𝑡4, 𝑐(𝑥, 𝑡4)) + 32𝐹(𝑥, 𝑡5, 𝑐(𝑥, 𝑡5)) + 12𝐹(𝑥, 𝑡6, 𝑐(𝑥, 𝑡6))
∫ 𝐹(𝑥, 𝜏, 𝑐(𝑥, 𝜏))𝑑𝜏 = [ ] 
90 (3-124) 𝑡 +32𝐹(𝑥, 𝑡7, 𝑐(𝑥, 𝑡4 7)) + 7𝐹(𝑥, 𝑡8, 𝑐(𝑥, 𝑡8))
𝑡𝑛−1
and, the generalized integral, ∫ 𝐹(𝑥, 𝜏, 𝑐(𝑥, 𝜏)) 𝑑𝜏, is considered: 
𝑡𝑛−5
65 
 
𝑡𝑛−1
∫ 𝐹(𝑥, 𝜏, 𝑐(𝑥, 𝜏))𝑑𝜏 = 
𝑡𝑛−5 (3-125) 
𝑡𝑛−1 − 𝑡𝑛−5 7𝐹(𝑥, 𝑡𝑛−5, 𝑐(𝑥, 𝑡𝑛−5)) + 32𝐹(𝑥, 𝑡𝑛−4, 𝑐(𝑥, 𝑡𝑛−4)) + 12𝐹(𝑥, 𝑡𝑛−3, 𝑐(𝑥, 𝑡𝑛−3))
[ ] 
90 +32𝐹(𝑥, 𝑡𝑛−2, 𝑐(𝑥, 𝑡𝑛−2)) + 7𝐹(𝑥, 𝑡𝑛−1, 𝑐(𝑥, 𝑡𝑛−1))
Incorporating the integrals back into Equation (3-100), and simplifying: 
(𝑥, 𝑡𝑛) = 𝑐(𝑥, 0) + 
7𝐹(𝑥, 𝑡0, 𝑐(𝑥, 𝑡0)) + 32𝐹(𝑥, 𝑡1, 𝑐(𝑥, 𝑡1)) + 12𝐹(𝑥, 𝑡 , 𝑐(𝑥, 𝑡 )) 2 2  
 +32𝐹(𝑥, 𝑡3, 𝑐(𝑥, 𝑡3)) + 7𝐹(𝑥, 𝑡4, 𝑐(𝑥, 𝑡4)) +  
  
4 ∆𝑡 7𝐹(𝑥, 𝑡4, 𝑐(𝑥, 𝑡4)) + 32𝐹(𝑥, 𝑡5, 𝑐(𝑥, 𝑡5)) + 12𝐹(𝑥, 𝑡6, 𝑐(𝑥, 𝑡6)) (3-126) 
( )  
90  +32𝐹(𝑥, 𝑡7, 𝑐(𝑥, 𝑡7)) + 7𝐹(𝑥, 𝑡8, 𝑐(𝑥, 𝑡8)) +⋯+  
 
 7𝐹(𝑥, 𝑡𝑛−5, 𝑐
(𝑥, 𝑡𝑛−5)) + 32𝐹(𝑥, 𝑡𝑛−4, 𝑐(𝑥, 𝑡𝑛−4)) + 12𝐹(𝑥, 𝑡
 
𝑛−3, 𝑐(𝑥, 𝑡𝑛−3)) 
{ +32𝐹(𝑥, 𝑡𝑛−2, 𝑐(𝑥, 𝑡𝑛−2)) + 7𝐹(𝑥, 𝑡𝑛−1, 𝑐(𝑥, 𝑡𝑛−1)) }
𝜕 𝜕2
Substituting back the function 𝐹(𝑥, 𝜏, 𝑐(𝑥, 𝜏)) = 𝑉𝛼𝐹 𝑐(𝑥, 𝜏) + 𝐷
𝛼
𝜕𝑥 𝐹
( )
𝜕𝑥2
𝑐 𝑥, 𝜏 : 
𝑐(𝑥, 𝑡𝑛) = 𝑐(𝑥, 0) + 
𝜕 𝜕2 𝜕 𝜕
2
7(𝑉𝛼 𝛼 𝛼 𝛼𝐹 𝑐(𝑥, 𝑡0) + 𝐷𝐹 𝑐(𝑥, 𝑡0)) + 32𝐹 (𝑉𝐹 𝑐(𝑥, 𝑡1) + 𝐷𝐹 𝑐(𝑥, 𝑡 ))
 
 𝜕𝑥 𝜕𝑥2 𝜕𝑥 𝜕𝑥2
1
 
 2 2  
 𝜕 𝜕 𝜕 𝜕+12(𝑉𝛼𝐹 𝑐(𝑥, 𝑡2) + 𝐷
𝛼
𝐹 𝑐(𝑥, 𝑡2)) + 32(𝑉
𝛼
𝐹 𝑐(𝑥, 𝑡 ) + 𝐷
𝛼 𝑐(𝑥, 𝑡 )) +  
 𝜕𝑥 𝜕𝑥2 𝜕𝑥
3 𝐹 𝜕𝑥2 3  
  
 𝜕 𝜕
2
7(𝑉𝛼𝐹 𝑐(𝑥, 𝑡 ) + 𝐷
𝛼
4 𝐹 𝑐(𝑥, 𝑡4))
 
 𝜕𝑥 𝜕𝑥2  
4 ∆𝑡 (3-127) 
( ) +⋯+  
90  𝜕 𝜕2 𝜕 𝜕2  
 7(𝑉𝛼𝐹 𝑐(𝑥, 𝑡
𝛼
𝑛−5) + 𝐷𝐹 𝑐(𝑥, 𝑡𝑛−5)) + 32(𝑉
𝛼
𝐹 𝑐(𝑥, 𝑡
𝛼
𝑛−4) + 𝐷𝐹 𝑐(𝑥, 𝑡𝑛−4))  
 𝜕𝑥 𝜕𝑥
2 𝜕𝑥 𝜕𝑥2  
 𝜕 𝜕2 𝜕 𝜕2  
 +12(𝑉𝛼𝐹 𝑐(𝑥, 𝑡𝑛−3) + 𝐷
𝛼
𝐹 𝑐(𝑥, 𝑡𝑛−3)) + 32(𝑉
𝛼
𝐹 𝑐(𝑥, 𝑡𝑛−2) + 𝐷
𝛼
𝐹 𝑐(𝑥, 𝑡𝑛−2)) 2 2
 𝜕𝑥 𝜕𝑥 𝜕𝑥 𝜕𝑥  
 𝜕 𝜕2  
 +7(𝑉𝛼𝐹 𝑐(𝑥, 𝑡𝑛−1) + 𝐷
𝛼
𝐹 𝑐(𝑥, 𝑡 ))  
{ 𝜕𝑥 𝜕𝑥
2 𝑛−1
}
Equation (3-127) is the numerical approximation of the fractal advection-dispersion equation, where 
the numerical integration of the fractal integral is performed using Boole’s Rule. 
The Simpson’s 3/8 Rule and Boole’s Rule of numerical integration are both methods from the Newton-
Cotes Quadrature Rules which are founded on assessing the integral at equally spaced points. The 
traditionally used Trapezoid Rule uses two points of calculation within each larger step of the 
numerical integration, Simpson’s 3/8 Rule uses four calculation points, and Boole’s Rule uses five 
calculation points. The accuracy of numerical integration is related to the size of the steps used, where 
the smaller the step the more accurate the solution. Thus, Simpson’s 3/8 Rule and Boole’s Rule will 
have a higher degree of accuracy because smaller sub-steps are considering within each step.  
66 
 
3.5 Numerical simulation for different fractal dimensions 
The fractal advection-dispersion equation was developed with the fractal derivative in space to 
account for anomalous diffusion associated with groundwater transport in fractured systems. To 
corroborate this theory, the equation is applied to a simple one-dimensional transport problem and 
simulated in the software programme Maple (Figure 2-1) (Appendix B). The velocity within the aquifer 
is constant at 0.05 m/d, the hydrodynamic dispersivity in the x-direction is 0.3 m2/d, and the initial 
contaminant concentration (C0) is 10 mg/l. The initial condition and boundary conditions for the 
defined problem are: 
 
The fractal advection-dispersion equation is applied to the defined problem, and the fractal dimension 
varied to investigate the influence this parameter has on the simulated transport (Figure 3-1, Figure 
3-2, and Figure 3-3). When the fractal dimension is 1 (𝛼 = 1), the equation reduces to the traditional 
advection-dispersion equation and thus forms the basis for comparison. The traditional advection-
dispersion simulates an expected movement along the x-directional line, with the contaminant 
reaching 50 m after 200 days. The concentration spreads out evenly from the source over time as 
anticipated. From this basecase, the fractal dimension is varied from 0.9 to 0.1, in increments of 0.1. 
When the fractal dimension is 0.9 (𝛼 = 0.9), the extent of the transport is increased, where the 
contaminant now reaches further than 50 m after 200 days. This trend is continued with each 
increment reduction in the fractal dimension, where the contaminant reaches 140 m after 200 days 
for the fractal dimension 0.7 (𝛼 = 0.7), and exceeds the 200 m line in 110 days with a fractal dimension 
of 0.6 (Figure 3-1). From the fractal dimension of 0.7 to 0.1, there is an exponential increase in the 
simulated transport along the x-directional line (Figure 3-2). For the fractal dimension 0.1 (𝛼 = 0.1), 
the contaminant reaches the 200 m line reach after just 10 days. The fractal dimensions above 0.6 
(𝛼 = 0.6) change the general orientation from perpendicular to the x-direction to parallel, 
representing the preferential pathway which fractures can provide. A constant groundwater velocity 
was applied to all models, and thus the simulation validates the use of this equation to develop new 
models to better simulate groundwater transport in fractured systems, where the specific 
characterisation of the fractured system is not available.  
To further investigate the influence of the fractal dimension, values greater than 1 are considered 
(Figure 3-3). The fractal dimension has the opposite influence when the fractal order is greater than 
1, where now the transport is impeded, slower than the traditional model, representing subdiffusion.  
67 
 
𝜶 = 𝟏 
10 mg/l 
0 mg/l 
𝜶 = 𝟎. 𝟗 𝜶 = 𝟎. 𝟖 
𝜶 = 𝟎. 𝟕 𝜶 = 𝟎. 𝟔 
 
Figure 3-1 Simulation results for the simple transport problem for variable fractal dimensions (𝟎. 𝟔 ≤ 𝜶 ≤
𝟏), 𝒙 is distance in meters, and 𝒕 is time in days. When 𝜶 = 𝟏, the equation simplifies to the traditional 
advection-dispersion equation and forms the basis of comparison. Changing the fractal dimension increases 
the extent of the transport plume along the one-dimensional line in the x-direction, representing 
superdiffusion. 
68 
 
𝜶 = 𝟎. 𝟓 𝜶 = 𝟎. 𝟒 
𝜶 = 𝟎. 𝟑 𝜶 = 𝟎. 𝟐 
𝜶 = 𝟎. 𝟏 
10 mg/l 
0 mg/l 
 
Figure 3-2 Simulation results for the simple transport problem for variable fractal dimensions (𝟎. 𝟏 ≤ 𝜶 ≤
𝟎. 𝟓). Fractal dimensions below 0.5 result in significant contaminant transport along the line. 
69 
 
𝜶 = 𝟏 
10 mg/l 
0 mg/l 
𝜶 = 𝟏. 𝟏 𝜶 = 𝟏. 𝟑 
𝜶 = 𝟏. 𝟓 𝜶 = 𝟐 
 
Figure 3-3 Simulation results for the simple transport problem for variable fractal dimensions (𝟏 ≤ 𝜶 ≤ 𝟐). 
When 𝜶 = 𝟏, the equation simplifies to the traditional advection-dispersion equation and forms the basis of 
comparison. Fractal dimensions greater than 1, impede transport along the x-directional line, representing 
subdiffusion. 
70 
 
3.6 Investigation of fractal velocity (𝑉𝛼𝐹 ) and fractal dispersivity (𝐷
𝛼
𝐹 )  
The simulation of variable fractal dimensions has validated the use of the model for fractured 
groundwater systems lacking detailed characterisation of heterogeneity. But, questions arise with 
respect to the fractal velocity and fractal dispersivity, in terms of the relationship between velocity, 
dispersivity and the fractal dimension. To understand this relationship, plots of the fractal velocity and 
fractal dispersivity with respect to the fractal dimension are evaluated on the same scale (Figure 3-5 
and Figure 3-4) and on a local scale (Figure 3-6 and Figure 3-7). For a fractal dimension of 1 (𝛼 = 1), 
the fractal velocity equals the defined constant velocity of 0.05 m/d. The introduction of the fractal 
order at 𝛼 = 0.9, slightly changes the set constant velocity to a variable velocity from 0.05 m/d to 0.81 
m/d at 50 m. A similar trend of exponential distributions are seen from 𝛼 = 0.8 to 𝛼 = 0.6, until a 
linear relationship over distance is found at 𝛼 = 0.5. From a fractal order of 0.5, an exponential trend 
is seen again up to 𝛼 = 0.1. 
𝛼 = 1 𝛼 = 0.9 
𝛼 = 0.8 𝛼 = 0.7 
 
Figure 3-4 Fractal velocity and dispersivity over space for varying fractal dimensions (𝟎. 𝟕 ≤ 𝜶 ≤ 𝟏) on the 
same scale. 
71 
 
Fractal velocity/dispersivity Fractal velocity/dispersivity 
𝛼 = 0.6 𝛼 = 0.5 
𝛼 = 0.4 𝛼 = 0.3 
𝛼 = 0.2 𝛼 = 0.1 
  
Figure 3-5 Fractal velocity and dispersivity over space for varying fractal dimensions (𝟎. 𝟏 ≤ 𝜶 ≤ 𝟎. 𝟔) on the 
same scale. 
72 
 
Fractal velocity/dispersivity 
Fractal velocity/dispersivity Fractal velocity/ dispersivity 
𝛼 = 1 𝛼 = 0.9 
𝛼 = 0.8 𝛼 = 0.7 
𝛼 = 0.6 𝛼 = 0.5 
 
Figure 3-6 Fractal velocity and dispersivity over space for varying fractal dimensions (𝟎. 𝟓 ≤ 𝜶 ≤ 𝟏) on a 
local scale for each plot. 
73 
 
Fractal velocity/dispersivity 
Fractal velocity/dispersivity Fractal velocity/dispersivity 
𝛼 = 0.4 
𝛼 = 0.3 
𝛼 = 0.2 𝛼 = 0.1 
 
Figure 3-7 Fractal velocity and dispersivity over space for varying fractal dimensions (𝟎. 𝟏 ≤ 𝜶 ≤ 𝟎. 𝟒) on an 
individual scale for each plot. 
From the plots of fractal velocity and fractal dispersivity, which show the same distributions over the 
fractal dimensions, it can be seen that there is an exponential increase in the fractal velocity and 
dispersivity as the fractal dimension decreases. However, from a fractal dimension of 0.5 to 0.1, the 
increase becomes extreme from a groundwater perspective, where typically groundwater moves 
slowly. From a practical interpretation, the use of a fractal dimensions below 𝛼 = 0.5 should be used 
with caution because of this exponential increase in velocity and hydrodynamic dispersivity.  
 
 
 
74 
 
Fractal velocity/dispersivity Fractal velocity/dispersivity 
3.7 Chapter summary 
It is discussed that the use of the classical advection-dispersion equation tends to inaccurately 
simulate observed contaminant transport because the information required to characterise a 
fractured system, to an appropriate level of detail, is often not available. This is due to the fact that 
heterogeneity is not explicitly incorporated into the classical advection-dispersion equation, but 
rather a constant velocity and dispersivity is considered at each point and heterogeneity is 
incorporated externally. In response to the current limitations of transport modelling using the 
advection-dispersion equation, especially in fractured media, a fractal advection-dispersion 
groundwater transport equation was developed and four methods to solve the fractal advection-
dispersion equation are described, namely forward finite differences and Crank-Nicolson finite 
differences for the fractal derivative formulation, and the Simpson 3/8 and Boole’s numerical 
integration for the fractal integral formulation.  
The developed fractal advection-dispersion equation is numerically simulated for a generic 
groundwater transport model to investigate the effects of varying the fractal dimension in a one-
dimensional groundwater flow system. The simulations provided evidence that the fractal formulation 
of the advection-dispersion equation could possibly model superdiffusion for fractal dimension less 
than 1, as well as subdiffusion for fractal dimensions greater than 1, without explicitly defining 
fractures or preferential pathways. Being able to model the effect of fractures or faults without 
explicitly including the location and specific details in the model has the potential to account for 
anomalous transport, especially where limited information is available on the preferential pathway 
causing the discrepancy.  
The relationship between the fractal velocity and dispersivity with the fractal dimension is evaluated, 
and the use of fractal dimensions above 0.5 are recommended for practical use, due to the supersonic 
increase in velocity and dispersivity for fractal dimensions 0.1 ≤ 𝛼 ≤ 0.5. 
 
 
  
75 
 
 
4  FRACTIONAL DERIVATIVES:  
 SINGULAR AND NON-SINGULAR 
 
A brief introduction to fractional calculus and the various fractional derivative definitions have been 
discussed in Section 1.2.2. To continue this discussion, the functions required for the definition of 
fractional integrals and derivatives are presented, followed by the fundamental fractional definitions, 
including Riemann-Liouville (RL), Caputo (C), Atangana-Baleanu in Caputo sense (ABC), and Atangana-
Baleanu in Riemann-Liouville (ABR). The kernel associated with each fractional definition is 
investigated from the perspective of a convolution of two functions.   
Derivatives and integrals are traditionally introduced as the mathematical representation of slopes 
and areas, yet fractional derivatives are mysterious, as they have no obvious geometric interpretation. 
Oldham and Spanier (1974) encourage that if one is able to move beyond with this pictorial 
representation of a classical derivative, fractional-order derivatives and integrals can become equally 
tangible as a new dimension in mathematics is revealed (Oldham and Spanier, 1974; Loverro, 2004).   
4.1 Special functions in fractional definitions 
The Gamma and Mittag-Leffler functions are well-established extensions of the factorial and 
exponential function, which service to explain the definitions and use of the fractional integral and 
derivative (Loverro, 2004; Herrmann, 2011).  
76 
 
4.1.1 Gamma function 
Euler’s Gamma function can be generalised as the factorial for all real numbers, and is intrinsically tied 
to fractional calculus. The definition of the gamma function is (Loverro, 2004; Petráš, 2010; Herrmann, 
2011): 
 ∞
Γ(𝑧) = ∫ 𝑢𝑧−1𝑒−𝑡 𝑑𝜏 (4-1) 
0
Applying integration by parts, 
 Γ(1 + 𝑧) = 𝑧 Γ(𝑧) (4-2) 
Applying direct integration Γ(1) = 1, 
 Γ(1 + 𝑛) = 𝑛! (4-3) 
or 
 Γ(𝑛) = (𝑛 − 1)! (4-4) 
4.1.2 Power Law function 
The power law can simply be stated as 
 𝑓(𝑥) = 𝑥𝑎 (4-5) 
where, 𝑎 is a constant known as the exponent of the power law.  
From a statistical perspective, the power law is a relationship between two quantities, where a change 
in one variable results in a proportional change in the other controlled by the exponential power.  
Power-law distributions are commonly found in nature and patterns, ranging from city population 
trends, magnitude of earthquakes, foraging patterns of various species, to describing the size of 
craters on the moon (Gutenberg and Richter, 1944; Neukum and Ivanov, 1994; Newman, 2005; Frank, 
2009; Humphries et al., 2010).  
4.1.3 Exponential function 
The exponential function can simply be stated as 
 𝑓(𝑥) = 𝑒𝑥 (4-6) 
where, 𝑒 is a constant, and in this case Euler’s constant of 2.71828. 
The exponential function describes the relationship where a quantity grows or decays at a rate 
proportional to its current value. The exponential pattern is a more common than the power law and 
is widely recognised in nature, from survival times for decaying atomic nuclei, the Boltzmann 
distribution of energies in statistical mechanics, to the frequency of Korean family names (Kim and 
Park, 2005; Newman, 2005; Frank, 2009).  
77 
 
The stretched exponential function has been proposed as a complement to the power law 
distributions, where Laherrere and Sornette (1998) showed stretched exponential functions to 
describe natural phenomena such as radio and light emissions from galaxies, oilfield reserve sizes, and 
Vostok temperature variations. Lee et al. (2001) demonstrated that the stretched exponential function 
could represent the observed autofluorescence decay profiles in biological tissue, producing high-
quality contrast and spatial maps.  
4.1.4 Mittag-Leffler function 
The Mittag-Leffler function plays an important role in the solution of non-integer order differential 
equations, and can be analogously compared to the exponential function, used in the solution of 
integer-order differential equations. The exponential function (𝑒𝑧) can be defined as (Loverro, 2004; 
Petráš, 2010; Herrmann, 2011): 
 ∞ 𝑧𝑛
𝑒𝑧 = ∑  (4-7) 
𝑛!
𝑛=0
Considering the defined gamma function (Γ) (Equation (4-3), the factorial 𝑛! can be replaced 
by Γ(1 + 𝑛): 
 ∞ 𝑧𝑛
𝑒𝑧 = ∑  (4-8) 
Γ(1 + 𝑛)
𝑛=0
The Mittag-Leffler function 𝐸𝛼(𝑧) can be defined as: 
 ∞ 𝑧𝑛
𝐸𝛼(𝑧) = ∑  (4-9) Γ(1 + 𝛼𝑛)
𝑛=0
where, 𝛼 denotes an arbitrary real number 𝛼 > 0. 
The generalised Mittag-Leffler function is a most natural generalisation of the exponential function, 
and can be expressed as (Shukla and Prajapati, 2007; Mathai and Haubold, 2008), 
 ∞ 𝑧𝑛
𝐸𝛼,𝛽(𝑧) = ∑  (4-10) Γ(𝛼𝑛 + 𝛽)
𝑛=0
The Mittag-Leffler function reduces to the exponential function when 𝛼 =  1, because the function is 
a direct generalisation of the exponential function. However, the Mittag-Leffler function interpolates 
1
between the pure exponential and a hypergeometric function ( ) for values of 0 < 𝛼 <  1 (Shukla 
1−𝑧
and Prajapati, 2007). 
78 
 
The versatility of the Mittag-Leffler function has been fully realised in the last two decades with 
applications over a wide range, from stress-strength analysis, growth-decay mechanisms, to 
production of melatonin in the body (Shukla and Prajapati, 2007; Sebastian and Gorenflo, 2016).  
4.2 Riemann-Liouville fractional definition 
The functions required for fractional derivatives have been discussed, and the various definitions of 
the fractional derivative and integral can be explored, starting with the Riemann-Liouville fractional 
definition. 
An integral can be interpreted as a generalisation of area in integer-order mathematical terms, and is 
referred to as the antiderivative or primitive. The integer-order integral is analysed and extended to 
the fractional operator. Let the integral operator be notionally represented by (Herrmann, 2011): 
 𝑥
  𝑎𝐼(𝑓(𝑥)) = ∫ 𝑓(𝜏) 𝑑𝜏 (4-11) 
𝑎
and, the multiple integral can be defined as: 
 𝑥𝑛 𝑥𝑛−1 𝑥1
 𝑛𝑎𝐼 (𝑓(𝑥)) = ∫  ∫ …∫ 𝑓(𝑥0) 𝑑𝑥0…𝑑𝑥𝑛−1 (4-12) 
𝑎 𝑎 𝑎
The multiple integral can be generalised for the 𝑛𝑡ℎ integration of the function 𝑓(𝑥) using Cauchy’s 
formula of repeated integration, on condition that 𝑓(𝑎) = 0: 
 1 𝑥
 𝐼𝑛𝑎 𝑓(𝑥) = ∫ (𝑥 − 𝜏)
𝑛−1 𝑓(𝜏) 𝑑𝜏 (4-13) 
(𝑛 + 1)! 𝑎
The Cauchy defined integral can be extended to the fractional case by replacing 𝑛 with 𝛼: 
 1 𝑥
 𝐼𝛼𝑓(𝑥) = ∫ (𝑥 − 𝜏)𝛼−1 𝑓(𝜏) 𝑑𝜏 (4-14) 𝑎 + Γ(𝛼) 𝑎
 1 𝑏
 𝑏𝐼
𝛼
−𝑓(𝑥) = ∫ (𝜏 − 𝑥)
𝛼−1 𝑓(𝜏) 𝑑𝜏 (4-15) 
Γ(𝛼) 𝑥
where, 𝑎 and 𝑏 denote the lower and upper boundary of the integral domain and may be arbitrarily 
chosen. Equation (4-14) is valid where 𝑥 > 𝑎, and Equation (4-15) is valid where 𝑥 < 𝑏. The choice of 
these two constants (𝑎, 𝑏) determines the value of the integral, and leads to a number of definitions 
for the fractional integral. The two most commonly applied definitions of the fractional integral 
include the Liouville and Riemann fractional integral definitions (Debnath, 2004; Loverro, 2004; 
Herrmann, 2011).  
The Liouville fractional integral is defined for 𝑎 = −∞ and 𝑏 = +∞ (Herrmann, 2011): 
79 
 
 1 𝑥
 𝐼𝛼𝑓(𝑥) = ∫ (𝑥 − 𝜏)𝛼−1𝐿 +  𝑓(𝜏) 𝑑𝜏 (4-16) Γ(𝛼) −∞
 1 +∞
 𝐿𝐼
𝛼 𝛼−1
−𝑓(𝑥) = ∫ (𝜏 − 𝑥)  𝑓(𝜏) 𝑑𝜏 Γ(𝛼) (4-17) 𝑥
The Riemann fractional integral is defined for 𝑎 = 0 and 𝑏 = 0 (Herrmann, 2011): 
 1 𝑥
 𝛼𝑅𝐼+𝑓(𝑥) = ∫ (𝑥 − 𝜏)
𝛼−1 𝑓(𝜏) 𝑑𝜏 (4-18) 
Γ(𝛼) 0
  1 0
 𝐼𝛼𝑅 −𝑓(𝑥) = ∫ (𝜏 − 𝑥)
𝛼−1 𝑓(𝜏) 𝑑𝜏 (4-19) 
Γ(𝛼) 𝑥
Herrmann (2011) considered the special function 𝑓(𝑥) = 𝑒𝑘𝑥 to compare the Liouville and Riemann 
fractional integral definitions. When applying the Liouville fractional integral, the function disappears 
for values of 𝑘 > 0 at the lower boundary of the integral domain, with no further contribution. 
Conversely, when applying the Riemann fractional integral, the function does not disappear 
because 𝑎 = 0, and a further contribution is maintained.  
The Liouville and Riemann fractional integral definitions are combined to form the general Riemann-
Liouville definition of fractional integral (Debnath, 2004; Atangana, 2016): 
 1 𝑥
 −𝛼𝑎𝐷𝑥 𝑓(𝑥) = 
𝛼
𝑎𝐼𝑥 𝑓(𝑥) = ∫ (𝑥 − 𝜏)
𝛼−1 𝑓(𝜏) 𝑑𝜏 (4-20) 
Γ(𝛼) 𝑎
where, 
             𝐷−𝛼𝑎  = 
𝛼
𝑎𝐼  is the Riemann-Liouville integral operator, 
 𝑎 = 0, the Riemann definition of the fractional integral is valid, 
 𝑎 = −∞, the Liouville definition of the fractional integral is valid 
The fractional integral and fractional derivative are inverse operations, and similar to how integer-
order derivatives are related to the integer-order integral, so do fractional derivatives share a 
relationship to fractional integrals. The derivative can be interpreted as the slope of a curve 
(geometric) or as a rate of change (physical) in integer-order mathematical terms. The integer-order 
derivative is analysed along with the fractional integral to extend the definition to the fractional 
derivative (Gootman, 1997; Loverro, 2004; Herrmann, 2011).  
Let the derivative to an arbitrary order (𝛼) order be denotes as follows: 
 𝑑𝑛
= 𝐷𝛼 (0-21) 
𝑑𝑥𝑛
80 
 
The fractional derivative can be divided into appropriate integer-order derivative and the remaining 
fractional order derivative (Herrmann, 2011): 
                             𝐷𝛼 = 𝐷𝑚𝐷𝛼−𝑚                          𝑚 ∈ ℕ 
𝑚 (4-22) 𝑑
=  𝐼𝑚−𝛼 
𝑑𝑥𝑚 𝑎
where, 
𝑑𝛼
 𝐷𝛼 = 𝐼−𝛼 =  
𝑑𝑥𝛼
Equation (4-22) states the fractional derivative can be represented or defined as a fractional integral 
followed by an integer-order derivative, where once a fractional integral is defined the fractional 
derivative is defined as well. This explains why the fractional derivative is referred to as the 
differintegral. The inverted sequence of operators leads to an alternative decomposition of the 
fractional derivative into an integer-order derivative followed by a fractional integral (Herrmann, 
2011): 
                            𝐷𝛼 = 𝐷𝛼−𝑚𝐷𝑚                          𝑚 ∈ ℕ 
𝑑𝑚 (4-23) 
=  𝑎𝐼
𝑚−𝛼  
𝑑𝑥𝑚
Equation (4-22) and Equation (4-23) serve to highlight the non-locality of fractional calculus, where 
the integer-order derivative is a local operator, but the fractional integral is a non-local operator. The 
fractional derivative is thus a non-local operator, because it is the inverse of the fractional integral, 
which is a non-local operator (Herrmann, 2011).  
The Liouville definition of the fractional derivative is obtained by applying the Liouville definition of 
the fractional integral (Equation (4-16) and Equation (4-17)) and the operator sequence in Equation 
(4-22) for the simple case (0 < 𝛼 < 1) (Herrmann, 2011): 
 𝑑
                         𝛼𝐿𝐷+𝑓(𝑥) =  𝐼
1−𝛼
𝐿 + 𝑓(𝑥) 𝑑𝑥
𝑥 (4-24) 𝑑 1
                           = ∫ (𝑥 − 𝜏)−𝛼  𝑓(𝜏) 𝑑𝜏 
𝑑𝑥 Γ(1 − 𝛼) −∞
and,   
 𝑑
                        𝛼𝐿𝐷−𝑓(𝑥) =  𝐼
1−𝛼
𝐿 − 𝑓(𝑥) 𝑑𝑥
+∞ (4-25) 𝑑 1
                           = ∫ (𝜏 − 𝑥)−𝛼  𝑓(𝜏) 𝑑𝜏 
𝑑𝑥 Γ(1 − 𝛼) 𝑥
The Riemann definition of the fractional derivative is obtained by applying the Riemann definition of 
the fractional integral (Equation (4-18) and Equation (4-19)) and the operator sequence in Equation 
(4-22) (Herrmann, 2011): 
81 
 
 𝑑
                                𝐷𝛼𝑅 +𝑓(𝑥) =  𝐼
1−𝛼𝑓(𝑥) 
𝑑𝑥 𝑅 + (4-26) 
𝑑 1 𝑥
                                         = ∫ (𝑥 − 𝜏)−𝛼 𝑓(𝜏) 𝑑𝜏 
𝑑𝑥 Γ(1 − 𝛼) 0
 and, 
 𝑑
                               𝛼𝑅𝐷−𝑓(𝑥) =  𝐼
1−𝛼𝑓(𝑥) 
𝑑𝑥 𝑅 −
(4-27) 
𝑑 1 0
                                      = ∫ (𝜏 − 𝑥)−𝛼  𝑓(𝜏) 𝑑𝜏 
𝑑𝑥 Γ(1 − 𝛼) 𝑥
The Liouville and Riemann fractional derivative definitions are combined to form the general Riemann-
Liouville definition of fractional derivative or differintegral for (𝑛 − 1 < 𝛼 < 𝑛) (Oldham and Spanier, 
1974; Kisela, 2008; Petráš, 2010; Atangana, 2016): 
 1 𝑑𝑛 𝑥
 𝑅𝐿𝐷𝛼𝑓(𝑥) =  ∫ (𝑥 − 𝜏)−𝛼−𝑛+1𝑎 𝑥  𝑓(𝜏) 𝑑𝜏 
(4-28) 
Γ(𝑛 − 𝛼) 𝑑𝑥𝑛 𝛼
where, 
             𝑅𝐿 𝛼 𝐷   is the Riemann-Liouville differintegral operator, 
 𝑎 = 0, the Riemann definition of the fractional integral is valid, 
 𝑎 = −∞, the Liouville definition of the fractional integral is valid. 
4.3 Caputo fractional definition 
The Caputo definition of the fractional derivative was developed in response to limitations of the 
Riemann-Liouville fractional derivative, in terms of an unusual initial condition, 𝑓𝛼(0). The Caputo 
fractional derivative is based on the inverted sequence of operators in Equation (4-23), where an 
alternative decomposition of the fractional derivative into a fractional integral is followed by an 
integer-order derivative, (𝑛 − 1 < 𝛼 < 𝑛) (Kisela, 2008; Petráš, 2010; Atangana, 2016): 
 𝑑𝑛
                 𝐶𝐷𝛼𝑎 𝑥 𝑓(𝑥) =  𝐼
𝑛−𝛼
𝑎 [ 𝑓(𝑥)𝑛 ] 𝑑𝑥 (4-29) 
1 𝑥 𝑑𝑛
                    =  ∫ 𝑓(𝜏)(𝑥 − 𝜏)−𝛼−𝑛+1  𝑑𝜏 
Γ(𝑛 − 𝛼) 𝑑𝜏𝑛𝛼
where, 
             𝐶 𝛼 𝐷   is the Caputo differintegral operator. 
The Caputo definition of the fractional derivative is different from the classical Riemann-Liouville 
definition, because it becomes unnecessary to define the fractional order initial condition with the 
Caputo definition. The Caputo definition of the fractional derivative can be applied either with the 
Liouville or Riemann definition of the fractional integral, where when 𝛼 = 0 the Riemann definition 
of the fractional integral is valid (Riemann-Caputo definition), and when 𝑎 = −∞ the Liouville 
82 
 
definition of the fractional integral is valid (Liouville-Caputo definition) (Petráš, 2010; Herrmann, 2011; 
Atangana, 2016).  
The standard Caputo definition fractional derivative for a system (0 < 𝛼 < 1) containing a singular 
power-kernel can be expressed as (Sun et al., 2017a): 
 1 𝑥 𝑑
 𝐶𝐷𝛼𝑓(𝑥) =  ∫  𝑓(𝜏)(𝑥 − 𝜏)−𝛼𝑎 𝑥  𝑑𝜏 (4-30) Γ(1 − 𝛼) 0 𝑑𝜏
The Caputo type fractional derivative is considered a successful tool to characterise anomalous 
dynamics, and mathematically this is produced from the standard power-law memory kernel. 
However, the singularity of the power-law kernel is a source of difficulties with numerical computation 
and applications of fractional partial differential equations. To overcome these difficulties, a number 
of authors have proposed new definitions of the fractional derivative, which replaces the singular 
power-law kernel. Caputo and Fabrizio (2015) define a fractional derivative where the singular power-
law kernel 𝑥−𝛼 is replaced with a non-singular exponential function (Caputo and Fabrizio, 2015; Sun 
et al., 2017a): 
 𝑀(𝛼) 𝑥 𝑑 𝛼(𝑥 − 𝜏) 
 𝐶𝐹1𝑎 𝐷
𝛼
𝑥 𝑓(𝑥) =  ∫  𝑓(𝜏) 𝑒𝑥𝑝 [− ]𝑑𝜏 (4-31) (1 − 𝛼) 𝛼 𝑑𝜏 1 − 𝛼 
where, 
  𝐶𝐹1 𝐷
 
 denotes the Caputo and Fabrizio type derivative, and 
 𝑀(𝛼) is a normalisation function where 𝑀(0) = 𝑀(1) = 1. 
Furthermore, Caputo and Fabrizio (2016) define a fractional derivative where the singular power-law 
kernel 𝑥−𝛼 is replaced with a Gaussian-function and Laplace operators (𝑛 = 1) (Caputo and Fabrizio, 
2016; Sun et al., 2017a): 
 1 + 𝛼2 𝑥 𝑑 𝛼(𝑥 − 𝜏)2 
 𝐶𝐹2 𝛼𝑎 𝐷𝑥 𝑓(𝑥) =  ∫  𝑓(𝜏) 𝑒𝑥𝑝 [− ]𝑑𝜏 (4-32) 
√𝜋2(1 − 𝛼) 𝛼 𝑑𝜏 1 − 𝛼 
An investigation on the application of the new Caputo-Fabrizio type derivatives was conducted by Sun 
et al. (2017), and found that these definitions of the fractional derivative have limitations for 
characterising complex anomalous relaxation and diffusion processes. Sun et al. (2017a) then propose 
an alternative notion with a stretched exponential-function kernel (0 < 𝛼 < 1); 𝑛 = 1: 
 𝑀(𝛼) 𝑥 𝑑 𝛼(𝑥 − 𝜏)𝛼  
 𝑆𝐸𝐷𝛼𝑎 𝑥 𝑓(𝑥) =  ∫  𝑓(𝜏) 𝑒𝑥𝑝 [− ]𝑑𝜏 
(4-33) 
(1 − 𝛼)1/𝛼 𝛼 𝑑𝜏 1 − 𝛼 
where, 
  𝑆𝐸𝐷   denotes the stretched exponential type Caputo definition fractional derivative operator. 
83 
 
4.4 Atangana-Baleanu fractional definitions 
An alternative approach to overcome the problems associated with the singular power-law kernel was 
performed by Atangana and Baleanu (2016). The Atangana and Baleanu definition of the fractional 
derivative replaces the singular power-law kernel in a similar manner to Caputo and Fabrizio (2015), 
but rather considers the Mittag-Leffler function (Equation (4-9)) instead of the exponential function. 
Atangana and Baleanu (2016) define two versions of the Atangana and Baleanu definition of the 
fractional derivative, namely the Atangana-Baleanu in Caputo sense (ABC) and the Atangana-Baleanu 
fractional derivative in Riemann-Liouville sense (ABR).  
Atangana and Baleanu (2016) state the Atangana-Baleanu fractional derivative definition will be 
helpful to discuss real world problems, and will have a great advantage when using the Laplace 
transform to solve physical problems with initial conditions. A number of applications of the Atangana-
Baleanu fractional derivative definition have proven its usefulness and its ability to characterise 
complex anomalous processes, ranging from applications in fundamental physics to geohydrology 
(Atangana, 2016; Atangana and Baleanu, 2016; Aldhaifallah et al., 2016; Abdeljawad and Baleanu, 
2016; Atangana and Koca, 2016; Zafar and Fetecau, 2016; Goufo et al., 2016; Omez-Aguilar and 
Atangana, 2017).  
The Atangana-Baleanu fractional derivative in Riemann-Liouville sense (ABR) is the derivative of a 
convolution of a given function not necessarily differentiable with the generalized Mittag-Leffler 
function (Atangana and Baleanu, 2016; Abdeljawad and Baleanu, 2016) (𝑛 = 1): 
 𝐴𝐵(𝛼) 𝑑 𝑡 𝛼 
𝐴𝐵𝑅 𝛼 ( ) ( )𝛼 (4-34)  𝑎 𝐷𝑡 𝑓 𝑡 =  ∫ 𝑓(𝜏) 𝐸𝛼 [− 𝑡 − 𝜏 ] 𝑑𝜏 (1 − 𝛼) 𝑑𝑡 𝛼 1 − 𝛼 
where, 
 𝐴𝐵𝑅𝐷   denotes the Atangana-Baleanu in Riemann sense fractional derivative operator, 
𝑧𝛼𝑛
𝐸𝛼  denotes the Mittag-Leffler function (𝐸𝛼(𝑧
𝛼) = ∑∞𝑛=0 ), and Γ(𝛼𝑛+1)
𝐴𝐵(𝛼) is a normalization function, where 
 𝛼       𝐴𝐵(𝛼) = 1 − 𝛼 +  
Γ(𝛼) (4-35) 
𝐴𝐵(0)  =  𝐴𝐵(1)  =  1 
The Atangana-Baleanu in Caputo sense (ABC) is the convolution of a local derivative of a given function 
with the generalized Mittag-Leffler function, and is defined as (Atangana and Baleanu, 2016; 
Aldhaifallah et al., 2016; Abdeljawad and Baleanu, 2016; Sun et al., 2017a): 
 𝐴𝐵(𝛼) 𝑡 𝑑 𝛼 (4-36) 
 𝐴𝐵𝐶 𝛼 𝛼𝑎 𝐷𝑥 𝑓(𝑡) =  ∫ 𝑓(𝜏) 𝐸𝛼 [− (𝑡 − 𝜏) ] 𝑑𝜏 (1 − 𝛼) 𝛼 𝑑𝜏 1 − 𝛼 
where, 
 𝐴𝐵𝐶   𝐷 denotes the Atangana-Baleanu in Caputo sense fractional derivative operator, 
84 
 
4.4.1 Properties 
Similarly to the local derivative, the properties of the non-local derivative are investigated by analysing 
the analytical (or exact) solution. Specifically, two fundamental integral transforms are considered, 
namely the Laplace transform and the Fourier transform.  
Laplace transform 
The Laplace transform of the arbitrary order α for the Riemann-Liouville and Caputo fractional 
derivatives, under the zero initial conditions is (Petráš, 2010): 
 ℒ{  𝐷𝛼𝑓(𝑡)}(𝓈) = 𝓈𝛼0 𝑡 𝐹(𝓈) (4-37) 
The Laplace transform of the fractional derivative is similar to the Laplace transform of an integer-
order derivative, except that the Laplace space variable (𝓈) is to the fractional order (𝛼). Considering 
the Atangana-Baleanu in Caputo sense (ABC) fractional derivative definition (Equation (4-36), let  
 𝑑
𝑣(𝑡) = 𝑓(𝑡) (4-38) 
𝑑𝜏
and 
 𝛼 𝑢(𝑡) = 𝐸 𝛼𝛼 [− 𝑡 ] (4-39) 1 − 𝛼 
Applying the convolution theorem (𝑐𝑜𝑛) for the functions 𝑣(𝑡) and 𝑢(𝑡), the following is obtained 
 𝐴𝐵(𝛼) 𝑑 𝛼 
 𝐴𝐵𝐶𝑎 𝐷
𝛼
𝑥 𝑓(𝑡) = [ 𝑓(𝑡) ∗ 𝐸 [− 𝑡
𝛼]] (4-40) 
1 − 𝛼 𝑑𝜏 𝛼 1 − 𝛼 
Considering, 
 ℒ(𝑣 ∗ 𝑢) = ℒ(𝑢) ∙ ℒ(𝑣) (4-41) 
Then,  
 𝐴𝐵(𝛼) 𝑑 𝛼 
ℒ{ 𝐴𝐵𝐶 𝛼𝑎 𝐷𝑥 𝑓(𝑡)}(𝓈) = ∙ ℒ { 𝑓(𝑡)} ∗ ℒ {𝐸 [− 𝑡
𝛼]}  (4-42) 
1 − 𝛼 𝑑𝜏 𝛼 1 − 𝛼 
Applying the standard Laplace transforms (Atangana and Koca, 2016), 
 𝐴𝐵(𝛼) 𝓈𝛼ℒ{𝑓(𝑡)}(𝓈) − 𝓈𝛼−1𝑓(0)
ℒ{ 𝐴𝐵𝐶𝐷𝛼𝑓(𝑡)}(𝓈) = ∙    (4-43) 𝑎 𝑥 1 − 𝛼 𝛼 𝛼𝓈 +
1 − 𝛼
where 
 ℒ{ 𝐴𝐵𝐶𝐷𝛼𝑎 𝑡 𝑓(𝑡) } represents the Laplace transform of function 𝑓 with respect to time 𝑡, 
𝓈 represents the transform domain or complex number frequency parameter (𝓈 = 𝜎 + 𝑖𝜐). 
Equation (4-43) is the Laplace transform of the ABC fractional derivative. From this analysis, it can 
been seen that the ABC fractional derivative produces a normal initial condition (𝑓(0)), similar to the 
Caputo fractional derivative.  
85 
 
The Laplace transform of the Atangana-Baleanu fractional derivative in the Riemann-Liouville sense 
((4-34) can be defined as (Atangana and Koca, 2016): 
𝐴𝐵(𝛼) 𝑑 𝑡 𝛼(𝑡 − 𝜏)𝛼  
ℒ{ 𝐴𝐵𝑅 𝛼𝑎 𝐷𝑡 𝑓(𝑡)}(𝓈) = ℒ {  ∫ 𝑓(𝜏) 𝐸𝛼 [− ] 𝑑𝜏} (1 − 𝛼) 𝑑𝑡 0 1 − 𝛼 
(4-44) 
𝐴𝐵(𝛼) 𝓈𝛼ℒ{𝑓(𝑡)}(𝓈)
=  
(1 − 𝛼) 𝛼𝓈𝛼 +
1 − 𝛼
where, 
 ℒ{ 𝐴𝐵𝑅 𝛼𝑎 𝐷𝑡 𝑓(𝑡) } represents the Laplace transform of function 𝑓 with respect to time 𝑡, 
𝓈 represents the transform domain or complex number frequency parameter (𝓈 = 𝜎 + 𝑖𝜐). 
Fourier transform 
The Fourier transform of the arbitrary order α for the Riemann-Liouville and Caputo fractional 
derivatives, under the zero initial conditions is (Kisela, 2008): 
 ℱ{  0𝐷
𝛼
𝑡 𝑓(𝑡)}(𝜐) = (𝑖𝜐)
𝛼𝐹(𝜐) (4-45) 
The Fourier transform of the fractional derivative is similar to the Fourier transform of an integer-
order derivative, except that the Fourier transform space variable (𝑖𝜐) is now to the fractional order 
(𝛼). The Fourier transform of the Atangana-Baleanu fractional derivative in Caputo sense (Equation 
(4-36)) has been defined as (Atangana and Koca, 2016): 
𝐴𝐵(𝛼) 𝑡 𝑑 𝛼(𝑡 − 𝜏)𝛼  
ℱ{ 𝐴𝐵𝐶𝑎 𝐷
𝛼
𝑥 𝑓(𝑡)}(𝜐) = ℱ {  ∫ 𝑓(𝜏) 𝐸𝛼 [− ] 𝑑𝜏} (𝜐) (1 − 𝛼) 0 𝑑𝜏 1 − 𝛼 
(4-46) 
𝐴𝐵(𝛼) 𝑖(−1 + 𝑒2𝑖𝛼)(−1 + 𝛼)𝛼|𝜐|𝛼−1(1 + 𝑠𝑔𝑛[𝜐])
           = (𝑗𝜐)ℱ{𝑓(𝑡)} ×
 ( 3𝑖𝛼 𝑖𝛼 𝑖𝛼 ) (1 − 𝛼)
2√2(𝑒 2 𝛼 − |𝜐|𝛼 + 𝛼|𝜐|𝛼 (𝛼 − 𝑒 2 |𝜐|𝛼 + 𝑒 2 |𝜐|𝛼))
where, 
 ℱ{ 𝐴𝐵𝐶𝐷𝛼𝑎 𝑡 𝑓(𝑡) } represents the Fourier transform of function 𝑓 with respect to time 𝑡, 
𝜐 represents the transform domain or complex number frequency parameter. 
The Fourier transform of the Atangana-Baleanu fractional derivative in Riemann sense (Equation (4-
34)) has been defined as (Atangana and Koca, 2016): 
𝐴𝐵(𝛼) 𝑑 𝑥 𝛼(𝑥 − 𝑢)𝛼  
ℱ{ 𝐴𝐵𝑅𝑎 𝐷
𝛼
𝑡 𝑓(𝑡)}(𝜐) = ℱ {  ∫ 𝑓(𝑥) 𝐸𝛼 [− ]𝑑𝑢} (𝜐) (1 − 𝛼) 𝑑𝑥 0 1 − 𝛼 
(4-47) 
𝐴𝐵(𝛼) 𝑖(−1 + 𝑒2𝑖𝛼(−1 + 𝛼)𝛼|𝜐|𝛼−1(1 + 𝑠𝑔𝑛[𝜐])
= ℱ{𝑓(𝑡)} × (− )  
(1 − 𝛼) 3𝑖𝛼 𝑖𝛼 𝑖𝛼
2√2(𝑒 2 𝛼 − |𝜐|𝛼 + 𝛼|𝜐|𝛼)(𝛼 − 𝑒 2 |𝜐|𝛼 + 𝑒 2 |𝜐|𝛼)
where, 
 ℱ{ 𝐴𝐵𝑅𝑎 𝐷
𝛼
𝑡 𝑓(𝑡) } represents the Fourier transform of function 𝑓 with respect to time 𝑡 
86 
 
In summary, the Laplace and Fourier transforms for the fractional derivatives are similar to those of 
the integer-order derivative, with the exception that the transform space is considered to the 
fractional order of the fractional derivative. The Laplace and Fourier transforms of the Atangana-
Baleanu fractional derivative in the Caputo and Riemann-Liouville senses are presented from literature 
to illustrate the fractional derivatives can be analytically solved by means of these integral transforms, 
although they become mathematically complex.  
4.4.2 Boundary analysis of fractional order 𝛼 
The fractional order is typically defined as 0 < 𝛼 < 1, yet the extreme cases of 𝛼 = 0 and 𝛼 = 1 are 
investigated for the Atangana-Baleanu fractional derivative. The Atangana-Baleanu fractional 
derivative in the Riemann-Liouville sense when 𝛼 = 0 becomes,  
 𝐴𝐵(0) 𝑑 𝑡 0 
 𝐴𝐵𝑅0 𝐷
0
𝑡 𝑓(𝑡) =  ∫ 𝑓(𝜏) 𝐸 − (𝑡 − 𝜏)
0 𝑑𝜏 (4-48) 
(1 − 0) 
[ ]
𝑑𝑡 𝛼0 1 − 0 
Simplifying, 
 𝑑 𝑡
 𝐴𝐵𝑅𝐷00 𝑡 𝑓(𝑡) =  ∫ 𝑓(𝜏) 𝑑𝜏 𝑑𝑡 0 (4-49) 
      =  𝑓(𝑡) 
Thus, when 𝛼 = 0, the Atangana-Baleanu fractional derivative in the Riemann-Liouville sense 
simplifies to the function 𝑓(𝑡).  
Before the scenario is considered where 𝛼 = 1, it is stated that as per the definition of the delta Dirac 
function (𝛿) (Caputo and Fabrizio, 2015): 
 1 1 
lim 𝑒𝑥𝑝 [− 𝑡] = 𝛿(𝑡) (4-50) 
𝑥→0𝑥  𝑥 
The Atangana-Baleanu fractional derivative in the Riemann-Liouville sense when 𝛼 → 1,  
 𝐴𝐵(𝛼) 𝑑 𝑡 𝛼 
lim  𝐴𝐵𝑅0 𝐷
𝛼
𝑡 𝑓(𝑡) = lim ∫ 𝑓(𝜏) 𝐸 [− (𝑡 − 𝜏)
𝛼] 𝑑𝜏 (4-51) 
𝛼→1 𝛼→1 (1 − 𝛼) 𝑑𝑡 𝛼0 1 − 𝛼 
It is known that the Mittag-Leffler function reduces to the exponential function when 𝛼 = 1 (Shukla 
and Prajapati, 2007), thus 
 𝑑 𝑡 1 1 
lim  𝐴𝐵𝑅0 𝐷
𝛼
𝑡 𝑓(𝑡) = ∫ 𝑓(𝜏) lim 𝑒𝑥𝑝 [− (𝑡 − 𝜏)] 𝑑𝜏 (4-52) 𝛼→1 𝑑𝑡 0 𝛼→1 (1 − 𝛼) 1 − 𝛼 
Let 
𝜀 = 1 − 𝛼 
 
Substituting in the newly defined variable (𝜀), 
87 
 
 𝑑 𝑡 1 1 
lim  𝐴𝐵𝑅0 𝐷
𝛼
𝑡 𝑓(𝑡) = ∫ 𝑓(𝜏) lim 𝑒𝑥𝑝 − (𝑡 − 𝜏)  𝑑𝜏  [ ] (4-53) 𝛼→1 𝑑𝑡 0 𝜀→0 𝜀 𝜀 
Applying the known property of the Delta Dirac function (𝛿), given in Equation (4-50) 
 𝑑 𝑡
lim  𝐴𝐵𝑅𝐷𝛼0 𝑡 𝑓(𝑡) = ∫ 𝑓(𝜏) 𝛿(𝑡 − 𝜏) 𝑑𝜏 (4-54) 
𝛼→1 𝑑𝑡 0
Now considering the property of the Delta Dirac function (Arfken and Weber, 1999), where 
 
∫𝐹(𝑡)𝛿(𝑡 − 𝑥)𝑑𝑡 = 𝐹(𝑥) (4-55) 
Therefore, 
 𝑑
lim  𝐴𝐵𝑅𝐷𝛼0 𝑡 𝑓(𝑡) = 𝑓(𝑡) (4-56) 
𝛼→1 𝑑𝑡
Equation (4-56) confirms that the ABR fractional derivative is valid for the boundary condition 𝛼 = 1. 
Following a similar procedure the ABC fractional derivative is also found to be valid for the boundary 
conditions.  
4.5 Analysis of kernels associated with fractional derivatives 
A kernel describes the fundamental component of a system. In computer science, the kernel computer 
program forms the core of a computers operating system and controls the entire system. Drawing an 
analogue with computers kernel, the kernel expressed for each fractional derivative in a similar way 
controls how a fractional derivative operates. The significant fractional derivative definitions have 
been given, and to facilitate a discussion on the implication of the different definitions, the kernel 
associated with each definition is analysed. 
To define the kernel for each fractional derivative definition, the convolution theorem is considered, 
which gives the inverse transform of the product of two transforms. Considering two functions 𝑣 
and 𝑢, which are piecewise continuous on 𝑡 ≥ 0, the Laplace transform is (Logan, 2006): 
 ℒ(𝑣 ∗ 𝑢)(𝓈) = 𝑉(𝓈)𝑈(𝓈) (4-57) 
The convolution of 𝑣(𝑡) and 𝑢(𝑡) is: 
 𝑡
(𝑣 ∗ 𝑢)(𝑡) ≡ ∫ 𝑣(𝜏) 𝑢(𝑡 − 𝜏) 𝑑𝜏 (4-58) 
0
and, the inverse Laplace transform is thus: 
 ℒ−1{𝑉(𝓈)𝑈(𝓈)} = (𝑣 ∗ 𝑢)(𝑡) (4-59) 
The Laplace transform is additive, but not multiplicative. This means that the Laplace transform of a 
product is not the product of the Laplace transforms. 
88 
 
The convolution theorem gives the transform required to get a product of Laplace transforms, i.e. the 
convolution (Logan, 2006).  
The Riemann-Liouville fractional derivative can be expressed for a first order derivative as 
 1 𝑑 𝑡
 𝑅𝐿𝐷𝛼𝑓(𝑡) =  ∫ 𝑓(𝜏) (𝑡 − 𝜏)−𝛼 𝑑𝜏 (4-60) 0 𝑡 Γ(1 − 𝛼) 𝑑𝜏 0
Let  
 𝑣(𝑡) = 𝑓(𝑡) (4-61) 
and 
 1
𝑢(𝑡) = 𝑡−𝛼 (4-62) 
Γ(1 − 𝛼)
Applying the convolution theorem for the functions 𝑣(𝑡) and 𝑢(𝑡), the following is obtained 
 𝑡    (𝑡 − 𝜏)−𝛼
(𝑣 ∗ 𝑢)(𝑡) = ∫ 𝑓(𝜏)  𝑑𝜏 (4-63) 
0 Γ(1 − 𝛼)
Taking the derivative,  
 𝑑 1 𝑑 𝑡
(𝑣 ∗ 𝑢)(𝑡) = ∫ 𝑓(𝜏) (𝑡 − 𝜏)−𝛼 𝑑𝜏 (4-64) 
𝑑𝑡 Γ(1 − 𝛼) 𝑑𝑡 0
Equation (4-64) is the Riemann-Liouville fractional derivative, and it can thus be concluded that the 
convolution of functions 𝑣(𝑡) and 𝑢(𝑡) is representative of the Riemann-Liouville fractional derivative 
and function 𝑢(𝑡) is an expression of the Riemann-Liouville kernel.  
From this perspective, it becomes clear why the Riemann-Liouville kernel is considered a singularity,  
 1
𝑡−𝛼 (4-65) 
Γ(1 − 𝛼)
where, for values where 𝑡 = 0, the kernel becomes undefined.  
The Caputo fractional derivative can be expressed for a first order derivative as 
 1 𝑡 𝑑
 𝐶 𝛼0𝐷𝑡 𝑓(𝑡) =  ∫ 𝑓(𝜏) (𝑡 − 𝜏)
−𝛼 𝑑𝜏 (4-66) 
Γ(1 − 𝛼) 0 𝑑𝜏
Following a similar procedure as for the Riemann-Liouville fractional derivative, the Caputo kernel is 
also a found to contain a singularity (Equation (4-65)). 
The Caputo-Fabrizio fractional derivative can be expressed for a first order derivative as 
 𝑀(𝛼) 𝑡 𝑑 𝛼(𝑡 − 𝜏) 
 𝐶𝐹𝐷𝛼0 𝑡 𝑓(𝑥) =  ∫ 𝑓(𝜏) 𝑒𝑥𝑝 [− ]𝑑𝜏 (4-67) (1 − 𝛼) 0 𝑑𝜏 1 − 𝛼 
89 
 
Following a similar procedure, the following is found:  
 𝑀(𝛼) 𝑡 𝑑 𝛼(𝑡 − 𝜏) 
(𝑣 ∗ 𝑢)(𝑡) = ∫ 𝑓(𝜏) 𝑒𝑥𝑝 [− ]  𝑑𝜏 (4-68) 
(1 − 𝛼) 0 𝑑𝜏 1 − 𝛼 
Equation (4-68) is the Caputo-Fabrizio fractional derivative, and it can thus be concluded that the 
convolution of functions 𝑣(𝑡) and 𝑢(𝑡) is representative of the Caputo-Fabrizio fractional derivative, 
and function 𝑢(𝑡) is an expression of the Caputo-Fabrizio kernel. From this perspective, the Caputo-
Fabrizio kernel is non-singular: 
𝑀(𝛼) 𝛼 
𝑒𝑥𝑝 [− 𝑡] 
(1 − 𝛼) 1 − 𝛼 
where, alpha (𝛼) is constrained 0 < 𝛼 < 1.  
The Atangana-Baleanu in Riemann sense (ABR) fractional derivative can be expressed for a first order 
derivative as 
 𝐴𝐵(𝛼) 𝑑 𝑡 𝛼 
 𝐴𝐵𝑅𝑎 𝐷
𝛼
𝑡 𝑓(𝑡) =  ∫ 𝑓(𝜏) 𝐸 [− (𝑡 − 𝜏)
𝛼] 𝑑𝜏 (4-69) 
(1 − 𝛼) 𝑑𝑡 𝛼𝛼 1 − 𝛼 
From a similar convolution perspective, the Atangana-Baleanu in Riemann sense (ABR) kernel is found 
to be non-singular: 
𝐴𝐵(𝛼) 𝛼 
 𝐸 − 𝑡𝛼  
(1 − 𝛼) 𝛼
[ ]
1 − 𝛼 
where, alpha (𝛼) is constrained 0 < 𝛼 < 1.  
The Atangana-Baleanu in Caputo sense (ABC) fractional derivative can be expressed for a first order 
derivative as 
 𝐴𝐵(𝛼) 𝑡 𝑑 𝛼 
 𝐴𝐵𝐶 𝛼𝑎 𝐷𝑥 𝑓(𝑡) =  ∫ 𝑓(𝜏) 𝐸 [− (𝑡 − 𝜏)
𝛼] 𝑑𝜏 (4-70) 
(1 − 𝛼) 𝑑𝜏 𝛼𝛼 1 − 𝛼 
From a similar convolution perspective, the Atangana-Baleanu in Caputo sense (ABC) kernel is the 
same as the ABR kernel and non-singular. The respective kernels for each fractional derivative are 
summarised in Table 4-1. The Riemann-Liouville and Caputo definitions of the fractional derivative 
contain a potential singularity within the power-law defined kernel, while the Caputo-Fabrizio and 
Atangana-Baleanu definitions of the fractional derivative do not contain any potential singularities 
within their respective exponential and Mittag-Leffler function defined kernels. The Riemann-Liouville 
and Caputo definitions do house a potential singularity, yet the formulation is non-local. Conversely, 
the Caputo-Fabrizio definition removes the potential singularity, but is considered a local operator 
because the variable (𝑡) has no associated fractional order (𝛼). From this perspective, the significance 
90 
 
of the Atangana-Baleanu definitions of the fractional derivative becomes clear, where the potential 
singularity is removed and the formulation retains a non-local operator.  
The fractional order exponent can be interpreted as an indication of the memory capacity in empirical 
systems (Du et al., 2013; Tateishi et al., 2017). Additionally, the fractional order exponent introduces 
a non-linear time dependence in the mean square displacement of the system (Metzler and Klafter, 
2000; Tateishi et al., 2017). Tateishi et al. (2017) explored this property by considering the relationship 
between the random walk concept and the diffusion equation. The usual random walk is characterised 
by Gaussian, Markovian and ergodic properties, which is related to a linear time dependence of the 
mean squared displacement (∆𝑥)2 ~ 𝑡. On the other hand, the continuous time random walk concept 
is characterised by non-Gaussian, non-Markovian and non-ergodic properties, which is related to non-
linear dependence of the mean squared displacement (∆𝑥)2 ~ 𝑡𝛼. Here, sub-diffusion is expressed 
where 𝛼 < 1, and superdiffusion where 𝛼 > 1 (Tateishi et al., 2017). From this perspective, it 
becomes clear why the Caputo-Fabrizio fractional derivative is local and has no memory component, 
because the exponential kernel has no fractional order exponent, which is an indication of the memory 
and non-linear time dependence. On the other hand, the Riemann-Liouville, Caputo and Atangana-
Baleanu fractional derivatives are non-local and have a memory because the fractional order exponent 
is present in the kernel and associated with time (𝑡).  
Table 4-1 Summary of kernels associated with each fractional derivative definition 
Fractional Singularity/ non-
Kernel Distribution Locality Memory 
derivative  Singularity 
Riemann-
Liouville 1
𝑡−𝛼
Non-
 Power Law Singularity Yes 
Γ(1 − 𝛼) Local 
Caputo 
𝑀(𝛼) 𝛼 
Caputo-Fabrizio 𝑒𝑥𝑝 [− 𝑡] Exponential Non-Singularity Local No 
(1 − 𝛼) 1 − 𝛼 
Atangana-
Baleanu in 
Riemann sense 
(ABR) 
𝐴𝐵(𝛼) 𝛼 
𝛼 Mittag- Non- 𝐸 [− 𝑡 ] 
 𝛼 Non-Singularity Yes (1 − 𝛼) 1 − 𝛼 Leffler Local 
Atangana-
Baleanu in 
Caputo sense 
(ABC) 
 
91 
 
The question is why the Riemann-Liouville and Caputo fractional derivatives contain a potential 
singularity within the power-law defined kernel, while Caputo-Fabrizio and Atangana-Baleanu 
derivatives do not. One possible explanation for this could be the order in which the fractional 
derivative and integral were formulated (Equation (4-23) and Equation (4-29)). The Riemann-Liouville 
fractional integral was first formulated, and from the integral, the derivative was derived. On the other 
hand, the Atangana-Baleanu fractional derivative was first formulated and then the integral was 
derived. Considering the natural properties of a derivative (discrete) and an integral (continuous), 
perhaps “squeezing” an integral into a derivative formulation could have triggered the singularity in 
the Riemann-Liouville and Caputo fractional definition.  
For further analysis of the various fractional derivative kernels, the associated statistical probability 
distributions are investigated. The statistical probability distribution of finding a particle in any one 
place is commonly correlated to simulating the transport of a contaminant or particle. The Riemann-
Liouville and Caputo fractional derivatives are associated with a power law distribution, the Caputo-
Fabrizio fractional derivative is associated with an exponential function distribution, and the 
Atangana-Baleanu fractional derivative is associated with a Mittag-Leffler function distribution (Table 
4-1). Investigating the statistical properties of each of these distributions, can provide an insight into 
how the kernels control the behaviour of each fractional derivative definition.  
The power law probability distribution, also called the Pareto distribution, uses the power law to 
describe certain phenomena (See Section 4.1.2). The various statistical properties of the Pareto 
distribution are discussed herein. The Pareto (type I) distribution can be defined by the survival 
function, where the probability of a random variable (𝑋) being greater than some number 𝑥 is 
expressed as (Arnold, 2008; Arnold, 2015a; Arnold, 2015b): 
 𝑥 𝛼𝑚
 ( ) 𝑥 ≥ 𝑥 ,
?̅?(𝑥) = Pr(𝑋 > 𝑥) = { 𝑚  (4-71) 𝑥
1 𝑥 < 𝑥𝑚,
where, 
 𝑥𝑚 is the minimum possible value of 𝑋 (scale parameter), and 
 𝛼 is the shape parameter (positive parameter). 
Based on this definition, the cumulative distribution function has been defined as (Arnold, 2015a):  
 𝑥 𝛼𝑚
1 − ( ) 𝑥 ≥ 𝑥 ,
𝐹𝑥(𝑥) = { 𝑥
𝑚   (4-72) 
0 𝑥 < 𝑥𝑚,
Considering the differentiation of the cumulative distribution function, the probability density 
function has been defined as (Arnold, 2015a):  
92 
 
 𝛼𝑥𝛼𝑚
𝑓(𝑥) = {  𝑥 ≥ 𝑥𝑥𝛼+1 𝑚
, (4-73) 
0 𝑥 < 𝑥𝑚,
Therefore, the expected value of a random variable has been defined as (Arnold, 2015a): 
 ∞ 𝛼 ≤ 1
𝐸(𝑋) = { 𝛼𝑥𝑚  (4-74) 
  𝛼 > 1
𝛼 − 1
The variance of a random variable has been defined as (Arnold, 2015a): 
 ∞ 𝛼 ∈ (1,2)
𝑉𝑎𝑟(𝑋) = { 𝑥 2𝑚 𝛼  (4-75) 
  ( ) 𝛼 > 2
𝛼 − 1 𝛼 − 1
From the properties of the power law distribution, it becomes clear that the expected value is 
undefined for values of 𝛼 ≤ 1, and the variance is undefined for values of 𝛼 ∈ (1,2). If the distribution 
is related back to the Riemann-Liouville and Caputo fractional derivative definitions, where the 
traditional values of the fractional order (𝛼) are 0 < 𝛼 < 1, there is a fundamental disagreement. This 
disparity could be the answer to the question posed, why the Riemann-Liouville and Caputo fractional 
derivatives contain a potential singularity within the power-law defined kernel.  
The exponential probability distribution uses the exponential function to describe certain phenomena 
(See Section 4.1.3). The various statistical properties of the exponential distribution are discussed 
herein. The general probability density function can be defined as (NIST/SEMATECH, 2003): 
 1
𝑓(𝑥) = 𝑒−(𝑥−𝜇)/𝛽 𝑥 ≥ 𝜇;  𝛽 > 0 (4-76) 
𝛽
where, 
 𝜇 is the location parameter, and  
1
 𝛽 is the scale parameter, often expressed as 𝜆, which equals .  
𝛽
The standard exponential distribution has been expressed for the case where 𝜇 = 0 and 𝛽 = 1: 
 𝑓(𝑥) = 𝑒−𝑥 𝑥 ≥ 0 (4-77) 
Based on this definition, the cumulative distribution function has been defined as (NIST/SEMATECH, 
2003): 
 𝐹(𝑥) = 1 − 𝑒−𝑥/𝛽 𝑥 ≥ 0;  𝛽 > 0 (4-78) 
The expected value of a random variable following an exponential distribution has been defined as:  
 1
𝐸(𝑋) = = 𝛽 (4-79) 
𝜆
93 
 
The variance of a random variable has been defined as: 
 1
𝑉𝑎𝑟(𝑋) = = 𝛽2 (4-80) 
𝜆2
From the properties of the exponential distribution, the parameter restrictions are only that 𝑥 ≥
0;  𝛽 > 0, which do not represent any fundamental disagreements when related back to the Caputo-
Fabrizio fractional derivative definition.  
The Mittag-Leffler probability distribution uses the Mittag-Leffler function to describe certain 
phenomena (See Section 4.1.4). The various statistical properties of the Mittag-Leffler distribution are 
discussed herein. A Mittag-Leffler statistical distribution was first explored by Pillai (1990), where the 
distribution function or cumulative density function was defined as (Mathai, 2010; Haubold et al., 
2011, Atangana and Gómez-Aguilar, 2018): 
 𝐺𝑥(𝑥) = 1 − 𝐸𝛼(−𝑥
𝛼)              0 < 𝛼 ≤ 1, 𝑥 > 0 (4-81) 
Differentiating with respect to 𝑥, the density function 𝑓(𝑥) is obtained: 
 ∞𝑑𝐺(𝑥) (−1)𝑛+1𝑥𝛼𝑛
𝑓(𝑥) = = ∑  (4-82) 
𝑑𝑥 Γ(1 + 𝛼𝑛)
𝑛=1
Rearranging and replacing 𝑛 with 𝑛 + 1: 
 ∞𝑑𝐺(𝑥) (−1)𝑛𝑥𝛼+𝛼𝑛−1
𝑓(𝑥) = = ∑ = 𝑥𝛼−1𝐸 (−𝑥𝛼)       0 < 𝛼 ≤ 1, 𝑥 > 0 (4-83) 
𝑑𝑥 Γ(1 + 𝛼𝑛) 𝛼,𝛼
𝑛=1
Considering the following probability density function (Atangana and Gómez-Aguilar, 2018), 
 1 ∞ sin [𝛼𝜋]
⋂𝛼 = ∫ exp[ −𝑦] 𝑑𝑦 𝑡 0 𝛼𝑡
𝛼 𝛼𝑡 −𝛼 (4-84) 
𝜋 [( ) + ( ) + 2cos( 𝜋𝛼)]
(1 − 𝛼)𝑦 (1 − 𝛼)𝑦
The tail probability distribution associated with the waiting time 𝑇 can be defined from the Equation 
(4-84) (Atangana and Gómez-Aguilar, 2018): 
 ∞ 𝛼
𝑃(𝑇 > 𝑡) = ∫ ⋂𝛼(𝑥)𝑑𝑥 = 𝐸𝛼 [− 𝑡
𝛼] (4-85) 
𝑡 1 − 𝛼
From Equation (4-85) the inter-arrival time density can be defined as (Atangana and Gómez-Aguilar, 
2018): 
 𝛼𝑡𝛼−1 𝛼
𝜒𝛼(𝑡) = 𝐸
𝛼
𝛼,𝛼 [− 𝑡 ]          𝑡 ≥ 0 (4-86) 1 − 𝛼 1 − 𝛼
It follows that when considering the n-arrival time, the corresponding probability density function can 
be defined as (Atangana and Gómez-Aguilar, 2018): 
94 
 
 𝛼𝑛+1 𝑡𝛼𝑛−1 𝛼
𝑓𝛼(𝑡) = 𝐸𝑛𝑛 𝛼 [− 𝑡
𝛼] 𝑛 (4-87) (1 − 𝛼) Γ(𝑛) 1 − 𝛼
The moment generating function can be defined from Equation (4-87) (Atangana and Gómez-Aguilar, 
2018): 
 𝛼 𝑗
∞ (( 𝑡𝛼(exp[−𝑝] )1 − 𝛼 − 1 ) )
Ω (4-88) 𝛼(𝑝, 𝑡) =∑  Γ(𝛼𝑗 + 1)
𝑗=0
Applying the classical formula, Equation (4-88) can be used to calculate the corresponding moment 
(Atangana and Gómez-Aguilar, 2018): 
 
 𝛼
𝑗
∞ (( 𝛼𝑙 (1 − 𝛼 𝑡 exp
[−𝑝] − 1))  
 )𝜕  
𝐸[𝑁(𝑡)]𝑙 = lim (−1)𝑛 ∑  (4-89) 
𝑝→0 𝑙 𝜕𝑝 Γ(𝛼𝑗 + 1)  
 𝑗=0  
{ }
Therefore, the mean of the corresponding process can be defined as (Atangana and Gómez-Aguilar, 
2018):  
 ∞ 𝛼𝑡𝛼
?̅? 𝛼 = ∑𝑛 𝑃𝛼𝑛 (𝑡) =  (4-90) 
1−𝛼 (1 − 𝛼)Γ(α + 1)
𝑛=0
The second moment, or variance, is thus given as (Atangana and Gómez-Aguilar, 2018): 
 1
  Γ(α)Γ (
𝛼 2
)   
∞   2   𝛼  𝛼 Γ (𝛼 + )  𝛼𝑡 𝛼𝑡
?̅?2
  
𝛼 = ∑𝑛2 𝑃𝛼𝑛 (𝑡) = 1 +
2
 − 1  (4-91) 
1−𝛼 (1 − 𝛼)Γ(α + 1) (1 − 𝛼)Γ(α + 1) 22𝛼−1   𝑛=0
      
{ [ ]}
From the properties of the Mittag-Leffler distribution, restrictions associated with the statistical 
properties do not form any fundamental disagreements with the values for 𝛼 in the Atangana-Baleanu 
fractional derivative definitions.  
The statistical properties associated with the respective kernel of each fractional derivative definition 
provides a potential explanation on the presence of a singularity in the Riemann-Liouville and Caputo 
fractional derivatives, while the Caputo-Fabrizio and Atangana-Baleanu contain non-singularities. The 
explanation being the distribution associated with the controlling kernel should not only satisfy the 
fractional derivative boundaries of the fractional order, but also the associated statistical boundaries. 
95 
 
The Riemann-Liouville and Caputo fractional derivatives have a disagreement between the fractional 
order boundaries and the statistical property boundaries.  
In a final analysis of the fractional derivative definitions, the results of the work performed by Tateishi 
et al. (2017) are considered. The usual diffusion equation was generalised by means of the Riemann-
Liouville, Caputo-Fabrizio and Atangana-Baleanu fractional derivatives, and the effect each kernel had 
on the simulated results analysed by a comparison with a continuous random walk framework. The 
results obtained by Tateishi et al. (2017) are in Table 4-2, where the Caputo-Fabrizio and Atangana-
Baleanu fractional derivatives modify the behaviour of the waiting time distribution, as well as 
introduce non-Gaussian distributions and different diffusive regimes depending on the time scale. The 
Caputo-Fabrizio fractional derivative has a waiting time distribution described by an exponential 
distribution, and the probability distribution displays a stationary state as well as saturated diffusion 
for long times. Tateishi et al. (2017) comments that this finding is remarkable because the fractional 
diffusion equation is solved without external force and subject to free diffusion boundary conditions. 
The Atangana-Baleanu fractional derivative produces a waiting time distribution described by stretch 
exponential for small times, and by a power law (similar to Riemann-Liouville) for long times. 
Additionally, the Atangana-Baleanu fractional derivative has a probability distribution indicating a 
cross-over from usual diffusion for small times, and sub-diffusive for long times. It was concluded that 
the new fractional derivatives definitions have the potential to incorporate different memory effects, 
which could change the way anomalous diffusion is modelled.  
Table 4-2 Summary of results obtained by Tateishi et al. (2017) on influence on different fractional 
derivative definitions on the Diffusion Equation   
Waiting time distribution / Mean square 
Fractional derivative Probability distribution 
kernel distribution displacement 
Power Law and 
Riemann-Liouville Power Law Non-Gaussian 
Scale invariant 
Cross-over from Cross-over from Gaussian to 
Caputo-Fabrizio Exponential  usual to confined non-Gaussian with steady 
diffusion state 
Mittag-Leffler (cross-over Cross-over from 
Cross-over from Gaussian to 
Atangana-Baleanu between a stretch usual to sub-
non-Gaussian 
Exponential to Power Law) diffusion 
 
96 
 
4.6 Fractional derivatives application to the Advection-Dispersion Equation 
Following on from the work performed by Tateishi et al. (2017), which found that the new fractional 
derivatives could be effective in modelling anomalous diffusion, the new fractional derivatives are 
applied to the advection-dispersion equation to model not only anomalous diffusion, but potentially 
anomalous advection in the form of preferential pathways in fractures within the groundwater 
system.  
It has been conceptualised that within an aquifer, water is moving within the porous media at a 
predictable rate, but the flow can also be faster than expected within unknown fractures or faults, 
and/or slower than expected in other areas. This discrepancy in the manner groundwater flows is 
correlated to the discrepancy in diffusion (anomalous diffusion), referred to as super-diffusion (faster 
than traditional methods predict) and sub-diffusion (slower than traditional methods predict).  
An analogy is drawn with the conceptual groundwater flow within a fractured aquifer, where super-
advection is defined as flow faster than traditional methods predict, and sub-advection as flow slower 
than traditional methods predict. For this reason, the fractional derivative will be applied to space for 
the advection term of the advection-dispersion equation.  
The fractional derivative will also be applied to time because of the properties of the fractional 
derivatives discussed, where the waiting time distribution is defined. Incorporating the fractional 
derivative for time, allows these features to be activated in the advection-dispersion equation 
solution, i.e. for the Caputo derivative a power law distribution and for the Atangana-Baleanu 
derivative a Mittag-Leffler distribution, including a cross-over between a stretch exponential to power 
Law.  
The traditional advection-dispersion equation, 
 𝜕 𝜕 𝜕2
𝑐(𝑥, 𝑡) = −𝑣𝑥 𝑐(𝑥, 𝑡) + 𝐷𝐿 𝑐(𝑥, 𝑡) (4-92) 𝜕𝑡 𝜕𝑥 𝜕𝑥2
Incorporating the fractional derivative for time and advection components, 
 𝜕2
𝐷𝛼 {𝑐(𝑥, 𝑡)} = −𝑣  𝐷𝛼{𝑐(𝑥, 𝑡)} + 𝐷 𝑐(𝑥, 𝑡) (4-93) 𝑡 𝑥 𝑥 𝐿 𝜕𝑥2
where, 𝐷𝛼𝑡  indicates the use of a fractional derivative with a fractional order 𝛼. 
This approach differs from the usual that incorporates the fractional derivative either in time (Liu et 
al., 2003; Povstenko, 2014; Rubbab et al., 2016), in space (but only for the diffusion/dispersion term) 
(Benson, 1998; Benson et al., 2000; Su et al., 2010; Wang and Wang, 2011), or in time and space 
(diffusion-dispersion term) (Huang and Liu, 2005; Povstenko, 2015; Javadi et al., 2016). 
97 
 
4.7 Chapter summary 
Fractional calculus is outlined by presenting the functions required for the definition of fractional 
integrals and derivatives, fundamental fractional derivative definitions, and an analysis of the kernels 
associated with each fractional definition. The kernel analysis found that the Riemann-Liouville and 
Caputo definitions of the fractional derivative contain a potential singularity within the power-law 
defined kernel, while the Caputo-Fabrizio and Atangana-Baleanu definitions of the fractional 
derivative do not contain any potential singularities within their respective exponential and Mittag-
Leffler function defined kernels. Yet, the Caputo-Fabrizio definition removes the potential singularity, 
but is considered a local operator, and thus the significance of the Atangana-Baleanu definitions of 
the fractional derivative is highlighted. Research performed by Tateishi et al. (2017) is evaluated, and 
it is found that the Atangana-Baleanu fractional derivative has a probability distribution indicating a 
cross-over from usual diffusion for small times and sub-diffusive for long times; and correlated to the 
potential to incorporate different memory effects, which could change the way anomalous diffusion 
is modelled. Lastly, a case for anomalous advection in fractured systems is made, with an application 
of the fractional derivative in space for the advective term and in time to activate the waiting time 
distribution properties. 
 
  
98 
 
 
5  FRACTIONAL ADVECTION- 
     DISPERSION EQUATION WITH    
     CAPUTO DERIVATIVE 
 
The applicability of fractional derivative in modelling anomalous diffusion has been established by 
several authors (Wyss, 1986; Schneider and Wyss, 1989; Metzler et al., 1994; Metzler and Klafter, 
2000; Meerschaert, 2012). Following this research, a case was made for the application of fractional 
derivatives to the advection-dispersion equation for modelling transport specifically in a fractured 
groundwater network (Benson, 1998; Benson et al., 2000). In Section 4.6, incorporating a fractional 
derivative for the advective term has been conceptualised.  
The developed fractional advection-dispersion equation requires a solution to be of use. A myriad of 
methods are available to solve fractional differentials, including Laplace transform methods, Fourier 
transform methods, Adomian’s decomposition method, Homotopy analysis method, finite difference 
methods, finite element methods and many more (Meerschaert and Tadjeran, 2004; Huang et al., 
2008; Jafari and Tajadodi, 2015; Pang et al., 2015; Li et a., 2017). The complex nature of the formulated 
fractional advection-dispersion equation results in an analytical solution being challenging, and thus 
numerical solutions will be employed (Sousa, 2009; Meerschaert and Tadjeran, 2004; Jafari and 
Tajadodi, 2015). Diethelm et al. (2005) reported a need for readily usable tools to numerically solve 
99 
 
fractional derivative problems. However, at that time only a limited collection of algorithms had been 
developed.  
Finite difference methods for numerically solving fractional partial differential equations will be 
considered for the developed fractional advection-dispersion equation. A review of available finite 
difference numerical schemes for fractional partial differential equations is performed. A finite 
difference numerical method for the space fractional advection-dispersion equation, with the 
fractional derivative applied to the dispersion term, was presented by Meerschaert and Tadjeran 
(2004). Making use of the standard Grünwald estimates, Meerschaert and Tadjeran (2004) describe 
unconditionally stable explicit and implicit Euler methods, and a Crank-Nicolson method. Lynch et al. 
(2003) proposed a simple explicit method and a semi-implicit method for the space fractional diffusion 
equation, which are analysed for efficiency and accuracy. The semi-implicit method was found to be 
more effective than the explicit method. Liu et al. (2007) considered the space-time fractional 
advection-dispersion equation, describing an explicit and implicit difference numerical solution 
method. The implicit difference method was found to be unconditionally stable and convergent, while 
the explicit difference method was conditionally stable and convergent. Sousa (2009) generalised 
explicit finite difference approximations for a fractional advection-diffusion problem, where the 
fractional derivative was used to replace the second order derivative in the problem, were described. 
One of these generalisations makes use of the finite difference upwind scheme for the advection term 
but with a classical derivative. An original numerical approximation of the space fractional advection-
dispersion equation was presented by Shen et al. (2014), where a fractional centred difference scheme 
was applied to the Riesz fractional derivatives. The scheme was found to have second-order accuracy, 
unconditionally stable, consistent and convergent. Since, the development and analysis of numerical 
solutions for the fractional advection-dispersion equation has remained relevant (Su et al., 2010; 
Wang and Wang, 2011; Sousa and Li, 2015; Liu and Hou, 2017; Owolabi and Atangana, 2017; Atangana 
and Owolabi, 2018; Fazio and Jannelli, 2018; Li and Rui, 2018; Liu and Li, 2018).  
Upwind finite difference numerical methods have been applied to fractional partial differential 
equations (Yuan, 2003; Yirang et al., 2014; Yuan et al., 2017), and following on this research, the 
augmented upwind schemes presented in Chapter 2 for the local advection-dispersion equation will 
be applied to the Fractional advection-dispersion equation with Caputo fractional derivative. These 
numerical schemes were found to be an improvement on the traditional upwind approach, and thus 
selected for application on the fractional advection-dispersion equation to investigate their suitability 
for anomalous advection in fractured systems.  
The numerical approximation schemes for the fractional advection-dispersion equation with Caputo 
fractional derivatives are developed using the new upwind advection Crank-Nicolson and weighted 
100 
 
upwind-downwind schemes, as well as the traditional upwind schemes. The traditional upwind 
schemes serve to form a base of comparison in terms of numerical stability for the developed 
schemes. 
5.1 Upwind numerical approximation schemes 
The augmented upwind schemes are applied to the fractional advection-dispersion equation for 
numerical approximation (Equation (4-93)). Applying the Caputo definition to the fractional advection-
dispersion equation, 
𝜕2
𝐶 𝛼 𝐶 𝛼 (5-1) 
0𝐷𝑡  (𝑐(𝑥, 𝑡)) = −𝑣𝑥 0𝐷𝑥 (𝑐(𝑥, 𝑡)) + 𝐷𝐿 (𝑐(𝑥, 𝑡)) 𝜕𝑥2
where, 
 1 𝑡 𝑑
 𝐶𝐷𝛼0 𝑡 𝑓(𝑥) =  ∫  𝑓(𝜏)(𝑡 − 𝜏)
−𝛼 𝑑𝜏  
Γ(1 − 𝛼) 0 𝑑𝜏
A forward difference scheme in time is applied to the Caputo fractional derivative to demonstrate the 
numerical approximation approach for the Caputo fractional derivative. The Caputo fractional 
derivative is considered for a specific time (𝑡𝑛), 
 1 𝑡𝑛 𝑑
 𝐶 𝛼 −𝛼 (5-2) 𝑎𝐷𝑡 𝑓(𝑡𝑛) =  ∫ 𝑓(𝜏) (𝑡𝑛 − 𝜏) 𝑑𝜏 Γ(1 − 𝛼) 0 𝑑𝜏
The time integer-order derivative 𝜏 is replaced with the forward differences approximation at specific 
points in time (𝑡), and a summation is used to express the integral performed for each time step, 
 𝑛−11 𝑡𝑘+1 𝑓
𝑘+1 − 𝑓𝑘
 𝐶𝑎𝐷
𝛼
𝑡 𝑓(𝑡𝑛) =  [∑  ∫
𝑖 𝑖  (𝑡𝑛 − 𝜏)
−𝛼 𝑑𝜏] (5-3) 
Γ(1 − 𝛼) 𝑡 ∆𝑡𝑘=0 𝑘
The approximation of the continuous 𝜏 function, results in two specific points of the function with 
respect to time (𝑡), which allows the approximated derivative to be taken out of the integral: 
 𝑛−11 𝑓
𝑘+1 − 𝑓𝑘 𝑡𝑘+1
 𝐶𝐷𝛼𝑓(𝑡 ) =  [∑  𝑖 𝑖 ∫  (𝑡𝑛 − 𝜏)−𝛼𝑎 𝑡 𝑛 𝑑𝜏] (5-4) Γ(1 − 𝛼) ∆𝑡
𝑘=0 𝑡𝑘
The fractional integral still requires approximation, let function 𝑦 represent (𝑡𝑛 − 𝜏), 𝑑𝑦 = −𝑑𝜏; and 
the integral is reversed to consider a single time step: 
 𝑛−1 𝑓𝑘+11 − 𝑓
𝑘 𝑡𝑛−𝑡𝑘+1
 𝐶 𝛼𝑎𝐷𝑡 𝑓(𝑡𝑛) =  [∑
𝑖 𝑖 {−∫  𝑦−𝛼 𝑑𝑦}] (5-5) 
Γ(1 − 𝛼) ∆𝑡
𝑘=0 𝑡𝑛−𝑡𝑘
Integrating, 
101 
 
𝑛−1
𝑓𝑘+1 − 𝑓𝑘1 𝑦1−𝛼
𝑡𝑛−𝑡𝑘+1
 𝐶𝐷𝛼𝑓(𝑡 ) =  [∑ 𝑖 𝑖𝑎 𝑡 𝑛 {− | }] (5-6) Γ(1 − 𝛼) ∆𝑡 1 − 𝛼
𝑘=0 𝑡𝑛−𝑡𝑘
𝑛−1 𝑘+1 𝑘
1 𝑓 − 𝑓 (𝑡 − 𝑡 )1−𝛼 − (𝑡 − 𝑡 )1−𝛼
 𝐶 𝛼
𝑛 𝑘 𝑛 𝑘+1
𝑎𝐷𝑡 𝑓(𝑡𝑛) =  [∑
𝑖 𝑖 { }] (5-7) 
Γ(1 − 𝛼) ∆𝑡 1 − 𝛼
𝑘=0
Considering the specific time can be represented as the number of time steps required to reach that 
time (𝑡𝑛 = ∆𝑡 ∙ 𝑛) and similarly, 𝑡𝑘 = ∆𝑡 ∙ 𝑘. Thus, the same can be applied to achieve 𝑡𝑛 − 𝑡𝑘 =
∆𝑡(𝑛 − 𝑘) and 𝑡𝑛 − 𝑡𝑘+1 = ∆𝑡(𝑛 − 𝑘 − 1): 
𝑛−1
1 𝑓𝑘+1𝑖 − 𝑓
𝑘
𝑖 (∆𝑡(𝑛 − 𝑘) )
1−𝛼 − (∆𝑡(𝑛 − 𝑘 − 1))1−𝛼
 𝐶 𝛼𝑎𝐷𝑡 𝑓(𝑡𝑛) =  [∑ {− }] (5-8) Γ(1 − 𝛼) ∆𝑡 1 − 𝛼
𝑘=0
Simplifying by moving the constant ∆𝑡, ∆𝑡1−𝛼 and 1 − 𝛼 out from the summation, and considering the 
properties of the gamma function, where Γ(2 − α) = (1 − α)Γ(1 − α) when using integration by 
parts and L’Hôpitals rule: 
𝑛−1
(∆𝑡)−𝛼
 𝐶 𝛼𝑎𝐷𝑡 𝑓(𝑡𝑛) =  [∑(𝑓
𝑘+1 − 𝑓𝑘) {(𝑛 − 𝑘)1−𝛼 − (𝑛 − 𝑘 − 1)1−𝛼}] (5-9) 
Γ(2 − 𝛼) 𝑖 𝑖
𝑘=0
Equation (5-9) is the numerically approximated Caputo fractional derivative, where the function is 
considered at two discrete points in time, yet additionally the fractional components are included to 
account for changes in between those two discrete points in time.  
5.1.1 Explicit upwind  
The numerical approximation of the Caputo fractional derivative with respect to time has been 
presented in Equations (5-2) to (5-9), with the resulting scheme applied specifically to the fractional 
advection dispersion equation: 
(∆𝑡)−𝛼
𝑛−1
 𝐶0𝐷
𝛼
𝑡 (𝑐(𝑥𝑚, 𝑡𝑛)) =  [∑(𝑐
𝑘+1 − 𝑐𝑘 ) {(𝑛 − 𝑘)1−𝛼 − (𝑛 − 𝑘 − 1)1−𝛼}] (5-10) 
Γ(2 − 𝛼) 𝑚 𝑚
𝑘=0
where, a function 𝛿𝛼𝑛,𝑘 is applied to simplify, 
𝛿𝛼 = (𝑛 − 𝑘)1−𝛼 − (𝑛 − 𝑘 − 1)1−𝛼𝑛,𝑘   
Thus, 
𝑛−1
(∆𝑡)−𝛼
 𝐶 𝛼 𝑘+1 𝑘 𝛼0𝐷𝑡 (𝑐(𝑥𝑚, 𝑡𝑛)) =  [∑(𝑐𝑚 − 𝑐 )𝛿Γ(2 − 𝛼) 𝑚 𝑛,𝑘
] (5-11) 
𝑘=0
Similarly, the Caputo fractional derivative with respect to space (explicit) and the upwind scheme is, 
102 
 
(∆𝑥)−𝛼
𝑚
 𝐶𝐷𝛼(𝑐(𝑥 , 𝑡 )) =  [∑(𝑐𝑛−1 − 𝑐𝑛−1) {(𝑚 − 𝑖)1−𝛼 − (𝑚 − 𝑖 − 1)1−𝛼0 𝑥 𝑚 𝑛 Γ(2 − 𝛼) 𝑖 𝑖−1
}] (5-12) 
𝑖=1
where, a function 𝛿𝛼𝑚,𝑖 is applied to simplify, 
𝛿𝛼𝑚,𝑖 = (𝑚 − 𝑖)
1−𝛼 − (𝑚 − 𝑖 − 1)1−𝛼  
Thus, 
(∆𝑥)−𝛼
𝑚
 𝐶𝐷𝛼0 𝑥 (𝑐(𝑥𝑚, 𝑡𝑛)) =  [∑(𝑐
𝑛−1 𝑛−1 𝛼
𝑖 − 𝑐𝑖−1 ) 𝛿𝑚,𝑖] (5-13) Γ(2 − 𝛼)
𝑖=1
And, the local second order derivative is approximated using the traditional finite difference approach: 
𝜕2 𝑐𝑛−1 − 2𝑐𝑛−1 + 𝑐𝑛−1
( ) 𝑚+1
𝑚 𝑚−1
𝐷𝐿 (𝑐 𝑥𝑚, 𝑡𝑛 ) = 𝐷𝐿 ( ) (5-14) 𝜕𝑥2 (∆𝑥)2
Substituting back into Equation (5-1), 
(∆𝑡)−𝛼
𝑛−1 𝑚
(∆𝑥)−𝛼
[∑(𝑐𝑘+1𝑚 − 𝑐
𝑘 𝛼
𝑚)𝛿𝑛,𝑘] + 𝑣𝑥 [∑(𝑐
𝑛−1 − 𝑐𝑛−1) 𝛿𝛼
Γ(2 − 𝛼) Γ(2 − 𝛼) 𝑖 𝑖−1 𝑚,𝑖
] 
(5-15) 
𝑘=0 𝑖=1
𝑐𝑛−1 − 2𝑐𝑛−1 + 𝑐𝑛−1𝑚+1 𝑚 𝑚−1
−𝐷𝐿 ( ) = 0 (∆𝑥)2
Reformulating the following can be obtained, 
(∆𝑡)−𝛼
𝑛−2
(∆𝑡)−𝛼
(𝑐𝑛 − 𝑐𝑛−1)𝛿  𝛼 +  (∑(𝑐𝑘+1𝑚 𝑚 𝑛,𝑛−1 𝑚 − 𝑐
𝑘
𝑚)𝛿
 𝛼
Γ(2 − 𝛼) Γ(2 − 𝛼) 𝑛,𝑘
)
𝑘=0
(∆𝑥)−𝛼
𝑚−1
(∆𝑥)−𝛼 (5-16) 
+ 𝑣 (𝑐𝑛−1 − 𝑐𝑛−1 )𝛿  𝛼  + 𝑣  [∑(𝑐𝑛−1 − 𝑐𝑛−1  𝛼𝑥 Γ(2 − 𝛼) 𝑚 𝑚−1 𝑛,𝑚 𝑥 Γ(2 − 𝛼) 𝑖 𝑖−1
) 𝛿𝑚,𝑖]
𝑖=1
𝑐𝑛−1𝑚+1 − 2𝑐
𝑛−1
𝑚 + 𝑐
𝑛−1
𝑚−1
− 𝐷𝐿 ( ) = 0 (∆𝑥)2
Rearranging, 
(∆𝑡)−𝛼 (∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷
( 𝛿  𝛼 ) 𝑐𝑛 = ( 𝛿  𝛼 + 𝑣 𝛿  𝛼
𝐿
𝑛,𝑛−1 𝑚 𝑛,𝑛−1 𝑥 𝑛,𝑚 − 2)𝑐
𝑛−1 
Γ(2 − 𝛼) Γ(2 − 𝛼) Γ(2 − 𝛼) (∆𝑥) 𝑚
(∆𝑥)−𝛼 𝐷
 𝛼 𝐿
𝐷
+(𝑣 𝛿 + )𝑐𝑛−1
𝐿
𝑥 𝑛,𝑚 𝑚−1 + ( )𝑐
𝑛−1  
Γ(2 − 𝛼) (∆𝑥)2 (∆𝑥)2 𝑚+1 (5-17) 
−𝛼 𝑛−2 −𝛼 𝑚−1(∆𝑡) (∆𝑥)
−  (∑(𝑐𝑘+1 − 𝑐𝑘 )𝛿  𝛼 ) − 𝑣  [∑(𝑐𝑛−1 − 𝑐𝑛−1) 𝛿  𝛼 ] 
Γ(2 − 𝛼) 𝑚 𝑚 𝑛,𝑘 𝑥 Γ(2 − 𝛼) 𝑖 𝑖−1 𝑚,𝑖
𝑘=0 𝑖=1
Equation (5-17) is the explicit upwind finite difference scheme for the one-dimensional, non-reactive 
fractional advection-dispersion equation with Caputo fractional derivatives. The numerical scheme 
can be simplified by substituting functions as followings,  
103 
 
𝑛−2 𝑚−1
𝑏 𝑐𝑛−1 + 𝑑 𝑐𝑛−1 + 𝑓 𝑐𝑛−1 − ℎ (∑(𝑐𝑘+1 − 𝑐𝑘3 𝑚 3 𝑚−1 3 𝑚+1 3 𝑚 𝑚)𝛿
 𝛼
𝑛,𝑘) − 𝑙3 (∑(𝑐
𝑛−1
𝑖 − 𝑐
𝑛−1  𝛼 (5-18) 
𝑖−1 ) 𝛿𝑚,𝑖) 
𝑘=0 𝑖=1
= 𝑎 𝑐𝑛3 𝑚 
where, 
(∆𝑡)−𝛼
𝑎3 = 𝛿
 𝛼
Γ(2 − 𝛼) 𝑛,𝑛−1
, 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷𝐿
𝑏3 = 𝛿
 𝛼 + 𝑣 𝛿  𝛼 − , 
Γ(2 − 𝛼) 𝑛,𝑛−1 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
(∆𝑥)−𝛼 𝐷𝐿
𝑑  𝛼3 = 𝑣𝑥 𝛿 + , Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
𝐷𝐿
𝑓3 = , (∆𝑥)2
(∆𝑡)−𝛼
ℎ3 = , Γ(2 − 𝛼)
(∆𝑥)−𝛼
𝑙3 = 𝑣𝑥  Γ(2 − 𝛼)
5.1.2 Implicit upwind  
Following the same approach, the following is obtained for the implicit upwind scheme, 
𝑛−1 𝑚
(∆𝑡)−𝛼 (∆𝑥)−𝛼
[∑(𝑐𝑘+1 𝑘 𝛼𝑚 − 𝑐𝑚)𝛿𝑛,𝑘] + 𝑣𝑥 [∑(𝑐
𝑛 𝑛 𝛼
Γ(2 − 𝛼) Γ(2 − 𝛼) 𝑖
− 𝑐𝑖−1) 𝛿𝑚,𝑖] 
(5-19) 
𝑘=0 𝑖=1
𝑐𝑛 𝑛𝑚+1 − 2𝑐𝑚 + 𝑐
𝑛
𝑚−1
−𝐷𝐿 ( ) = 0 (∆𝑥)2
Reformulating the following can be obtained, 
𝑛−2
(∆𝑡)−𝛼 (∆𝑡)−𝛼
(𝑐𝑛 − 𝑐𝑛−1)𝛿  𝛼 +  (∑(𝑐𝑘+1 − 𝑐𝑘 )𝛿  𝛼 ) 
Γ(2 − 𝛼) 𝑚 𝑚 𝑛,𝑛−1 Γ(2 − 𝛼) 𝑚 𝑚 𝑛,𝑘
𝑘=0
(∆𝑥)−𝛼
𝑚−1
(∆𝑥)−𝛼 (5-20) 
+𝑣 (𝑐𝑛𝑥 𝑚 − 𝑐
𝑛
𝑚−1)𝛿
 𝛼 𝑛 𝑛  𝛼
Γ(2 − 𝛼) 𝑛,𝑚
 + 𝑣𝑥  [∑(𝑐 − 𝑐Γ(2 − 𝛼) 𝑖 𝑖−1
) 𝛿𝑚,𝑖] 
𝑖=1
𝑐𝑛𝑚+1 − 2𝑐
𝑛
𝑚 + 𝑐
𝑛
𝑚−1
−𝐷𝐿 ( ) = 0 (∆𝑥)2
Rearranging, 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷 (∆𝑡)−𝛼
( 𝛿  𝛼  𝛼
𝐿 𝑛  𝛼 𝑛−1
Γ(2 − 𝛼) 𝑛,𝑛−1
+ 𝑣𝑥 𝛿Γ(2 − 𝛼) 𝑛,𝑚
−
2)𝑐(∆𝑥) 𝑚
= ( 𝛿
Γ(2 − 𝛼) 𝑛,𝑛−1
)𝑐𝑚  
(∆𝑥)−𝛼 𝐷 𝐷
+(𝑣 𝛿  𝛼
𝐿 𝑛 𝐿
𝑥 𝑛,𝑚 + )𝑐𝑚−1 + ( )𝑐
𝑛  
Γ(2 − 𝛼) (∆𝑥)2 (∆𝑥)2 𝑚+1 (5-21) 
𝑛−2 𝑚−1
(∆𝑡)−𝛼 (∆𝑥)−𝛼
−  (∑(𝑐𝑘+1𝑚 − 𝑐
𝑘 )𝛿  𝛼 ) − 𝑣  [∑(𝑐𝑛𝑚 𝑛,𝑘 𝑥 𝑖 − 𝑐
𝑛
𝑖−1) 𝛿
 𝛼 ] 
Γ(2 − 𝛼) Γ(2 − 𝛼) 𝑚,𝑖
𝑘=0 𝑖=1
104 
 
Equation (5-21) is the implicit upwind finite difference scheme for the fractional advection-dispersion 
equation with Caputo fractional derivatives. The numerical scheme is simplified by substituting the 
following functions (Equation (5-18)),  
𝑛−2 𝑚−1
𝑎 𝑛−13𝑐𝑚 = 𝑏 𝑐
𝑛
3 𝑚 − 𝑑3𝑐
𝑛
𝑚−1 − 𝑓
𝑛 𝑘+1 𝑘  𝛼 𝑛 𝑛  𝛼
3𝑐𝑚+1 + ℎ3 (∑(𝑐𝑚 − 𝑐𝑚)𝛿𝑛,𝑘) + 𝑙 3 (∑(𝑐𝑖 − 𝑐𝑖−1) 𝛿𝑚,𝑖) (5-22) 
𝑘=0 𝑖=1
5.1.3 Upwind advection Crank-Nicolson scheme 
The first-order upwind advection Crank-Nicolson finite difference scheme is applied, where the time 
component remains the same as with the first-order implicit and explicit schemes (Equation (5-10)), 
but the space components change to, 
𝑚
(∆𝑥)−𝛼 ∑[0.5(𝑐𝑛−1 − 𝑐𝑛−1 𝑛 𝑛
 𝐶𝐷𝛼(𝑐(𝑥 , 𝑡 )) =  [ 𝑖 𝑖−1
) + 0.5(𝑐𝑖 − 𝑐𝑖−1)]]   (5-23) 0 𝑥 𝑚 𝑛 Γ(2 − 𝛼) 𝑖=1
{(𝑚 − 𝑖)1−𝛼 − (𝑚 − 𝑖 − 1)1−𝛼}
Simplifying using the function 𝛿𝛼𝑚,𝑖, 
𝑚
(∆𝑥)−𝛼
 𝐶𝐷𝛼0 𝑥 (𝑐(𝑥𝑚, 𝑡𝑛)) =  [∑(0.5(𝑐
𝑛−1 − 𝑐𝑛−1) + 0.5(𝑐𝑛 − 𝑐𝑛𝑖 𝑖−1 𝑖 𝑖−1)) 𝛿
𝛼
𝑚,𝑖] (5-24) Γ(2 − 𝛼)
𝑖=1
Substituting back into Equation (5-1), 
(∆𝑡)−𝛼
𝑛−1 𝑚
(∆𝑥)−𝛼
[∑(𝑐𝑘+1 − 𝑐𝑘 )𝛿𝛼 ] + 𝑣 [∑(0.5(𝑐𝑛−1𝑚 𝑚 𝑛,𝑘 𝑥 𝑖 − 𝑐
𝑛−1
𝑖−1 ) + 0.5(𝑐
𝑛 − 𝑐𝑛 )) 𝛿𝛼 ] 
Γ(2 − 𝛼) Γ(2 − 𝛼) 𝑖 𝑖−1 𝑚,𝑖
𝑘=0 𝑖=1 (5-25) 
𝑐𝑛−1𝑚+1 − 2𝑐
𝑛−1 + 𝑐𝑛−1𝑖𝑚 𝑚−1
−𝐷𝐿 ( ) = 0 (∆𝑥)2
Reformulating the following can be obtained, 
𝑛−2
(∆𝑡)−𝛼 (∆𝑡)−𝛼
(𝑐𝑛 − 𝑐𝑛−1)𝛿  𝛼 +  (∑(𝑐𝑘+1 𝑘  𝛼
Γ(2 − 𝛼) 𝑚 𝑚 𝑛,𝑛−1 Γ(2 − 𝛼) 𝑚
− 𝑐𝑚)𝛿𝑛,𝑘)
𝑘=0
(∆𝑥)−𝛼
+ 𝑣 (0.5(𝑐𝑛−1 − 𝑐𝑛−1𝑥 𝑚 𝑚−1) + 0.5(𝑐
𝑛 − 𝑐𝑛𝑚 𝑚−1)) 𝛿
 𝛼
Γ(2 − 𝛼) 𝑛,𝑚
  
(5-26) 
𝑚−1
(∆𝑥)−𝛼
+𝑣  [∑ (0.5(𝑐𝑛−1𝑥 𝑖 − 𝑐
𝑛−1
𝑖−1 ) + 0.5(𝑐
𝑛
𝑖 − 𝑐
𝑛
𝑖−1)) 𝛿
 𝛼
𝑚,𝑖] Γ(2 − 𝛼)
𝑖=1
𝑐𝑛−1 − 2𝑐𝑛−1 𝑛−1𝑚+1 𝑚 + 𝑐𝑚−1
−𝐷𝐿 ( ) = 0 (∆𝑥)2
Rearranging, 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 (5-27) 
( 𝛿  𝛼  𝛼 𝑛
Γ(2 − 𝛼) 𝑛,𝑛−1
+ 0.5𝑣𝑥 𝛿𝑛,𝑚)𝑐𝑚 = Γ(2 − 𝛼)
105 
 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷
( 𝛿  𝛼 − 0.5𝑣  𝛿  𝛼
𝐿
− )𝑐𝑛−1
Γ(2 − 𝛼) 𝑛,𝑛−1 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2 𝑚
(∆𝑥)−𝛼 𝐷𝐿 (∆𝑥)
−𝛼
+ (0.5 𝑣 𝛿  𝛼 𝑛−1  𝛼 𝑛𝑥 Γ(2 − 𝛼) 𝑛,𝑚
+ )𝑐𝑚−1 + 0.5𝑣(∆𝑥)2 𝑥
(  𝛿 ) 𝑐
Γ(2 − 𝛼) 𝑛,𝑚 𝑚−1
−𝛼 𝑛−2𝐷𝐿 (∆𝑡)
+ ( ) 𝑐𝑛−1 −  (∑(𝑐𝑘+1 𝑘  𝛼
(∆𝑥)2 𝑚+1 Γ(2 − 𝛼) 𝑚
− 𝑐𝑚)𝛿𝑛,𝑘)
𝑘=0
𝑚−1
(∆𝑥)−𝛼
− 𝑣𝑥  (∑ (0.5(𝑐
𝑛−1 − 𝑐𝑛−1𝑖 𝑖−1 ) + 0.5(𝑐
𝑛 − 𝑐𝑛  𝛼
Γ(2 − 𝛼) 𝑖 𝑖−1
)) 𝛿𝑚,𝑖) 
𝑖=1
Equation (5-27) is the upwind advection Crank-Nicolson finite difference scheme for the one-
dimensional, non-reactive fractional advection-dispersion equation (Caputo). The numerical scheme 
can be simplified by substituting functions as follows,  
𝑝 𝑐𝑛−13 𝑚 + 𝑞
𝑛−1
3𝑐𝑚−1 + 𝑟3𝑐
𝑛 𝑛−1
𝑚−1 + 𝑓3𝑐𝑚+1 (5-28) 
𝑛−2 𝑚−1
−ℎ  (∑(𝑐𝑘+1 − 𝑐𝑘 )𝛿  𝛼 ) − 𝑙  (∑ (0.5(𝑐𝑛−13 𝑚 𝑚 𝑛,𝑘 3 𝑖 − 𝑐
𝑛−1
𝑖−1 ) + 0.5(𝑐
𝑛 𝑛  𝛼
𝑖 − 𝑐𝑖−1)) 𝛿𝑚,𝑖) = 𝑜3𝑐
𝑛
𝑚 
𝑘=0 𝑖=1
where, 
(∆𝑡)−𝛼 (∆𝑥)−𝛼
𝑜3 = 𝛿
 𝛼  𝛼
Γ(2 − 𝛼) 𝑛,𝑛−1
+ 0.5𝑣𝑥 𝛿 , Γ(2 − 𝛼) 𝑛,𝑚
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷𝐿
𝑝 = 𝛿  𝛼3 𝑛,𝑛−1 − 0.5𝑣  𝛿
 𝛼
𝑥 − , Γ(2 − 𝛼) Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
(∆𝑥)−𝛼 𝐷
 𝛼 𝐿𝑞3 = 0.5𝑣𝑥  𝛿 + , Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
(∆𝑥)−𝛼
𝑟3 = 0.5𝑣
 𝛼
𝑥  𝛿 , Γ(2 − 𝛼) 𝑛,𝑚
5.1.4 Explicit upwind-downwind weighted scheme  
The first-order explicit upwind-downwind weighted scheme, as described in Section 2.2.5, is applied 
and results in the following changes in the space fractional derivative approximation, where the ratio 
of upwind to downwind is controlled by 𝜃, where 0 ≤ 𝜃 ≤ 1, 
𝑚
(∆𝑥)−𝛼
 𝐶 𝛼0𝐷𝑥 (𝑐(𝑥 , 𝑡 )) =  [∑[𝜃(𝑐
𝑛−1 − 𝑐𝑛−1𝑚 𝑛 𝑖 𝑖−1 ) + (1 − 𝜃)(𝑐
𝑛−1 − 𝑐𝑛−1 𝛼𝑖+1 𝑖 )] 𝛿𝑚,𝑖] (5-29) Γ(2 − 𝛼)
𝑖=1
Substituting into Equation (5-1), 
𝑛−1
(∆𝑡)−𝛼
[∑(𝑐𝑘+1𝑚 − 𝑐
𝑘
𝑚)𝛿
𝛼
Γ(2 − 𝛼) 𝑛,𝑘
]
𝑘=0
(∆𝑥)−𝛼
𝑚
(5-30) 
+ 𝑣𝑥 [∑[𝜃(𝑐
𝑛−1 𝑛−1 𝑛−1 𝑛−1 𝛼
Γ(2 − 𝛼) 𝑖
− 𝑐𝑖−1 ) + (1 − 𝜃)(𝑐𝑖+1 − 𝑐𝑖 )] 𝛿𝑚,,𝑖] 
𝑖=1
𝑐𝑛−1 𝑛−1 𝑛−1𝑚+1 − 2𝑐𝑖𝑚 + 𝑐𝑚−1
−𝐷𝐿 ( ) = 0 (∆𝑥)2
106 
 
Reformulating the following can be obtained, 
𝑛−2
(∆𝑡)−𝛼 (∆𝑡)−𝛼
(𝑐𝑛 − 𝑐𝑛−1)𝛿  𝛼 𝑘+1 𝑘  𝛼
Γ(2 − 𝛼) 𝑚 𝑚 𝑛,𝑛−1
+  (∑(𝑐𝑚 − 𝑐𝑚)𝛿𝑛,𝑘) Γ(2 − 𝛼)
𝑘=0
(∆𝑥)−𝛼
+𝑣 (𝜃(𝑐𝑛−1 − 𝑐𝑛−1 ) + (1 − 𝜃)(𝑐𝑛−1 − 𝑐𝑛−1)) 𝛿  𝛼𝑥   Γ(2 − 𝛼) 𝑚 𝑚−1 𝑚+1 𝑚 𝑛,𝑚 (5-31) 
(∆𝑥)−𝛼
𝑚−1
+𝑣  [∑ (𝜃(𝑐𝑛−1 − 𝑐𝑛−1) + (1 − 𝜃)(𝑐𝑛−1 𝑛−1𝑥 𝑖 𝑖−1 𝑖+1 − 𝑐𝑖 )) 𝛿
 𝛼
Γ(2 − 𝛼) 𝑚,𝑖
] 
𝑖=1
𝑐𝑛−1 − 2𝑐𝑛−1 + 𝑐𝑛−1𝑚+1 𝑚 𝑚−1
−𝐷𝐿 ( ) = 0 (∆𝑥)2
Rearranging, 
(∆𝑡)−𝛼 (∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷
( 𝛿  𝛼 ) 𝑐𝑛 = ( 𝛿  𝛼 + 𝑣 (2𝜃 − 1) 𝛿  𝛼
𝐿
− )𝑐𝑛−1 
Γ(2 − 𝛼) 𝑛,𝑛−1 𝑚 Γ(2 − 𝛼) 𝑛,𝑛−1 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2 𝑚
(∆𝑥)−𝛼 𝐷𝐿 (∆𝑥)
−𝛼 𝐷𝐿
+(𝑣𝑥𝜃 𝛿
 𝛼
𝑛,𝑚 + 2)𝑐
𝑛−1
𝑚−1 + (𝑣𝑥(1 − 𝜃) 𝛿
 𝛼 𝑛−1
Γ(2 − 𝛼) (∆𝑥) Γ(2 − 𝛼) 𝑛,𝑚
+ )𝑐
(∆𝑥)2 𝑚+1
𝑛−2 (5-32) 
(∆𝑡)−𝛼
−  (∑(𝑐𝑘+1𝑚 − 𝑐
𝑘
𝑚)𝛿
 𝛼
𝑛,𝑘)Γ(2 − 𝛼)
𝑘=0
𝑚−1
(∆𝑥)−𝛼
− 𝑣𝑥  [∑ (𝜃(𝑐
𝑛−1
𝑖 − 𝑐
𝑛−1
𝑖−1 ) + (1 − 𝜃)(𝑐
𝑛−1
𝑖+1 − 𝑐
𝑛−1  𝛼
Γ(2 − 𝛼) 𝑖
)) 𝛿𝑚,𝑖] 
𝑖=1
Equation (5-32) is the explicit upwind-downwind weighted finite difference scheme for the fractional 
advection-dispersion equation (Caputo). The numerical scheme can be simplified by substituting 
functions as shown,  
𝑛−2
𝑠 𝑐𝑛−1 + 𝑢 𝑐𝑛−13 𝑚 3 𝑚−1 + 𝑣
𝑛−1 𝑘+1 𝑘  𝛼
3𝑐𝑚+1 − ℎ3 (∑(𝑐𝑚 − 𝑐𝑚)𝛿𝑛,𝑘) 
𝑘=0 (5-33) 
𝑚−1
−𝑙 [∑ (𝜃 (𝑐𝑛−1 − 𝑐𝑛−1) + (1 − 𝜃)(𝑐𝑛−1 − 𝑐𝑛−1)) 𝛿  𝛼 ] = 𝑎 𝑐𝑛3 𝑖 𝑖−1 𝑖+1 𝑖 𝑚,𝑖 3 𝑚 
𝑖=1
where, 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷
𝑠 = 𝛿  𝛼  𝛼
𝐿
3 Γ(2 − 𝛼) 𝑛,𝑛−1
+ 𝑣𝑥(2𝜃 − 1) 𝛿𝑛,𝑚 −  Γ(2 − 𝛼) (∆𝑥)2
(∆𝑥)−𝛼 𝐷𝐿
𝑢3 = 𝑣
 𝛼
𝑥𝜃 𝛿 +  Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
(∆𝑥)−𝛼 𝐷𝐿
𝑣3 = 𝑣𝑥(1 − 𝜃) 𝛿
 𝛼 +  
Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
5.1.5 Implicit upwind-downwind weighted scheme  
Similarly, the implicit upwind-downwind weighted scheme, presented in Section 2.2.6, is applied, 
where the ratio of upwind to downwind is controlled by 𝜃, where 0 ≤ 𝜃 ≤ 1, 
107 
 
(∆𝑡)−𝛼
𝑛−1 𝑚
(∆𝑥)−𝛼
[∑(𝑐𝑘+1𝑚 − 𝑐
𝑘
𝑚)𝛿
𝛼
𝑛,𝑘] + 𝑣𝑥 [∑[𝜃(𝑐
𝑛
𝑖 − 𝑐
𝑛
𝑖−1) + (1 − 𝜃)(𝑐
𝑛 𝑛 𝛼
Γ(2 − 𝛼) Γ(2 − 𝛼) 𝑖+1
− 𝑐𝑖 )] 𝛿𝑚,,𝑖] 
(5-34) 
𝑘=0 𝑖=1
𝑐𝑛 − 2𝑐𝑛 + 𝑐𝑛𝑚+1 𝑖𝑚 𝑚−1
−𝐷𝐿 ( ) = 0 (∆𝑥)2
Reformulating the following can be obtained, 
(∆𝑡)−𝛼 (∆𝑡)−𝛼
𝑛−2
(𝑐𝑛 𝑛−1  𝛼 𝑘+1 𝑘  𝛼
Γ(2 − 𝛼) 𝑚
− 𝑐𝑚 )𝛿𝑛,𝑛−1 +  (∑(𝑐 − 𝑐 )𝛿 )Γ(2 − 𝛼) 𝑚 𝑚 𝑛,𝑘
𝑘=0
(∆𝑥)−𝛼
+ 𝑣 (𝜃(𝑐𝑛 − 𝑐𝑛 ) + (1 − 𝜃)(𝑐𝑛 − 𝑐𝑛  𝛼𝑥 Γ(2 − 𝛼) 𝑚 𝑚−1 𝑚+1 𝑚
))𝛿𝑛,𝑚 +  
(∆𝑥)−𝛼
𝑚−1
𝑐𝑛𝑚+1 − 2𝑐
𝑛
𝑚 + 𝑐
𝑛
𝑚−1
𝑣𝑥  [∑(𝜃(𝑐
𝑛
𝑖 − 𝑐
𝑛
𝑖−1) + (1 − 𝜃)(𝑐
𝑛 𝑛  𝛼
Γ(2 − 𝛼) 𝑖+1
− 𝑐𝑖 )) 𝛿𝑚,𝑖] − 𝐷𝐿 ( 2 ) = 0 (∆𝑥)
𝑖=1 (5-35) 
Rearranging, 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷𝐿 (∆𝑡)
−𝛼
( 𝛿  𝛼𝑛,𝑛−1 + 𝑣𝑥(2𝜃 − 1) 𝛿
 𝛼 − )𝑐𝑛 = ( 𝛿  𝛼 )𝑐𝑛−1 
Γ(2 − 𝛼) Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2 𝑚 Γ(2 − 𝛼) 𝑛,𝑛−1 𝑚
(∆𝑥)−𝛼 𝐷 (∆𝑥)−𝛼𝐿 𝐷𝐿
+(𝑣𝑥𝜃 𝛿
 𝛼 𝑛
𝑛,𝑚 + )𝑐𝑚−1 + (𝑣𝑥(1 − 𝜃) 𝛿
 𝛼 + )𝑐𝑛  
Γ(2 − 𝛼) (∆𝑥)2 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2 𝑚+1 (5-36) 
𝑛−2 𝑚−1
(∆𝑡)−𝛼 (∆𝑥)−𝛼
−  (∑(𝑐𝑘+1 𝑘  𝛼 𝑛 𝑛 𝑛 𝑛𝑚 − 𝑐𝑚)𝛿𝑛,𝑘) − 𝑣𝑥  [∑(𝜃(𝑐𝑖 − 𝑐𝑖−1) + (1 − 𝜃)(𝑐𝑖+1 − 𝑐𝑖 )) 𝛿
 𝛼 ] 
Γ(2 − 𝛼) Γ(2 − 𝛼) 𝑚,𝑖
𝑘=0 𝑖=1
Equation (5-36) is the implicit upwind-downwind weighted finite difference scheme for the fractional 
advection-dispersion equation with Caputo fractional derivative. The numerical scheme is simplified 
with the following functions (Equations (5-33) and (5-18)),  
𝑛−2 𝑚−1
𝑎 𝑐𝑛−1 − ℎ (∑(𝑐𝑘+1 − 𝑐𝑘 )𝛿  𝛼 ) − 𝑙 [∑(𝜃 (𝑐𝑛 𝑛 𝑛 𝑛  𝛼3 𝑚 3 𝑚 𝑚 𝑛,𝑘 3 𝑖 − 𝑐𝑖−1) + (1 − 𝜃)(𝑐𝑖+1 − 𝑐𝑖 )) 𝛿𝑚,𝑖] (5-37) 
𝑘=0 𝑖=1
= 𝑠 𝑐𝑛 𝑛3 𝑚 − 𝑢3𝑐𝑚−1 − 𝑣3𝑐
𝑛
𝑚+1 
This concludes the formulation of the numerical approximations schemes to be investigated for the 
fractional advection-dispersion equation with Caputo fractional derivative. In the following section, 
the numerical stability of each scheme will be analysed. 
5.2 Numerical stability analysis 
The numerical stability method used in Section 2.3 for the local operator numerical approximation 
schemes, will also be applied to the developed numerical approximation schemes for the fractional 
advection-dispersion equation with Caputo fractional derivative. The numerical stability for the 
upwind schemes are evaluated to validate their use in solving the fractional advection-dispersion 
equation with a Caputo fractional definition.  
108 
 
5.2.1 Explicit upwind  
The developed finite difference explicit upwind numerical scheme presented in Section 5.1.1 is 
applied. Substituting induction method terms results in: 
𝑛−2
𝑏 ?̂? 𝑒𝑗𝑘𝑖𝑥3 𝑛−1 + 𝑑
𝑗𝑘𝑖(𝑥−∆𝑥)
3?̂?𝑛−1𝑒 + 𝑓3?̂? 𝑒
𝑗𝑘𝑖𝑥(𝑥+∆𝑥) − ℎ (∑(?̂? 𝑒𝑗𝑘𝑚𝑥 − ?̂? 𝑒𝑗𝑘𝑚𝑥  𝛼𝑛−1 3 𝑘+1 𝑘 )𝛿𝑛,𝑘)
𝑘=0
𝑚−1 (5-38) 
− 𝑙 (∑(?̂? 𝑒𝑗𝑘𝑖𝑥 − ?̂? 𝑒𝑗𝑘𝑖(𝑥−∆𝑥)) 𝛿  𝛼3 𝑛−1 𝑛−1 𝑚,𝑖) = 𝑎 ?̂? 𝑒
𝑗𝑘𝑖𝑥
3 𝑛  
𝑖=1
Multiple out, 
𝑏 ?̂? 𝑒𝑗𝑘𝑖𝑥3 𝑛−1 + 𝑑3?̂?
𝑗𝑘𝑖𝑥 −𝑗𝑘𝑖∆𝑥 𝑗𝑘𝑖𝑥 𝑗𝑘𝑖∆𝑥
𝑛−1𝑒 𝑒 + 𝑓3?̂?𝑛−1𝑒 𝑒  
𝑛−2 𝑚−1
−ℎ (∑(?̂? 𝑗𝑘𝑚𝑥 𝑗𝑘𝑚𝑥  𝛼3 𝑘+1𝑒 − ?̂?𝑘𝑒 )𝛿𝑛,𝑘) − 𝑙3 (∑(?̂? 𝑒
𝑗𝑘𝑖𝑥 − ?̂? 𝑒𝑗𝑘𝑖𝑥𝑒−𝑗𝑘𝑖∆𝑥) 𝛿  𝛼𝑛−1 𝑛−1 𝑚,𝑖) (5-39) 
𝑘=0 𝑖=1
= 𝑎 ?̂? 𝑒𝑗𝑘𝑖𝑥3 𝑛  
Divide by 𝑒𝑗𝑘𝑖𝑥, 
𝑎3?̂?𝑛 = 𝑏
−𝑗𝑘𝑖∆𝑥 𝑗𝑘𝑖∆𝑥
3?̂?𝑛−1 + 𝑑3?̂?𝑛−1𝑒 + 𝑓3?̂?𝑛−1𝑒  
𝑛−2 𝑚−1
(5-40) 
−ℎ3 (∑(?̂?𝑘+1 − ?̂?
 𝛼 −𝑗𝑘𝑖∆𝑥  𝛼
𝑘)𝛿𝑛,𝑘) − 𝑙3 (∑(?̂?𝑛−1 − ?̂?𝑛−1𝑒 )𝛿𝑚,𝑖) 
𝑘=0 𝑖=1
The induction numerical stability analysis is performed in two parts; firstly it is proved for a set ∀ 𝑛 > 1,  
|?̂?𝑛| < |?̂?𝑜|  
If 𝑛 = 1, then 
𝑚−1
𝑎 ?̂? = 𝑏 ?̂? + 𝑑 ?̂? 𝑒−𝑗𝑘𝑖∆𝑥 + 𝑓 𝑗𝑘𝑖∆𝑥 −𝑗𝑘𝑖∆𝑥  𝛼 (5-41) 3 1 3 0 3 0 3?̂?0𝑒 − 𝑙3 (∑(?̂?0 − ?̂?0𝑒 )𝛿𝑚,𝑖) 
𝑖=1
A subset for 𝑚 is now considered, where 𝑚 = 1, 
𝑎3?̂?1 = 𝑏 ?̂? + 𝑑 ?̂? 𝑒
−𝑗𝑘𝑖∆𝑥
3 0 3 0 + 𝑓 ?̂?
𝑗𝑘𝑖∆𝑥
3 0𝑒  (5-42)  
 
Simplifying, 
𝑎 ?̂? = (𝑏 + 𝑑 𝑒−𝑗𝑘𝑖∆𝑥 + 𝑓 𝑒𝑗𝑘𝑖∆𝑥3 1 3 3 3 )?̂?0 (5-43) 
Rearranging, 
?̂?1 𝑏 + 𝑑 𝑒
−𝑗𝑘𝑖∆𝑥 + 𝑓 𝑒𝑗𝑘𝑖∆𝑥3 3 3
=  (5-44) 
?̂?0 𝑎3
Apply a norm on both sides, 
|?̂?1| |𝑏 | + |−𝑑 𝑒
−𝑗𝑘𝑖∆𝑥
3 3 | + |−𝑓 𝑒
𝑗𝑘𝑖∆𝑥
3 |
=  
|?̂? (5-45) 0| |𝑎3|
The condition thus becomes, 
109 
 
|?̂?1|
< 1 
|?̂?  0|
Remembering |𝑒𝑛| = 1, the condition becomes 
|𝑏3| + |𝑑3| + |𝑓3|
< 1  
|𝑎3|
It can be concluded that |?̂?1| < |?̂?𝑜|, when 
|𝑏3| + |𝑑3| + |𝑓3|
< 1 
|𝑎3|
The term is expanded using the simplification terms associated with Equation (5-18), 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 𝛼 2𝐷 (∆𝑥)
−𝛼 𝐷 𝐷
| 𝛿𝑛,𝑛−1 + 𝑣𝑥 𝛿
 𝛼
𝑛,𝑚 −
𝐿
2| + |𝑣 𝛿
 𝛼 𝐿
𝑥 𝑛,𝑚 + | + |
𝐿 |
Γ(2 − 𝛼) Γ(2 − 𝛼) (∆𝑥) Γ(2 − 𝛼) (∆𝑥)2 (∆𝑥)2
−𝛼 < 1 ( ) (5-46) ∆𝑡
| 𝛿  𝛼
Γ(2 − 𝛼) 𝑛,𝑛−1
|
The assumption is made where, 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷
𝛿  𝛼 + 𝑣 𝛿  𝛼
𝐿
>  (5-47) 
Γ(2 − 𝛼) 𝑛,𝑛−1 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
Then, 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷 (∆𝑥)−𝛼 𝐷 𝐷
𝛿  𝛼 + 𝑣 𝛿  𝛼 𝐿  𝛼 𝐿 𝐿
Γ(2 − 𝛼) 𝑛,𝑛−1 𝑥 Γ(2 − 𝛼) 𝑛,𝑚
− + 𝑣
(∆𝑥)2 𝑥
𝛿𝑛,𝑚 + +Γ(2 − 𝛼) (∆𝑥)2 (∆𝑥)2
(∆𝑡)−𝛼
< 1 (5-48) 
𝛿  𝛼
Γ(2 − 𝛼) 𝑛,𝑛−1
Simplifying, under this assumption (Equation (5-47)), the explicit upwind numerical scheme for the 
fractional advection-dispersion equation with Caputo fractional derivative is unstable. 
When the complementary assumption is made where, 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷
𝛿  𝛼
𝐿
+ 𝑣 𝛿  𝛼 <  (5-49) 
Γ(2 − 𝛼) 𝑛,𝑛−1 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
Then, 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷 (∆𝑥)−𝛼
− 𝛿  𝛼 − 𝑣 𝛿  𝛼 + 𝐿  𝛼
𝐷𝐿 𝐷𝐿
Γ(2 − 𝛼) 𝑛,𝑛−1 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
+ 𝑣𝑥 𝛿𝑛,𝑚 + +Γ(2 − 𝛼) (∆𝑥)2 (∆𝑥)2
−𝛼 < 1 (∆𝑡) (5-50) 
𝛿  𝛼
Γ(2 − 𝛼) 𝑛,𝑛−1
Simplifying, 
2𝐷𝐿 (∆𝑡)
−𝛼
< 𝛿  𝛼  
(∆𝑥)2 Γ(2 − 𝛼) 𝑛,𝑛−1 (5-51) 
Under this assumption, the finite difference first-order upwind (explicit) numerical scheme for the 
fractional advection-dispersion equation with Caputo fractional derivative is conditionally stable. 
110 
 
A subset for 𝑚 is now considered for all 𝑚 > 1, 
𝑚−1
𝑎3?̂?1 = 𝑏3?̂? + 𝑑 ?̂? 𝑒
−𝑗𝑘𝑖∆𝑥
0 3 0 + 𝑓3?̂?
𝑗𝑘𝑖∆𝑥
0𝑒 − 𝑙 (∑(?̂? − ?̂? 𝑒
−𝑗𝑘𝑖∆𝑥) 𝛿  𝛼 ) (5-52) 3 0 0 𝑚,𝑖
𝑖=1
Simplifying, 
𝑚−1
𝑎 ?̂? = (𝑏 −𝑗𝑘𝑖∆𝑥 𝑗𝑘𝑖∆𝑥3 1 3 + 𝑑3𝑒 + 𝑓3𝑒 − 𝑙3(1 − 𝑒
−𝑗𝑘𝑖∆𝑥) ∑ 𝛿  𝛼𝑚,𝑖) ?̂?0 (5-53) 
𝑖=1
Remembering 𝛿  𝛼 = (𝑚 − 𝑖)1−𝛼𝑚,𝑖 − (𝑚 − 𝑖 − 1)
1−𝛼, and substitute back into Equation (5-53), 
𝑚−1
𝑎3?̂?1 = (𝑏3 + 𝑑 𝑒
−𝑗𝑘𝑖∆𝑥
3 + 𝑓 𝑒
𝑗𝑘𝑖∆𝑥
3 − 𝑙3(1 − 𝑒
−𝑗𝑘𝑖∆𝑥) ∑(𝑚 − 𝑖)1−𝛼 − (𝑚 − 𝑖 − 1)1−𝛼) ?̂?0 (5-54) 
𝑖=1
where, expanding the summation the following is obtained 
𝑚−1
∑((𝑚 − 𝑖)1−𝛼 − (𝑚 − 𝑖 − 1)1−𝛼) = {𝑚1−𝛼 − (𝑚 − 1)1−𝛼} + {(𝑚 − 1)1−𝛼 − (𝑚 − 2)1−𝛼} 
𝑖=1
                     +{(𝑚 − 2)1−𝛼 − (𝑚 − 3)1−𝛼} 
                                                                         +{(𝑚 − 3)1−𝛼 − (𝑚 − 4)1−𝛼} + ⋯+ {−(−1)1−𝛼}  
= 𝑚1−𝛼 − (−1)1−𝛼 
Simplifying and substituting,  
𝑎3?̂?1 = (𝑏 + 𝑑 𝑒
−𝑗𝑘𝑖∆𝑥 + 𝑓 𝑒𝑗𝑘𝑖∆𝑥 − 𝑙 (1 − 𝑒−𝑗𝑘𝑖∆𝑥3 3 3 3 )(𝑚
1−𝛼 − (−1)1−𝛼)) ?̂?0 (5-55) 
Let a new function simplify to, 
𝜙 = 𝑘𝑖∆𝑥 
where, 
𝑒−𝑗𝜙 = 𝑒−𝑗𝑘𝑖∆𝑥 
Remembering Euler’s formula for complex numbers,  
   1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙 = 1 − 𝑒−𝑖𝜙 
Substituting back into Equation (5-55): 
𝑎 ?̂? = (𝑏 + 𝑑 𝑒−𝑗𝑘𝑖∆𝑥 + 𝑓 𝑒𝑗𝑘𝑖∆𝑥3 1 3 3 3 − 𝑙3(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)(𝑚
1−𝛼 − (−1)1−𝛼)) ?̂?0 (5-56) 
A norm is applied on both sides, 
|𝑎 ||?̂? | = (|𝑏 | + |𝑑 𝑒−𝑗𝑘𝑖∆𝑥3 1 3 3 | + |𝑓 𝑒
𝑗𝑘𝑖∆𝑥
3 |  
(5-57) 
+ |−𝑙3| (|1 − cos𝜙| + 𝑖 |𝑠𝑖𝑛 𝜙|)|𝑚
1−𝛼 − (−1)1−𝛼|)|?̂?0| 
Remembering |𝑒𝑛| = 1,  
|𝑎 1−𝛼3||?̂?1| = (|𝑏3| + |𝑑3| + |𝑓3| + |𝑙3| (|1 − cos𝜙| + 𝑖 |𝑠𝑖𝑛 𝜙|)|𝑚 − (−1)
1−𝛼|)|?̂?0| (5-58) 
and, 
|1 − cos𝜙| + 𝑖 |𝑠𝑖𝑛 𝜙| = (1 − cos𝜙)2 + 𝑠𝑖𝑛2 𝜙 
                                                          = 1 − 2 cos𝜙 + cos2𝜙 + 𝑠𝑖𝑛2 𝜙 
                       = 2 − 2 cos𝜙 
111 
 
Thus, 
|𝑎3||?̂?1| = (|𝑏3| + |𝑑3| + |𝑓3| + |𝑙3| (2 − 2 cos𝜙)|𝛽𝑚|)|?̂?0| (5-59) 
where, 
𝛽 = 𝑚1−𝛼𝑚 − (−1)
1−𝛼
 
Rearranging, 
|?̂?1| |𝑏3| + |𝑑3| + |𝑓3| + |𝑙3|(2 − 2 cos𝜙)|𝛽𝑚|
=  (5-60) 
|?̂?0| |𝑎3|
The condition required can be expressed as, 
|?̂?1|
< 1  
|?̂?0|
The condition becomes, 
|𝑏3| + |𝑑3| + |𝑓3| + |𝑙3|(2 − 2 cos𝜙)|𝛽𝑚|
< 1 (5-61) 
|𝑎3|
The term is expanded using the simplification terms associated with Equation (5-18), 
(∆𝑡)−𝛼  𝛼 (∆𝑥)
−𝛼
 𝛼 2𝐷 (∆𝑥)
−𝛼 𝐷 𝐷
| 𝛿 + 𝑣 𝛿 𝐿  𝛼 𝐿 𝐿
Γ(2 − 𝛼) 𝑛,𝑛−1 𝑥 Γ(2 − 𝛼) 𝑛,𝑚
− 2| + |𝑣𝑥 𝛿𝑛,𝑚 +(∆𝑥) Γ(2 − 𝛼) (∆𝑥)2
| + | |
(∆𝑥)2
( )
(∆𝑥)−𝛼
+ |𝑣𝑥 | (2 − 2 cos𝜙)(|𝛽Γ(2 − 𝛼) 𝑚
|) (5-62) 
(∆𝑡)−𝛼
< 1 
| 𝛿  𝛼 |
Γ(2 − 𝛼) 𝑛,𝑛−1
The assumption is made where, 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷
𝛿  𝛼  𝛼
𝐿
𝑛,𝑛−1 + 𝑣𝑥 𝛿𝑛,𝑚 >  (5-63) Γ(2 − 𝛼) Γ(2 − 𝛼) (∆𝑥)2
Then, 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷 (∆𝑥)−𝛼 𝐷 𝐷
𝛿  𝛼 + 𝑣 𝛿  𝛼 𝐿  𝛼 𝐿 𝐿
Γ(2 − 𝛼) 𝑛,𝑛−1 𝑥 Γ(2 − 𝛼) 𝑛,𝑚
− + 𝑣
(∆𝑥)2 𝑥
𝛿𝑛,𝑚 + +Γ(2 − 𝛼) (∆𝑥)2 (∆𝑥)2
(
(∆𝑥)−𝛼
)
+𝑣𝑥 (2 − 2 cos𝜙)𝛽Γ(2 − 𝛼) 𝑚 (5-64) 
< 1 
(∆𝑡)−𝛼
𝛿  𝛼
Γ(2 − 𝛼) 𝑛,𝑛−1
Simplifying, under this assumption (Equation (5-63)), the finite difference first-order upwind (explicit) 
numerical scheme for the fractional advection-dispersion equation with Caputo fractional derivative 
is unstable. 
When the opposite assumption is made, 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷𝐿
𝛿  𝛼𝑛,𝑛−1 + 𝑣 𝛿
 𝛼 <  (5-65) 
Γ(2 − 𝛼) Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
Then, 
112 
 
(∆𝑡)−𝛼  𝛼 (∆𝑥)
−𝛼
 𝛼 2𝐷𝐿 (∆𝑥)
−𝛼
 𝛼 𝐷𝐿 𝐷− 𝛿𝑛,𝑛−1 − 𝑣 𝛿𝑛,𝑚 + 2 + 𝑣 𝛿 + +
𝐿
Γ(2 − 𝛼) Γ(2 − 𝛼) (∆𝑥) Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2 (∆𝑥)2
( )
(∆𝑥)−𝛼
+𝑣 (2 − 2 cos𝜙)𝛽
Γ(2 − 𝛼) 𝑚 (5-66) 
(∆𝑡)−𝛼
< 1 
𝛿  𝛼
Γ(2 − 𝛼) 𝑛,𝑛−1
Simplifying, 
2𝐷 −𝛼𝐿 (∆𝑡)
<  
(∆𝑥)2 Γ(2 − 𝛼) (5-67) 
Under this assumption, the finite difference first-order upwind (explicit) numerical scheme for the 
fractional advection-dispersion equation with Caputo fractional derivative is conditionally stable. 
Furthermore, making the assumption that |?̂?𝑛−1| < |?̂?𝑜| is true for all time steps, the second part of 
the numerical stability analysis is to demonstrate for a set ∀ 𝑛 ≥ 1, that 
|?̂?𝑛| < |?̂?𝑜|  
Rearranging Equation (5-40) for ?̂?𝑛, 
𝑚−1
𝑎3?̂?𝑛 = (𝑏3 + 𝑑 𝑒
−𝑗𝑘𝑖∆𝑥
3 + 𝑓 𝑒
𝑗𝑘𝑖∆𝑥 − 𝑙 (1 − 𝑒−𝑗𝑘𝑖∆𝑥3 3 )(∑ 𝛿
 𝛼
𝑚,𝑖)) ?̂?𝑛−1 
𝑖=1 (5-68) 
𝑛−2
−ℎ3 (∑(?̂?𝑘+1 − ?̂?𝑘)𝛿
 𝛼
𝑛,𝑘) 
𝑘=0
Following a similar process as previously (Equation (5-56)), 
𝑎3?̂?
−𝑗𝑘𝑖∆𝑥 𝑗𝑘𝑖∆𝑥
𝑛 = (𝑏3 + 𝑑3𝑒 + 𝑓3𝑒 − 𝑙3(1 − cos𝜙 + 𝑖 sin𝜙)(𝑚
1−𝛼 − (−1)1−𝛼)) ?̂?𝑛−1 
𝑛−2 (5-69) 
−ℎ3 (∑(?̂?
 𝛼
𝑘+1 − ?̂?𝑘)𝛿𝑛,𝑘) 
𝑘=0
Taking the norm on both sides, 
(𝑏 + 𝑑 𝑒−𝑗𝑘𝑖∆𝑥3 3 + 𝑓 𝑒
𝑗𝑘𝑖∆𝑥 − 𝑙 1−𝛼 1−𝛼
3 3
(1 − cos𝜙 + 𝑖 sin𝜙)(𝑚 − (−1) )) ?̂?𝑛−1
|𝑎3?̂?𝑛| = |
𝑛−2
| || (5-70) 
−ℎ (∑(?̂? − ?̂? )𝛿  𝛼3 𝑘+1 𝑘 𝑛,𝑘)
𝑘=0
Thus, 
|𝑎3||?̂?𝑛| < |𝑏3 + 𝑑3𝑒
−𝑗𝑘𝑖∆𝑥 + 𝑓 𝑒𝑗𝑘𝑖∆𝑥 − 𝑙 1−𝛼 1−𝛼
3 3
(1 − cos𝜙 + 𝑖 sin𝜙)(𝑚 − (−1) )||?̂?𝑛−1| 
𝑛−2
(5-71) 
+|ℎ | |∑(?̂? − ?̂? )𝛿  𝛼3 𝑘+1 𝑘 𝑛,𝑘| 
𝑘=0
Remembering that it has been proved that for a set ∀ 𝑛 ≥ 1,   
|?̂?𝑛−1| < |?̂?𝑜| 
Thus, 
113 
 
|𝑎 ||?̂? | < |𝑏 + 𝑑 𝑒−𝑗𝑘𝑖∆𝑥 + 𝑓 𝑒𝑗𝑘𝑖∆𝑥3 𝑛 3 3 − 𝑙3(1 − cos𝜙 + 𝑖 sin𝜙)(𝑚
1−𝛼 − (−1)1−𝛼)||?̂?
3 𝑛−1
|
𝑛−2
+ |ℎ3| |∑(?̂?𝑘+1 − ?̂?
 𝛼
𝑘)𝛿𝑛,𝑘|    
𝑘=0
(5-72) 
        < |𝑏3 + 𝑑
−𝑗𝑘𝑖∆𝑥 𝑗𝑘𝑖∆𝑥
3𝑒 + 𝑓 𝑒 − 𝑙3(1 − cos𝜙 + 𝑖 sin𝜙)(𝑚
1−𝛼 − (−1)1−𝛼)||?̂?
3 0
|
𝑛−2
+ |ℎ3| |∑(?̂?𝑘+1 − ?̂?𝑘)𝛿
 𝛼
𝑛,𝑘| 
𝑘=0
and, it can be inferred that, 
|𝑎3||?̂?𝑛| < |𝑏
−𝑗𝑘𝑖∆𝑥 𝑗𝑘𝑖∆𝑥
3 + 𝑑3𝑒 + 𝑓 𝑒 − 𝑙3(1 − cos𝜙 + 𝑖 sin𝜙)(𝑚
1−𝛼 − (−1)1−𝛼)||?̂?0| 3
𝑛−2 (5-73) 
+|ℎ3| |∑(?̂?𝑘+1 − ?̂?𝑘)𝛿
 𝛼
𝑛,𝑘| 
𝑘=0
The remaining summation is considered at the upper limit, 
𝑛−2 𝑛−2
?̂?𝑘
|∑(?̂?𝑘+1 − ?̂? )𝛿
 𝛼
𝑘 𝑛,𝑘| < ∑|?̂?𝑘+1| (|1 − |)𝛿
 𝛼
?̂? 𝑛,𝑘
 (5-74) 
𝑘+1
𝑘=0 𝑘=0
Subset 𝑘 will follow the same assumption made for a set ∀ 𝑛 ≥ 1, where 
𝑛−2 𝑛−2
?̂?𝑘
∑?̂?𝑘+1 (1 − )𝛿
 𝛼
𝑛,𝑘 < |?̂?0|∑ 𝛿
 𝛼
?̂? 𝑛,𝑘
 (5-75) 
𝑘+1
𝑘=0 𝑘=0
Substituting back into Equation (5-73), 
|𝑎3||?̂?𝑛| < |𝑏
−𝑗𝑘𝑖∆𝑥 𝑗𝑘𝑖∆𝑥 1−𝛼 1−𝛼
3 + 𝑑3𝑒 + 𝑓 𝑒 − 𝑙3(1 − cos𝜙 + 𝑖 sin𝜙)(𝑚 − (−1) )||?̂?0| 3
𝑛−2 (5-76) 
+|ℎ3||?̂? |∑ 𝛿
 𝛼
0 𝑛,𝑘 
𝑘=0
Expanding the summation, 
𝑛−2 𝑛−2
∑𝛿  𝛼 = ∑((𝑛 − 𝑘)1−𝛼𝑛,𝑘 − (𝑛 − 𝑘 − 1)
1−𝛼) 
𝑘=0 𝑘=0
                 = {𝑛1−𝛼 − (𝑛 − 1)1−𝛼} + {(𝑛 − 1)1−𝛼 − (𝑛 − 2)1−𝛼} + {(𝑛 − 2)1−𝛼 − (𝑛 − 3)1−𝛼} (5-77) 
+ {(𝑛 − 3)1−𝛼 − (𝑛 − 4)1−𝛼} + ⋯+ {21−𝛼 − 11−𝛼} 
                  = 𝑛1−𝛼 + 21−𝛼 − 11−𝛼 
Substituting back into Equation (5-76) and simplifying, 
|𝑎 ||?̂? | < |𝑏 + 𝑑 𝑒−𝑗𝑘𝑖∆𝑥 + 𝑓 𝑒𝑗𝑘𝑖∆𝑥3 𝑛 3 3 − 𝑙3(1 − cos𝜙 + 𝑖 sin𝜙)𝛽 ||?̂?0| + |ℎ3||?̂?0|𝛽  (5-78) 3 𝑚 𝑛
where, 
𝛽 = 𝑛1−𝛼𝑛 + 2
1−𝛼 − 11−𝛼 
𝛽𝑚 = 𝑚
1−𝛼 − (−1)1−𝛼 
Simplifying and rearranging, 
114 
 
|?̂? | |𝑏 + 𝑑 𝑒−𝑗𝑘𝑖∆𝑥𝑛 3 3 + 𝑓 𝑒
𝑗𝑘𝑖∆𝑥 − 𝑙3(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)𝛽 | + |ℎ3|𝛽
< 3 𝑚 𝑛 (5-79) 
|?̂?0| |𝑎3|
Remembering |𝑒𝑛| = 1 and simplifying, 
|?̂?𝑛| |𝑏3| + |𝑑3| + |𝑓 | + |𝑙3|(|1 − cos𝜙| + 𝑖 |𝑠𝑖𝑛 𝜙|)𝛽 + |ℎ3|𝛽
< 3 𝑚 𝑛 
|?̂?0| |𝑎3|
(5-80) 
|?̂?𝑛| |𝑏3| + |𝑑3| + |𝑓 | + |𝑙3|(2 − 2 cos𝜙)𝛽 + |ℎ3|𝛽
< 3 𝑚 𝑛 
|?̂?0| |𝑎3|
Thus, the solution will be stable when, 
|𝑏| + |𝑑3| + |𝑓 | + |𝑙3|(2 − 2 cos𝜙)𝛽 + |ℎ3|𝛽3 𝑚 𝑛 < 1 (5-81) 
|𝑎3|
The term is expanded using the simplification terms associated with Equation (5-18), 
(∆𝑡)−𝛼  𝛼 (∆𝑥)
−𝛼
 𝛼 2𝐷𝐿 (∆𝑥)
−𝛼
| 𝛿 + 𝑣 𝛿 − | + |𝑣 𝛿 𝛼
𝐷𝐿
Γ(2 − 𝛼) 𝑛,𝑛−1 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2 𝑥 Γ(2 − 𝛼) 𝑛,𝑚
+
(∆𝑥)2
|
( )
𝐷 (∆𝑥)−𝛼 (∆𝑡)−𝛼
+| 𝐿 2| + |𝑣𝑥 | (2 − 2 cos𝜙)𝛽 + 𝛽(∆𝑥) Γ(2 − 𝛼) 𝑚
| | (5-82) 
Γ(2 − 𝛼) 𝑛
−𝛼 < 1 (∆𝑡)
| 𝛿 𝛼 |
Γ(2 − 𝛼) 𝑛,𝑛−1
The assumption is made where, 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷
𝛿  𝛼
𝐿
+ 𝑣 𝛿  𝛼
Γ(2 − 𝛼) 𝑛,𝑛−1 𝑥 Γ(2 − 𝛼) 𝑛,𝑚
>  (5-83) 
(∆𝑥)2
Then, 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷 (∆𝑥)−𝛼
𝛿 𝛼  𝛼 𝐿  𝛼
𝐷𝐿
Γ(2 − 𝛼) 𝑛,𝑛−1
+ 𝑣𝑥 𝛿 − + 𝑣Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2 𝑥
𝛿
Γ(2 − 𝛼) 𝑛,𝑚
+
(∆𝑥)2
(
𝐷 (∆𝑥)−𝛼
)
𝐿 (∆𝑡)
−𝛼
+ 2 + 𝑣𝑥 (2 − 2 cos𝜙)𝛽 + 𝛽(∆𝑥) Γ(𝛼) 𝑚 Γ(2 − 𝛼) 𝑛 (5-84) 
(∆𝑡)−𝛼
< 1 
𝛿 𝛼
Γ(2 − 𝛼) 𝑛,𝑛−1
Simplifying, under this assumption (Equation (5-83)), the explicit upwind numerical scheme for the 
fractional advection-dispersion equation with Caputo fractional derivative is unstable. 
When the complementary assumption is made where, 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷𝐿
𝛿  𝛼  𝛼 (5-85) 
Γ(2 − 𝛼) 𝑛,𝑛−1
+ 𝑣𝑥 𝛿 <  Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
Then, 
(∆𝑡)−𝛼 −𝛼 −𝛼
− 𝛿 𝛼
(∆𝑥)
− 𝑣 𝛿 𝛼
2𝐷𝐿 (∆𝑥) 𝐷
𝑛,𝑛−1 𝑥 + + 𝑣 𝛿
 𝛼 + 𝐿
Γ(2 − 𝛼) Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
( −𝛼 )𝐷𝐿 (∆𝑥) (∆𝑡)
−𝛼
+ 2 + 𝑣𝑥 (2 − 2 cos𝜙)𝛽 + 𝛽(∆𝑥) Γ(2 − 𝛼) 𝑚 Γ(2 − 𝛼) 𝑛
(5-86) 
(∆𝑡)−𝛼
< 1 
𝛿 𝛼
Γ(2 − 𝛼) 𝑛,𝑛−1
115 
 
Simplifying, 
(∆𝑡)−𝛼 4𝐷 −𝛼𝐿 (∆𝑥)
(2+ 𝛽 ) <  + 𝑣 (2 − 2 cos𝜙)𝛽  (5-87) 𝑛 Γ(2 − 𝛼) (∆𝑥)2 𝑥 Γ(2 − 𝛼) 𝑚
where, 
𝛿  𝛼
1−𝛼
𝑛,𝑛−1 = (𝑛 − (𝑛 − 1)) − (𝑛 − (𝑛 − 1) − 1)
1−𝛼 
                                                          = 11−𝛼 
Under this assumption (Equation (5-85)), the finite difference first-order upwind (explicit) numerical 
scheme for the fractional advection-dispersion equation with Caputo fractional derivative is 
conditionally stable. 
It has been proved by means of the induction method that the explicit upwind finite difference scheme 
for the one-dimensional, non-reactive fractional advection-dispersion equation (Caputo) has the 
following stability criterion, 
(∆𝑡)−𝛼 2𝐷𝐿
<  
Γ(2 − 𝛼) (∆𝑥)2
 
(∆𝑡)−𝛼 (∆𝑥)−𝛼
𝛽 < 𝑣 (2𝛿  𝛼  + (2 − 2 cos𝜙)𝛽 ) 
Γ(2 − 𝛼) 𝑛 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 𝑚
and, 
(∆𝑡)−𝛼 4𝐷𝐿 (∆𝑥)
−𝛼
(2+ 𝛽𝑛) <  + 𝑣 (2 − 2 cos𝜙)𝛽  Γ(2 − 𝛼) (∆𝑥)2 𝑥 Γ(2 − 𝛼) 𝑚
According to the stability analysis method applied, under these conditions, the error of the 
approximation is not propagated throughout the solution, but rather decreases with each time step. 
5.2.2 Implicit upwind  
The developed implicit upwind numerical scheme discussed in Section 5.1.2 (Equation (5-22)) is 
applied, and substituting induction method terms provides: 
𝑏 ?̂? 𝑒𝑗𝑘𝑖𝑥 = 𝑎 ?̂? 𝑒𝑗𝑘𝑖𝑥 + 𝑑 ?̂? 𝑒𝑗𝑘𝑖(𝑥−∆𝑥) + 𝑓 ?̂? 𝑒𝑗𝑘𝑖𝑥(𝑥+∆𝑥)3 𝑛 3 𝑛−1 3 𝑛 3 𝑛  
𝑛−2 𝑚−1
(5-88) 
−ℎ3 (∑(?̂?𝑘+1𝑒
𝑗𝑘𝑚𝑥 − ?̂?𝑘𝑒
𝑗𝑘𝑚𝑥)𝛿  𝛼 ) − 𝑙 (∑(?̂? 𝑒𝑗𝑘𝑖𝑥 − ?̂? 𝑒𝑗𝑘𝑖(𝑥−∆𝑥)  𝛼𝑛,𝑘 3 𝑛 𝑛 ) 𝛿𝑚,𝑖) 
𝑘=0 𝑖=1
Multiple out, 
𝑏 𝑗𝑘𝑖𝑥 𝑗𝑘𝑖𝑥3?̂?𝑛𝑒 = 𝑎3?̂?𝑛−1𝑒 + 𝑑 ?̂? 𝑒
𝑗𝑘𝑖𝑥𝑒−𝑗𝑘𝑖∆𝑥 + 𝑓 ?̂? 𝑒𝑗𝑘𝑖𝑥𝑒𝑗𝑘𝑖∆𝑥3 𝑛 3 𝑛  
𝑛−2 𝑚−1 (5-89) 
−ℎ3 (∑(?̂?𝑘+1𝑒
𝑗𝑘𝑚𝑥 − ?̂? 𝑒𝑗𝑘𝑚𝑥)𝛿  𝛼 ) − 𝑙 (∑(?̂? 𝑒𝑗𝑘𝑖𝑥 − ?̂? 𝑒𝑗𝑘𝑖𝑥𝑒−𝑗𝑘𝑖∆𝑥  𝛼𝑘 𝑛,𝑘 3 𝑛 𝑛 ) 𝛿𝑚,𝑖) 
𝑘=0 𝑖=1
Divide by 𝑒𝑗𝑘𝑖𝑥, 
116 
 
𝑏 ?̂? = 𝑎 ?̂? + 𝑑 ?̂? 𝑒−𝑗𝑘𝑖∆𝑥 + 𝑓 ?̂? 𝑒𝑗𝑘𝑖∆𝑥3 𝑛 3 𝑛−1 3 𝑛 3 𝑛  
𝑛−2 𝑚−1
(5-90) 
−ℎ3 (∑(?̂?
 𝛼 −𝑗𝑘𝑖∆𝑥  𝛼
𝑘+1 − ?̂?𝑘)𝛿𝑛,𝑘) − 𝑙3 (∑(?̂?𝑛 − ?̂?𝑛𝑒 )𝛿𝑚,𝑖) 
𝑘=0 𝑖=1
The induction numerical stability analysis is performed in two parts; firstly it is proved for a set ∀ 𝑛 > 1,  
|?̂?𝑛| < |?̂?𝑜|  
If 𝑛 = 1, then 
𝑚−1
𝑏 ?̂? = 𝑎 ?̂? + 𝑑 ?̂? 𝑒−𝑗𝑘𝑖∆𝑥 + 𝑓 ?̂? 𝑒𝑗𝑘𝑖∆𝑥3 1 3 0 3 1 3 1 − 𝑙3 (∑(?̂?1 − ?̂?1𝑒
−𝑗𝑘𝑖∆𝑥) 𝛿  𝛼 ) (5-91) 𝑚,𝑖
𝑖=1
A subset for 𝑚 is now considered, where 𝑚 = 1, 
𝑏3?̂?1 = 𝑎3?̂?0 + 𝑑 ?̂? 𝑒
−𝑗𝑘𝑖∆𝑥
3 1 + 𝑓3?̂?
𝑗𝑘𝑖∆𝑥
1𝑒  (5-92)  
 
Simplifying, 
(𝑏3 − 𝑑 𝑒
−𝑗𝑘𝑖∆𝑥
3 − 𝑓 𝑒
𝑗𝑘𝑖∆𝑥
3 )?̂?1 = 𝑎3?̂?0 (5-93) 
Rearranging, 
?̂?1 𝑎3
=  
−𝑗𝑘 ∆𝑥 𝑗𝑘 ∆𝑥 (5-94) ?̂?0 (𝑏3 − 𝑑3𝑒 𝑖 − 𝑓3𝑒 𝑖 )
Applying a norm, 
|?̂?1| |𝑎3|
=  
|?̂?0| (|𝑏3| + |−𝑑 𝑒
−𝑗𝑘𝑖∆𝑥| + |−𝑓 𝑒𝑗𝑘𝑖∆𝑥|) (5-95) 3 3
The condition thus becomes: 
|?̂?1|
< 1 
|?̂?  0|
Remembering |𝑒𝑛| = 1, it can be concluded that |?̂?1| < |?̂?𝑜|, when 
|𝑎3|
< 1 
(|𝑏3| + |𝑑3| + |𝑓3|)
The term is expanded using the simplification terms associated with Equation (5-22), 
(∆𝑡)−𝛼
| 𝛿  𝛼
Γ(2 − 𝛼) 𝑛,𝑛−1
|
−𝛼 < 1 (∆𝑡)  𝛼 (∆𝑥)
−𝛼
 𝛼 2𝐷𝐿 (∆𝑥)
−𝛼
 𝛼 𝐷 𝐷 (5-96) (| 𝛿 + 𝑣 𝛿 − | + |𝑣 𝛿 + 𝐿 | + | 𝐿 |)
Γ(2 − 𝛼) 𝑛,𝑛−1 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2 (∆𝑥)2
The assumption is made where, 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷
 𝛼  𝛼 𝐿𝛿 + 𝑣 𝛿 >  (5-97) 
Γ(2 − 𝛼) 𝑛,𝑛−1 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
Then, 
117 
 
(∆𝑡)−𝛼
𝛿  𝛼
Γ(2 − 𝛼) 𝑛,𝑛−1
(∆𝑡)−𝛼 −𝛼 −𝛼
< 1 (5-98) 
𝛿  𝛼
(∆𝑥) 2𝐷 (∆𝑥) 𝐷 𝐷
𝑛,𝑛−1 + 𝑣 𝛿
 𝛼 − 𝐿  𝛼 𝐿 𝐿
Γ(2 − 𝛼) 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
+ 𝑣𝑥 𝛿 + +Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2 (∆𝑥)2
Simplifying, 
(∆𝑥)−𝛼
2𝑣 𝛿  𝛼𝑥 Γ(2 − 𝛼) 𝑛,𝑚
> 0 (5-99) 
Under this assumption (Equation (5-97)), the finite difference first-order upwind (implicit) numerical 
scheme for the fractional advection-dispersion equation with Caputo fractional derivative is 
unconditionally stable. 
When the corresponding assumption is made where, 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷
𝛿  𝛼  𝛼
𝐿
Γ(2 − 𝛼) 𝑛,𝑛−1
+ 𝑣𝑥 𝛿Γ(2 − 𝛼) 𝑛,𝑚
<  (5-100) 
(∆𝑥)2
Then, 
(∆𝑡)−𝛼
𝛿  𝛼
Γ(2 − 𝛼) 𝑛,𝑛−1
< 1 
(∆𝑡)−𝛼 −𝛼 −𝛼 (5-101) 
− 𝛿  𝛼
(∆𝑥)  𝛼 2𝐷𝐿 (∆𝑥)− 𝑣 𝛿 + + 𝑣 𝛿  𝛼
𝐷 𝐷
+ 𝐿 + 𝐿
Γ(2 − 𝛼) 𝑛,𝑛−1 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2 (∆𝑥)2
Simplifying, 
(∆𝑡)−𝛼 2𝐷𝐿
<  
Γ(2 − 𝛼) (∆𝑥)2 (5-102) 
Under this assumption (Equation (5-100)), the implicit upwind numerical scheme for the fractional 
advection-dispersion equation with Caputo fractional derivative is conditionally stable. 
A subset for 𝑚 is now considered for all 𝑚 > 1, 
𝑚−1
𝑏 −𝑗𝑘𝑖∆𝑥 𝑗𝑘𝑖∆𝑥 −𝑗𝑘𝑖∆𝑥  𝛼 (5-103) 3?̂?1 = 𝑎3?̂?0 + 𝑑3?̂?1𝑒 + 𝑓3?̂?1𝑒 − 𝑙3 (∑(?̂?1 − ?̂?1𝑒 ) 𝛿𝑚,𝑖) 
𝑖=1
Simplifying, 
𝑚−1
(𝑏 − 𝑑 𝑒−𝑗𝑘𝑖∆𝑥 − 𝑓 𝑒𝑗𝑘𝑖∆𝑥 + 𝑙  (1 − 𝑒−𝑗𝑘𝑖∆𝑥) ∑ 𝛿  𝛼3 3 3 3 𝑚,𝑖) ?̂?1 = 𝑎3?̂?0 (5-104) 
𝑖=1
Following the same procedure as in Section 5.2.1, 
(|𝑏3| + |𝑑3| + |𝑓3| + |𝑙3| (2 − 2 cos𝜙)|𝛽𝑚|)|?̂?1| = |𝑎3||?̂?0| (5-105) 
Rearranging, 
|?̂?1| |𝑎3|
=  
|?̂?0| (|𝑏3| + |𝑑3| + |𝑓3| + |𝑙3| (2 − 2 cos𝜙)|𝛽𝑚|)
(5-106) 
The condition thus becomes, 
118 
 
|?̂?1|
< 1  
|?̂?0|
And, substituting the following is found, 
|𝑎3|
< 1 (5-107) 
(|𝑏3| + |𝑑3| + |𝑓3| + |𝑙| (2 − 2 cos𝜙)(|𝛽𝑚|))
The term is expanded using the simplification terms associated with Equation (5-22), 
(∆𝑡)−𝛼
| 𝛿  𝛼 |
Γ(2 − 𝛼) 𝑛,𝑛−1
< 1 
(∆𝑡)−𝛼  𝛼 (∆𝑥)
−𝛼
 𝛼 2𝐷𝐿 (∆𝑥)
−𝛼 𝐷 𝐷
| 𝛿 + 𝑣 𝛿 − | + |𝑣 𝛿  𝛼 + 𝐿 | + | 𝐿 |
Γ(2 − 𝛼) 𝑛,𝑛−1 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2 (∆𝑥)2 (5-108) 
( )
(∆𝑥)−𝛼
+ |𝑣𝑥 | (2 − 2 cos𝜙)(|𝛽 |)Γ(2 − 𝛼) 𝑚
The assumption is made where, 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷𝐿
𝛿  𝛼 + 𝑣 𝛿  𝛼 >  (5-109) 
Γ(2 − 𝛼) 𝑛,𝑛−1 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
Then, 
(∆𝑡)−𝛼
𝛿  𝛼
Γ(2 − 𝛼) 𝑛,𝑛−1
< 1 
(∆𝑡)−𝛼  𝛼 (∆𝑥)
−𝛼
 𝛼 2𝐷 (∆𝑥)
−𝛼
𝛿 𝐿  𝛼
𝐷𝐿 𝐷𝐿
Γ(2 − 𝛼) 𝑛,𝑛−1
+ 𝑣𝑥 𝛿 − + 𝑣 𝛿 + +Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2 (∆𝑥)2 (5-110) 
(
(∆𝑥)−𝛼
)
+𝑣𝑥 (2 − 2 cos𝜙)𝛽Γ(2 − 𝛼) 𝑚
Simplifying, 
(∆𝑥)−𝛼
𝑣 (2𝛿  𝛼𝑥 𝑛,𝑚  + (2 − 2 cos𝜙)𝛽Γ(2 − 𝛼) 𝑚
) > 0 (5-111) 
Under this assumption (Equation (5-109)), the implicit upwind numerical scheme for the fractional 
advection-dispersion equation with Caputo fractional derivative is unconditionally stable. 
When the complementary assumption is made where, 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷𝐿
𝛿  𝛼  𝛼
Γ(2 − 𝛼) 𝑛,𝑛−1
+ 𝑣𝑥 𝛿Γ(2 − 𝛼) 𝑛,𝑚
<  (5-112) 
(∆𝑥)2
Then, 
(∆𝑡)−𝛼
𝛿  𝛼
Γ(2 − 𝛼) 𝑛,𝑛−1
−𝛼 −𝛼 −𝛼 < 1 (∆𝑡) (∆𝑥)
− 𝛿  𝛼 − 𝑣 𝛿  𝛼
2𝐷𝐿 (∆𝑥)  𝛼 𝐷𝐿 𝐷𝐿
Γ(2 − 𝛼) 𝑛,𝑛−1 𝑥 𝑛,𝑚
+ + 𝑣 𝛿 + +
Γ(2 − 𝛼) (∆𝑥)2 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2 (∆𝑥)2 (5-113) 
( )
(∆𝑥)−𝛼
+𝑣𝑥 (2 − 2 cos𝜙)(𝛽 )Γ(2 − 𝛼) 𝑚
Simplifying, 
(∆𝑡)−𝛼 2𝐷𝐿 (∆𝑥)
−𝛼
< + 𝑣𝑥 (1 − 1 cos𝜙)𝛽  Γ(2 − 𝛼) (∆𝑥)2 Γ(2 − 𝛼) 𝑚 (5-114) 
119 
 
Under this assumption (Equation (5-112)), the finite difference first-order upwind (implicit) numerical 
scheme for the fractional advection-dispersion equation with Caputo fractional derivative is 
conditionally stable. 
It is assumed that |?̂?𝑛−1| < |?̂?𝑜| is true for all time steps is made, and the second part of the numerical 
stability analysis is to demonstrate for a set ∀ 𝑛 ≥ 1, that 
|?̂?𝑛| < |?̂?𝑜|  
Rearranging Equation (5-90) for ?̂?𝑛, 
𝑚−1
(𝑏 − 𝑑 𝑒−𝑗𝑘𝑖∆𝑥3 3 − 𝑓3𝑒
𝑗𝑘𝑖∆𝑥 + 𝑙3(1 − 𝑒
−𝑗𝑘𝑖∆𝑥)(∑ 𝛿  𝛼𝑚,𝑖)) ?̂?𝑛 
𝑖=1 (5-115) 
𝑛−2
= 𝑎 ?̂?  𝛼3 𝑛−1 − ℎ3 (∑(?̂?𝑘+1 − ?̂?𝑘)𝛿𝑛,𝑘) 
𝑘=0
Following a similar process as previously (Equation (5-105)), 
(𝑏 −𝑗𝑘𝑖∆𝑥 𝑗𝑘𝑖∆𝑥3 − 𝑑3𝑒 − 𝑓 𝑒 + 𝑙3(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)(𝑚
1−𝛼 − (−1)1−𝛼)) ?̂?𝑛 3
𝑛−2 (5-116) 
= 𝑎3?̂?
 𝛼
𝑛−1 − ℎ3 (∑(?̂?𝑘+1 − ?̂?𝑘)𝛿𝑛,𝑘) 
𝑘=0
Taking the norm on both sides, 
|(𝑏 − 𝑑 𝑒−𝑗𝑘𝑖∆𝑥 − 𝑓 𝑒𝑗𝑘𝑖∆𝑥3 3 + 𝑙 (1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)(𝑚
1−𝛼 − (−1)1−𝛼) ?̂?  
3 3 ) 𝑛|
𝑛−2 (5-117) 
= |𝑎3?̂?𝑛−1 − ℎ3 (∑(?̂? − ?̂? )𝛿
 𝛼
𝑘+1 𝑘 𝑛,𝑘)| 
𝑘=0
Thus, 
|𝑏3 − 𝑑 𝑒
−𝑗𝑘𝑖∆𝑥
3 − 𝑓 𝑒
𝑗𝑘𝑖∆𝑥 + 𝑙3(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)(𝑚
1−𝛼 − (−1)1−𝛼)||?̂?𝑛| 3
𝑛−2
(5-118) 
< |𝑎  𝛼3||?̂?𝑛−1| + |ℎ3| |∑(?̂?𝑘+1 − ?̂?𝑘)𝛿𝑛,𝑘| 
𝑘=0
Remembering that it has been proved that for a set ∀ 𝑛 ≥ 1,   
|?̂?𝑛−1| < |?̂?𝑜| 
Thus, 
|𝑏 −𝑗𝑘𝑖∆𝑥3 − 𝑑3𝑒 − 𝑓 𝑒
𝑗𝑘𝑖∆𝑥 + 𝑙3(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)(𝑚
1−𝛼 − (−1)1−𝛼)||?̂?𝑛| 3
𝑛−2 𝑛−2
(5-119) 
< |𝑎3||?̂?𝑛−1| + |ℎ3| |∑(?̂?
 𝛼  𝛼
𝑘+1 − ?̂?𝑘)𝛿𝑛,𝑘| < |𝑎3||?̂?0| + |ℎ3| |∑(?̂?𝑘+1 − ?̂?𝑘)𝛿𝑛,𝑘| 
𝑘=0 𝑘=0
and, it can be inferred that, 
|𝑏 − 𝑑 𝑒−𝑗𝑘𝑖∆𝑥3 3 − 𝑓 𝑒
𝑗𝑘𝑖∆𝑥 + 𝑙3(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)(𝑚
1−𝛼 − (−1)1−𝛼)||?̂?𝑛| 3
𝑛−2
(5-120) 
< |𝑎3||?̂? | + |ℎ | |∑(?̂? − ?̂? )𝛿
 𝛼
0 3 𝑘+1 𝑘 𝑛,𝑘| 
𝑘=0
120 
 
The remaining summation is considered at the upper limit and simplifying as in Section 5.2.1, 
|𝑏 − 𝑑 𝑒−𝑗𝑘𝑖∆𝑥3 3 − 𝑓 𝑒
𝑗𝑘𝑖∆𝑥 + 𝑙3(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)𝛽 ||?̂?𝑛| < |𝑎3||?̂?0| + |ℎ3||?̂?0|𝛽  3 𝑚 𝑛 (5-121) 
Simplifying and rearranging, 
|?̂?𝑛| |𝑎3| + |ℎ3|𝛽
< 𝑛  
|?̂?0|
(5-122) 
||𝑏 − 𝑑 𝑒−𝑗𝑘𝑖∆𝑥 − 𝑓 𝑒𝑗𝑘𝑖∆𝑥3 3 + 𝑙3(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)𝛽 |3 𝑚 |
Remembering |𝑒𝑛| = 1 and simplifying, 
|?̂?𝑛| |𝑎3| + |ℎ3|𝛽
< 𝑛  (5-123) 
|?̂?0| |𝑏3| + |𝑑3| + |𝑓 | + |𝑙 |(2 − 2 cos𝜙)𝛽3 3 𝑚
Thus, the solution will be stable when, 
|𝑎3| + |ℎ3|𝛽𝑛 < 1 (5-124) 
|𝑏3| + |𝑑3| + |𝑓 | + |𝑙3|(2 − 2 cos𝜙)𝛽3 𝑚
The term is expanded using the simplification terms associated with Equation (5-22), 
(∆𝑡)−𝛼  𝛼 (∆𝑡)
−𝛼
| 𝛿𝑛,𝑛−1| + | |𝛽Γ(2 − 𝛼) Γ(2 − 𝛼) 𝑛
(∆𝑡)−𝛼
< 1 
 𝛼 (∆𝑥)−𝛼  𝛼 2𝐷𝐿 (∆𝑥)
−𝛼
| 𝛿 + 𝑣 𝛿 − | + |𝑣 𝛿 𝛼
𝐷
+ 𝐿 | (5-125) 
Γ(2 − 𝛼) 𝑛,𝑛−1 𝑥 Γ(𝛼) 𝑛,𝑚 (∆𝑥)2 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
( −𝛼 )𝐷 (∆𝑥)
+ | 𝐿 | + |𝑣 | (2 − 2 cos𝜙)𝛽
(∆𝑥)2 𝑥 Γ(2 − 𝛼) 𝑚
The assumption is made where, 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷𝐿
𝛿  𝛼  𝛼 (5-126) 
Γ(2 − 𝛼) 𝑛,𝑛−1
+ 𝑣𝑥 𝛿Γ(2 − 𝛼) 𝑛,𝑚
>  
(∆𝑥)2
Then, 
(∆𝑡)−𝛼 (∆𝑡)−𝛼
𝛿 𝛼𝑛,𝑛−1 + 𝛽Γ(2 − 𝛼) Γ(2 − 𝛼) 𝑛
< 1 
(∆𝑡)−𝛼 −𝛼 𝛼 (∆𝑥)  𝛼 2𝐷𝐿 (∆𝑥)
−𝛼 𝐷
𝛿 + 𝑣 𝛿 − + 𝑣 𝛿 𝛼 + 𝐿
Γ(2 − 𝛼) 𝑛,𝑛−1 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2 (5-127) 
(
𝐷 (∆𝑥)−𝛼
)
+ 𝐿 2 + 𝑣𝑥 (2 − 2 cos𝜙)𝛽(∆𝑥) Γ(2 − 𝛼) 𝑚
Simplifying, 
(∆𝑡)−𝛼 (∆𝑥)−𝛼
𝛽𝑛 < 𝑣
 𝛼
𝑥 (2𝛿  + (2 − 2 cos𝜙)𝛽 ) Γ(2 − 𝛼) Γ(2 − 𝛼) 𝑛,𝑚 𝑚 (5-128) 
Under this assumption (Equation (5-126)), the implicit upwind numerical scheme for the fractional 
advection-dispersion equation with Caputo fractional derivative is conditionally stable. 
When the opposite assumption is made where, 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷
 𝛼  𝛼 𝐿𝛿 + 𝑣 (5-129) 
Γ(2 − 𝛼) 𝑛,𝑛−1 𝑥
𝛿 <  
Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
Then, 
121 
 
(∆𝑡)−𝛼 (∆𝑡)−𝛼
𝛿 𝛼 + 𝛽
Γ(2 − 𝛼) 𝑛,𝑛−1 Γ(2 − 𝛼) 𝑛
< 1 
(∆𝑡)−𝛼  𝛼 (∆𝑥)−𝛼  𝛼 2𝐷𝐿 (∆𝑥)
−𝛼 𝐷
− 𝛿  𝛼 𝐿
Γ(2 − 𝛼) 𝑛,𝑛−1
− 𝑣𝑥 𝛿Γ(2 − 𝛼) 𝑛,𝑚
+ 2 + 𝑣𝑥 𝛿 +( ) Γ(2 − 𝛼) 𝑛,𝑚 ( )2 (5-130) ∆𝑥 ∆𝑥
(
𝐷 (∆𝑥)−𝛼
)
+ 𝐿 2 + 𝑣𝑥 (2 − 2 cos𝜙)𝛽(∆𝑥) Γ(2 − 𝛼) 𝑚
Simplifying, 
(∆𝑡)−𝛼 4𝐷 (∆𝑥)−𝛼𝐿
(2+ 𝛽𝑛) <  + 𝑣𝑥 (2 − 2 cos𝜙)𝛽  Γ(2 − 𝛼) (∆𝑥)2 Γ(2 − 𝛼) 𝑚 (5-131) 
Under this assumption (Equation (5-129)), the implicit upwind numerical scheme for the fractional 
advection-dispersion equation with Caputo fractional derivative is conditionally stable. 
It has been proven by means of the induction method that the first-order implicit upwind finite 
difference scheme for the fractional advection-dispersion equation (Caputo) has the following stability 
criterion, 
(∆𝑡)−𝛼 2𝐷𝐿
<  
Γ(2 − 𝛼) (∆𝑥)2
 
(∆𝑡)−𝛼 (∆𝑥)−𝛼
𝛽 < 𝑣  𝛼
Γ(2 − 𝛼) 𝑛 𝑥
(2𝛿
Γ(2 − 𝛼) 𝑛,𝑚
 + (2 − 2 cos𝜙)𝛽𝑚) 
and 
(∆𝑡)−𝛼 4𝐷𝐿 (∆𝑥)
−𝛼
(2+ 𝛽𝑛) <  + 𝑣Γ(2 − 𝛼) (∆𝑥)2 𝑥
(2 − 2 cos𝜙)𝛽𝑚 Γ(2 − 𝛼)
Under these conditions, the error of the approximation decreases with each time step, as according 
to the induction method, where for all values of n, |?̂?𝑛+1| < |?̂?𝑜|. 
5.2.3 Upwind advection Crank-Nicolson scheme 
The developed upwind advection Crank-Nicolson numerical scheme offered in Section 5.1.3 (Equation 
(5-28) is applied, and the substituting induction method terms results in, 
𝑜 ?̂? 𝑒𝑗𝑘𝑖𝑥 = 𝑝 ?̂? 𝑒𝑗𝑘𝑖𝑥 + 𝑞 ?̂? 𝑒𝑗𝑘𝑖(𝑥−∆𝑥) − 𝑟 𝑗𝑘𝑖(𝑥−∆𝑥)3 𝑛 3 𝑛−1 3 𝑛−1 3?̂?𝑛𝑒  
𝑛−2
+𝑓 ?̂? 𝑒𝑗𝑘𝑖(𝑥+∆𝑥) − ℎ  (∑(?̂? 𝑒𝑗𝑘𝑚𝑥 − ?̂? 𝑒𝑗𝑘𝑚𝑥)𝛿  𝛼3 𝑛−1 3 𝑘+1 𝑘 𝑛,𝑘) 
𝑘=0 (5-132) 
𝑚−1
−𝑙3 (∑ (0.5(?̂? 𝑒
𝑗𝑘𝑖𝑥 − ?̂? 𝑒𝑗𝑘𝑖(𝑥−∆𝑥)) + 0.5(?̂? 𝑒𝑗𝑘𝑖𝑥 − ?̂? 𝑒𝑗𝑘𝑖(𝑥−∆𝑥)  𝛼𝑛−1 𝑛−1 𝑛 𝑛 )) 𝛿𝑚,𝑖) 
𝑖=1
Multiple out, 
 
122 
 
𝑜 ?̂? 𝑒𝑗𝑘𝑖𝑥 = 𝑝 ?̂? 𝑒𝑗𝑘𝑖𝑥3 𝑛 3 𝑛−1 + 𝑞3?̂?
𝑗𝑘𝑖𝑥 −𝑗𝑘𝑖∆𝑥 𝑗𝑘𝑖𝑥 −𝑗𝑘𝑖∆𝑥
𝑛−1𝑒 𝑒 − 𝑟3?̂?𝑛𝑒 𝑒  
𝑛−2
+𝑓 ?̂? 𝑒𝑗𝑘𝑖𝑥3 𝑛−1 𝑒
𝑗𝑘𝑖∆𝑥 − ℎ3 (∑(?̂? 𝑒
𝑗𝑘𝑚𝑥
𝑘+1 − ?̂?
𝑗𝑘𝑚𝑥
𝑘𝑒 )𝛿
 𝛼
𝑛,𝑘) 
(5-133) 
𝑘=0
𝑚−1
−𝑙 (∑ (0.5(?̂? 𝑒𝑗𝑘𝑖𝑥 − ?̂? 𝑒𝑗𝑘𝑖𝑥𝑒−𝑗𝑘𝑖∆𝑥) + 0.5(?̂? 𝑒𝑗𝑘𝑖𝑥 − ?̂? 𝑒𝑗𝑘𝑖𝑥𝑒−𝑗𝑘𝑖∆𝑥  𝛼3 𝑛−1 𝑛−1 𝑛 𝑛 )) 𝛿𝑚,𝑖) 
𝑖=1
Divide by 𝑒𝑗𝑘𝑖𝑥, 
𝑛−2
𝑜3?̂?𝑛 = 𝑝
−𝑗𝑘𝑖∆𝑥
3?̂?𝑛−1 + 𝑞3?̂?𝑛−1𝑒 − 𝑟 ?̂? 𝑒
−𝑗𝑘𝑖∆𝑥 + 𝑓 ?̂? 𝑒𝑗𝑘𝑖∆𝑥3 𝑛 3 𝑛−1 − ℎ3 (∑(?̂?
 𝛼
𝑘+1 − ?̂?𝑘)𝛿𝑛,𝑘) 
𝑘=0
𝑚−1 (5-134) 
−𝑙 (∑ (0.5(?̂? − ?̂? 𝑒−𝑗𝑘𝑖∆𝑥) + 0.5(?̂? − ?̂? 𝑒−𝑗𝑘𝑖∆𝑥)) 𝛿  𝛼3 𝑛−1 𝑛−1 𝑛 𝑛 𝑚,𝑖) 
𝑖=1
The induction numerical stability analysis is performed in two parts; firstly it is proved for a set ∀ 𝑛 > 1,  
|?̂?  𝑛| < |?̂?𝑜| 
If 𝑛 = 1, then 
𝑜 −𝑗𝑘𝑖∆𝑥 −𝑗𝑘𝑖∆𝑥 𝑗𝑘𝑖∆𝑥3?̂?1 = 𝑝3?̂?0 + 𝑞3?̂?0𝑒 − 𝑟3?̂?1𝑒 + 𝑓3?̂?0𝑒  
𝑚−1
(5-135) 
− 𝑙3 (∑ (0.5(?̂?
−𝑗𝑘𝑖∆𝑥
0 − ?̂?0𝑒 ) + 0.5(?̂?
−𝑗𝑘𝑖∆𝑥  𝛼
1 − ?̂?1𝑒 )) 𝛿𝑚,𝑖) 
𝑖=1
A subset for 𝑚 is now considered, where 𝑚 = 1, 
𝑜 ?̂? = 𝑝 ?̂? + 𝑞 ?̂? 𝑒−𝑗𝑘𝑖∆𝑥 − 𝑟 ?̂? 𝑒−𝑗𝑘𝑖∆𝑥 + 𝑓 ?̂? 𝑒𝑗𝑘𝑖∆𝑥 (5-136) 3 1 3 0 3 0 3 1 3 0
Simplifying, 
(𝑜3 + 𝑓𝑒
−𝑗𝑘𝑖∆𝑥)?̂? = (𝑝 + 𝑞 𝑒−𝑗𝑘𝑖∆𝑥 + 𝑓 𝑒𝑗𝑘𝑖∆𝑥)?̂?  (5-137) 1 3 3 3 0
Rearranging, 
?̂? 𝑝 + 𝑞 𝑒−𝑗𝑘𝑖∆𝑥 + 𝑓 𝑒𝑗𝑘𝑖∆𝑥1 3 3 3
=  (5-138) 
?̂?0 𝑜3 + 𝑟3𝑒
−𝑗𝑘𝑖∆𝑥
Applying the norm, 
|?̂? −𝑗𝑘𝑖∆𝑥 𝑗𝑘𝑖∆𝑥1| |𝑝3| + |𝑞3𝑒 | + |𝑓3𝑒 |
=  
|?̂? | |𝑜 | + |𝑟 𝑒−𝑗𝑘𝑖∆𝑥| (5-139) 0 3 3
The condition required can be expressed as, 
|?̂?1|
< 1  
|?̂?0|
Remembering |𝑒𝑛| = 1, it can be established that |?̂?1| < |?̂?𝑜|, when 
|𝑝3| + |𝑞3| + |𝑓3|
< 1 
|𝑜3| + |𝑟3|
The term is expanded using the simplification terms associated with Equation (5-28), 
123 
 
(∆𝑡)−𝛼  𝛼 (∆𝑥)
−𝛼 2𝐷 (∆𝑥)−𝛼
| 𝛿 − 0.5𝑣  𝛿  𝛼 − 𝐿 | + |0.5𝑣  𝛿  𝛼
𝐷𝐿 𝐷
𝑛,𝑛−1 𝑥 + | + |
𝐿 |
Γ(2 − 𝛼) Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2 (∆𝑥)2
−𝛼 −𝛼 −𝛼 < 1 (∆𝑡) (5-140) 
| 𝛿  𝛼
(∆𝑥) (∆𝑥)
𝑛,𝑛−1 + 0.5𝑣𝑥 𝛿
 𝛼
𝑛,𝑚| + |0.5𝑣  𝛿
 𝛼
Γ(2 − 𝛼) Γ(2 − 𝛼) 𝑥 Γ(2 − 𝛼) 𝑛,𝑚
|
The assumption is made where, 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷
𝛿  𝛼 > 0.5𝑣  𝛿  𝛼
𝐿
+  (5-141) 
Γ(2 − 𝛼) 𝑛,𝑛−1 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
Then, 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 𝛼  𝛼 2𝐷𝐿 (∆𝑥)
−𝛼
 𝛼 𝐷𝐿 𝐷𝛿𝑛,𝑛−1 − 0.5𝑣𝑥 𝛿𝑛,𝑚 − + 0.5𝑣 𝛿
𝐿
Γ(2 − 𝛼) Γ(2 − 𝛼) (∆𝑥)2 𝑥 Γ(2 − 𝛼) 𝑛,𝑚
+ 2 +(∆𝑥) (∆𝑥)2
< 1 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 (∆𝑥)−𝛼 (5-142) 
𝛿  𝛼  𝛼  𝛼
Γ(2 − 𝛼) 𝑛,𝑛−1
+ 0.5𝑣𝑥 𝛿Γ(2 − 𝛼) 𝑛,𝑚
+ 0.5𝑣𝑥 𝛿Γ(2 − 𝛼) 𝑛,𝑚
Simplifying, 
(∆𝑥)−𝛼
𝑣 𝛿  𝛼𝑥 > 0 Γ(2 − 𝛼) 𝑛,𝑚 (5-143) 
Under this assumption (Equation (5-141)), the upwind advection Crank-Nicolson scheme for the 
fractional advection-dispersion equation with Caputo fractional derivative is unconditionally stable. 
The complementary assumption is made, 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷
𝛿  𝛼
𝐿
𝑛,𝑛−1 < 0.5𝑣𝑥  𝛿
 𝛼 (5-144) 
Γ(2 − 𝛼) Γ(2 − 𝛼) 𝑛,𝑚
+  
(∆𝑥)2
Then, 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 𝛼  𝛼 2𝐷𝐿 (∆𝑥)
−𝛼 𝐷
− 𝛿 + 0.5𝑣  𝛿  𝛼 𝐿
𝐷𝐿
Γ(2 − 𝛼) 𝑛,𝑛−1 𝑥 Γ(2 − 𝛼) 𝑛,𝑚
+ + 0.5𝑣  𝛿 + +
(∆𝑥)2 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2 (∆𝑥)2
−𝛼 −𝛼 −𝛼 < 1 (∆𝑡) (5-145) 
𝛿  𝛼
(∆𝑥) (∆𝑥)
𝑛,𝑛−1 + 0.5𝑣𝑥 𝛿
 𝛼 + 0.5𝑣  𝛿  𝛼
Γ(2 − 𝛼) Γ(2 − 𝛼) 𝑛,𝑚 𝑥 Γ(2 − 𝛼) 𝑛,𝑚
Simplifying, 
2𝐷𝐿 (∆𝑡)
−𝛼
<  
(∆𝑥)2 Γ(2 − 𝛼) (5-146) 
Under the assumption made in Equation (5-144), the upwind advection Crank-Nicolson numerical 
scheme for the fractional advection-dispersion equation with Caputo fractional derivative is 
conditionally stable. 
A subset for 𝑚 is now considered for all 𝑚 > 1, 
𝑜 ?̂? = 𝑝 ?̂? + 𝑞 ?̂? 𝑒−𝑗𝑘𝑖∆𝑥 − 𝑟 ?̂? 𝑒−𝑗𝑘𝑖∆𝑥 + 𝑓 ?̂? 𝑒𝑗𝑘𝑖∆𝑥3 1 3 0 3 0 3 1 3 0  
𝑚−1
(5-147) 
− 𝑙 −𝑗𝑘𝑖∆𝑥3 (∑ (0.5(?̂?0 − ?̂?0𝑒 ) + 0.5(?̂?
−𝑗𝑘𝑖∆𝑥  𝛼
1 − ?̂?1𝑒 )) 𝛿𝑚,𝑖) 
𝑖=1
Simplifying, 
124 
 
𝑚−1
(𝑜3 + 𝑟3𝑒
−𝑗𝑘𝑖∆𝑥 + 0.5𝑙3(1 − 𝑒
−𝑗𝑘𝑖∆𝑥) ∑ 𝛿  𝛼𝑚,𝑖) ?̂?1 
𝑖=1
𝑚−1 (5-148) 
= (𝑝 + 𝑞 𝑒−𝑗𝑘𝑖∆𝑥 + 𝑓 𝑒𝑗𝑘𝑖∆𝑥 − 0.5𝑙 (1 − 𝑒−𝑗𝑘𝑖∆𝑥3 3 3 3 ) ∑ 𝛿
 𝛼
𝑚,𝑖) ?̂?0 
𝑖=1
Following the same procedure as in Section 5.2.1,  
(𝑜 + 𝑟 𝑒−𝑗𝑘𝑖∆𝑥3 3 + 0.5𝑙3(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)(𝑚
1−𝛼 − (−1)1−𝛼)) ?̂?1 
(5-149) 
= (𝑝 + 𝑞 𝑒−𝑗𝑘𝑖∆𝑥 + 𝑓 𝑒𝑗𝑘𝑖∆𝑥3 3 3 − 0.5𝑙3(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)(𝑚
1−𝛼 − (−1)1−𝛼)) ?̂?0 
Applying a norm on both sides, 
(|𝑜3| + |𝑟 𝑒
−𝑗𝑘𝑖∆𝑥
3 | + 0.5|𝑙3|(|1 − cos𝜙| + 𝑖 |𝑠𝑖𝑛 𝜙|)(|𝑚
1−𝛼 − (−1)1−𝛼|)) |?̂?1| 
(5-150) 
= (|𝑝3| + |𝑞 𝑒
−𝑗𝑘𝑖∆𝑥| + |𝑓 𝑒𝑗𝑘𝑖∆𝑥3 3 | + 0.5|𝑙3|(|1 − cos𝜙| + 𝑖 |𝑠𝑖𝑛 𝜙|)(|𝑚
1−𝛼 − (−1)1−𝛼|)) |?̂?0| 
Remembering |𝑒𝑛| = 1 and the procedure followed in Section 5.2.1,  
(|𝑜3| + |𝑟3| + 0.5|𝑙3|(2 − 2 cos𝜙)𝛽𝑚)|?̂?1| 
(5-151) 
= (|𝑝3| + |𝑞3| + |𝑓3| + 0.5|𝑙3|(2 − 2 cos𝜙)𝛽𝑚)|?̂?0| 
Rearranging, 
|?̂?1| |𝑝3| + |𝑞3| + |𝑓3| + 0.5|𝑙3|(2 − 2 cos𝜙)𝛽𝑚
=  
|?̂?0| (|𝑜3| + |𝑟3| + 0.5|𝑙 |(2 − 2 cos𝜙)𝛽 ) (5-152) 3 𝑚
The condition required can be expressed as: 
|?̂?1|
< 1  
|?̂?0|
Thus, the condition becomes, 
|𝑝3| + |𝑞3| + |𝑓3| + 0.5|𝑙3|(2 − 2 cos𝜙)𝛽𝑚
< 1 
(|𝑜3| + |𝑟3| + 0.5|𝑙3|(2 − 2 cos𝜙)𝛽𝑚) (5-153) 
The term is expanded using the simplification terms associated with Equation (5-28), 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷 (∆𝑥)−𝛼 𝐷
| 𝛿  𝛼  𝛼 𝐿  𝛼 𝐿
Γ(2 − 𝛼) 𝑛,𝑛−1
− 0.5𝑣𝑥  𝛿 − | + |0.5𝑣  𝛿 + |Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
( )
𝐷𝐿 (∆𝑥)
−𝛼
+ | 2| + 0.5 |𝑣𝑥 | (2 − 2 cos𝜙)𝛽(∆𝑥) Γ(2 − 𝛼) 𝑚
< 1 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 (∆𝑥)−𝛼 (5-154) 
| 𝛿  𝛼 + 0.5𝑣 𝛿  𝛼 | + |0.5𝑣  𝛼
Γ(2 − 𝛼) 𝑛,𝑛−1 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 𝑥
 𝛿 |
Γ(2 − 𝛼) 𝑛,𝑚
( )
(∆𝑥)−𝛼
+0.5 |𝑣𝑥 | (2 − 2 cos𝜙)𝛽Γ(2 − 𝛼) 𝑚
The assumption is made where, 
125 
 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷
𝛿  𝛼
𝐿
𝑛,𝑛−1 > 0.5𝑣 𝛿
 𝛼 +  
Γ(2 − 𝛼) 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
(5-155) 
Then, 
(∆𝑡)−𝛼  𝛼 (∆𝑥)
−𝛼
 𝛼 2𝐷𝐿 (∆𝑥)
−𝛼 𝐷
𝛿 − 0.5𝑣  𝛿 − + 0.5𝑣  𝛿  𝛼 + 𝐿
Γ(2 − 𝛼) 𝑛,𝑛−1 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
( )
𝐷𝐿 (∆𝑥)
−𝛼
+ 2 + 0.5𝑣𝑥 (2 − 2 cos𝜙)𝛽(∆𝑥) Γ(𝛼) 𝑚
< 1 
(∆𝑡)−𝛼 −𝛼 −𝛼 (5-156) 
𝛿  𝛼
(∆𝑥)  𝛼 (∆𝑥)  𝛼
Γ(2 − 𝛼) 𝑛,𝑛−1
+ 0.5𝑣𝑥 𝛿 + 0.5𝑣  𝛿Γ(2 − 𝛼) 𝑛,𝑚 𝑥 Γ(2 − 𝛼) 𝑛,𝑚
( )
(∆𝑥)−𝛼
+0.5𝑣𝑥 (2 − 2 cos𝜙)𝛽Γ(2 − 𝛼) 𝑚
Simplifying, 
(∆𝑥)−𝛼
𝑣𝑥 𝛿
 𝛼
Γ(2 − 𝛼) 𝑛,𝑚
> 0 (5-157) 
Under the assumption made in Equation (5-155), the upwind advection Crank-Nicolson scheme for 
the fractional advection-dispersion equation with Caputo fractional derivative is unconditionally 
stable. 
When the complementary assumption is made where, 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷
𝛿  𝛼
𝐿
𝑛,𝑛−1 < 0.5𝑣 𝛿
 𝛼
Γ(2 − 𝛼) 𝑥 Γ(2 − 𝛼) 𝑛,𝑚
+  (5-158) 
(∆𝑥)2
Then, 
(∆𝑡)−𝛼 −𝛼 −𝛼
− 𝛿  𝛼
(∆𝑥) 2𝐷 (∆𝑥) 𝐷
+ 0.5𝑣  𝛿  𝛼 + 𝐿 + 0.5𝑣  𝛿  𝛼 + 𝐿
Γ(2 − 𝛼) 𝑛,𝑛−1 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
( )
𝐷𝐿 (∆𝑥)
−𝛼
+ 2 + 0.5𝑣𝑥 (2 − 2 cos𝜙)𝛽(∆𝑥) Γ(2 − 𝛼) 𝑚
−𝛼 < 1 (∆𝑡)  𝛼 0.5(∆𝑥)
−𝛼 −𝛼 (5-159) 
𝛿 + 𝑣 𝛿  𝛼
0.5(∆𝑥)  𝛼
Γ(2 − 𝛼) 𝑛,𝑛−1 𝑥 𝑛,𝑚
+ 𝑣𝑥  𝛿Γ(2 − 𝛼) Γ(2 − 𝛼) 𝑛,𝑚
( )
(∆𝑥)−𝛼
+0.5𝑣𝑥 (2 − 2 cos𝜙)𝛽Γ(2 − 𝛼) 𝑚
Simplifying, 
2𝐷𝐿 (∆𝑡)
−𝛼
 <   
(∆𝑥)2 Γ(2 − 𝛼) (5-160) 
Under this assumption (Equation (5-158)), the upwind advection Crank-Nicolson scheme for the 
fractional advection-dispersion equation with Caputo fractional derivative is conditionally stable. 
Secondly, it is assumed that |?̂?𝑛−1| < |?̂?𝑜| is true for all time steps, and the second component of the 
numerical stability analysis is to demonstrate for a set ∀ 𝑛 ≥ 1, that 
|?̂?𝑛| < |?̂?𝑜|  
Rearranging Equation (5-134) for ?̂?𝑛, 
126 
 
𝑚−1
(𝑜 + 𝑟 𝑒−𝑗𝑘𝑖∆𝑥 +  𝑙 (0.5(1 − 𝑒−𝑗𝑘𝑖∆𝑥) ∑ 𝛿  𝛼3 3 𝑚,𝑖)) ?̂?𝑛
𝑖=1
𝑚−1
= (𝑝3 + 𝑞3𝑒
−𝑗𝑘𝑖∆𝑥 + 𝑓 𝑒𝑗𝑘𝑖∆𝑥 − 𝑙 (0.5(1 − 𝑒−𝑗𝑘𝑖∆𝑥) ∑ 𝛿  𝛼 )) ?̂?  (5-161) 3 3 𝑚,𝑖 𝑛−1
𝑖=1
𝑛−2
−ℎ  𝛼3 (∑(?̂?𝑘+1 − ?̂?𝑘)𝛿𝑛,𝑘) 
𝑘=0
Following a similar process as previously, 
(𝑜 + 𝑟 𝑒−𝑗𝑘𝑖∆𝑥3 3 + 𝑙3 (0.5(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)(𝑚
1−𝛼 − (−1)1−𝛼))) ?̂?𝑛 
= (𝑝3 + 𝑞 𝑒
−𝑗𝑘𝑖∆𝑥
3 + 𝑓3𝑒
𝑗𝑘𝑖∆𝑥 − 𝑙 1−𝛼 1−𝛼3 (0.5(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)(𝑚 − (−1) ))) ?̂?𝑛−1 (5-162) 
𝑛−2
−ℎ3 (∑(?̂?𝑘+1 − ?̂?𝑘)𝛿
 𝛼
𝑛,𝑘) 
𝑘=0
Taking the norm on both sides, 
|(𝑜 + 𝑟 𝑒−𝑗𝑘𝑖∆𝑥3 3 + 𝑙3(0.5(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)(𝑚
1−𝛼 − (−1)1−𝛼))) ?̂?𝑛| 
= |(𝑝 + 𝑞 𝑒−𝑗𝑘𝑖∆𝑥 + 𝑓 𝑒𝑗𝑘𝑖∆𝑥 − 𝑙3(0.5(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)(𝑚
1−𝛼 − (−1)1−𝛼))
3 3 3 ) ?̂?𝑛−1 (5-163) 
𝑛−2
− ℎ3 (∑(?̂?𝑘+1 − ?̂?𝑘)𝛿
 𝛼
𝑛,𝑘)| 
𝑘=0
Thus, 
|𝑜3 + 𝑟3𝑒
−𝑗𝑘𝑖∆𝑥 + 𝑙3 (0.5(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)(𝑚
1−𝛼 − (−1)1−𝛼))| |?̂?𝑛| 
< |𝑝3 + 𝑞
−𝑗𝑘𝑖∆𝑥
3𝑒 + 𝑓3𝑒
𝑗𝑘𝑖∆𝑥 − 𝑙3 (0.5(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)(𝑚
1−𝛼 − (−1)1−𝛼))| |?̂?𝑛−1| 
(5-164) 
𝑛−2
+|ℎ3| |∑(?̂? − ?̂? )𝛿
 𝛼
𝑘+1 𝑘 𝑛,𝑘| 
𝑘=0
Remembering that it has been proved that for a set ∀ 𝑛 ≥ 1,   
|?̂?𝑛−1| < |?̂?𝑜| 
Thus, 
|𝑜 + 𝑟 𝑒−𝑗𝑘𝑖∆𝑥3 3 + 𝑙3 (0.5(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)(𝑚
1−𝛼 − (−1)1−𝛼))| |?̂?𝑛| 
< |𝑝 + 𝑞 𝑒−𝑗𝑘𝑖∆𝑥3 3 + 𝑓3𝑒
𝑗𝑘𝑖∆𝑥 − 𝑙3 (0.5(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)(𝑚
1−𝛼 − (−1)1−𝛼))| |?̂?𝑛−1| (5-165) 
𝑛−2
+|ℎ3| |∑(?̂? − ?̂? )𝛿
 𝛼
𝑘+1 𝑘 𝑛,𝑘| 
𝑘=0
127 
 
< |𝑝 + 𝑞 𝑒−𝑗𝑘𝑖∆𝑥 + 𝑓 𝑒𝑗𝑘𝑖∆𝑥 − 𝑙 (0.5(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)(𝑚1−𝛼 − (−1)1−𝛼3 3 3 3 ))| |?̂?0| 
𝑛−2
+|ℎ3| |∑(?̂?
 𝛼
𝑘+1 − ?̂?𝑘)𝛿𝑛,𝑘| 
𝑘=0
and, it can be inferred that, 
|𝑜 + 𝑟 𝑒−𝑗𝑘𝑖∆𝑥3 3 + 𝑙3 (0.5(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)(𝑚
1−𝛼 − (−1)1−𝛼))| |?̂?𝑛| 
< |𝑝3 + 𝑞 𝑒
−𝑗𝑘𝑖∆𝑥
3 + 𝑓 𝑒
𝑗𝑘𝑖∆𝑥
3 − 𝑙3 (0.5(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)(𝑚
1−𝛼 − (−1)1−𝛼))| |?̂?0| (5-166) 
𝑛−2
+|ℎ3| |∑(?̂?𝑘+1 − ?̂?𝑘)𝛿
 𝛼
𝑛,𝑘| 
𝑘=0
A simplification is performed as in Section 5.2.1, 
|𝑜 + 𝑟 −𝑗𝑘𝑖∆𝑥3 3𝑒 + 𝑙3(0.5(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)𝛽𝑚)||?̂?𝑛| 
< |𝑝 + 𝑞 𝑒−𝑗𝑘𝑖∆𝑥 + 𝑓 𝑒𝑗𝑘𝑖∆𝑥
(5-167) 
3 3 3 − 𝑙3(0.5(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)𝛽𝑚)||?̂?0|+ |ℎ3||?̂?0|𝛽𝑛 
Simplifying and rearranging, 
|?̂?𝑛| |𝑝 | + |𝑞 | + |𝑓 | +  0.5|𝑙3|(2 − 2 cos𝜙)𝛽 + |ℎ3|𝛽
< 3 3 3 𝑚 𝑛 
|?̂? | |𝑜 | + |𝑟 | + |𝑙 |(2 − 2 cos 𝜃)𝛽 (5-168) 0 3 3 3 𝑚
Thus, the solution will be stable when, 
|𝑝 | + |𝑞 |+ |𝑓 |+  0.5|𝑙3|(2 − 2 cos𝜙)𝛽 + |ℎ3|𝛽3 3 3 𝑚 𝑛 < 1 (5-169) 
|𝑜3| + |𝑟3| + |𝑙3|(2 − 2 cos𝜙)𝛽𝑚
The term is expanded using the simplification terms associated with Equation (5-28), 
(∆𝑡)−𝛼  𝛼 (∆𝑥)
−𝛼
 𝛼 2𝐷 (∆𝑥)
−𝛼 𝐷
| 𝛿 𝐿𝑛,𝑛−1 − 0.5𝑣𝑥  𝛿𝑛,𝑚 − 2|+ |0.5𝑣𝑥  𝛿
 𝛼 + 𝐿 |
Γ(2 − 𝛼) Γ(2 − 𝛼) (∆𝑥) Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
( )
𝐷𝐿 (∆𝑥)
−𝛼 (∆𝑡)−𝛼
+ | 2|+  0.5 |𝑣𝑥 | ((2 − 2 cos𝜙)𝛽 ) + | |𝛽(∆𝑥) Γ(2 − 𝛼) 𝑚 Γ(2 − 𝛼) 𝑛
< 1 (5-170) 
(∆𝑡)−𝛼  𝛼 (∆𝑥)−𝛼 (∆𝑥)−𝛼| 𝛿 + 0.5𝑣 𝛿 𝛼 | + |0.5𝑣  𝛿 𝛼 |
Γ(2 − 𝛼) 𝑛,𝑛−1 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 𝑥 Γ(2 − 𝛼) 𝑛,𝑚
( )
(∆𝑥)−𝛼
+ |𝑣𝑥 | (2 − 2 cos𝜙)𝛽Γ(2 − 𝛼) 𝑚
The assumption is made where, 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷
𝛿  𝛼  𝛼
𝐿
Γ(2 − 𝛼) 𝑛,𝑛−1
> 𝑣𝑥 𝛿 +  (5-171) Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
Then, 
128 
 
(∆𝑡)−𝛼 −𝛼 −𝛼
𝛿 𝛼
0.5(∆𝑥)  𝛼 2𝐷𝐿 0.5(∆𝑥) 𝐷
𝑛,𝑛−1 − 𝑣𝑥  𝛿𝑛,𝑚 − 2 + 𝑣𝑥  𝛿
 𝛼 𝐿
Γ(2 − 𝛼) Γ(2 − 𝛼) (∆𝑥) Γ(2 − 𝛼) 𝑛,𝑚
+
(∆𝑥)2
( )
𝐷 (∆𝑥)−𝛼 (∆𝑡)−𝛼
+ 𝐿 2 +  0.5𝑣𝑥 ((2 − 2 cos𝜙)𝛽𝑚) + 𝛽(∆𝑥) Γ(2 − 𝛼) Γ(2 − 𝛼) 𝑛
< 1 
(∆𝑡)−𝛼  𝛼 0.5(∆𝑥)−𝛼 0.5(∆𝑥)−𝛼 (5-172) 𝛿 + 𝑣 𝛿 𝛼 + 𝑣  𝛿 𝛼
Γ(2 − 𝛼) 𝑛,𝑛−1 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 𝑥 Γ(2 − 𝛼) 𝑛,𝑚
( )
(∆𝑥)−𝛼
+𝑣𝑥 (2 − 2 cos𝜙)𝛽Γ(2 − 𝛼) 𝑚
Simplifying, 
(∆𝑡)−𝛼 (∆𝑥)−𝛼
𝛽 < 𝑣 (𝛿  𝛼𝑛 𝑥 𝑛,𝑚  + 0.5𝛽𝑚(2 − 2 cos𝜙)) Γ(2 − 𝛼) Γ(2 − 𝛼) (5-173) 
The assumption made in Equation (5-171), leads to a conditionally stable result for the upwind 
advection Crank-Nicolson scheme. 
When the complementary assumption is made where, 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷
𝛿  𝛼
𝐿
𝑛,𝑛−1 < 𝑣 𝛿
 𝛼 +  (5-174) 
Γ(2 − 𝛼) 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
Then, 
(∆𝑡)−𝛼  𝛼 (∆𝑥)
−𝛼 2𝐷 (∆𝑥)−𝛼 𝐷
− 𝛿𝑛,𝑛−1 + 0.5𝑣𝑥  𝛿
 𝛼 𝐿  𝛼 𝐿
Γ(2 − 𝛼) Γ(2 − 𝛼) 𝑛,𝑚
+ 2 + 0.5𝑣(∆𝑥) 𝑥
 𝛿 +
Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
( )
𝐷𝐿 (∆𝑥)
−𝛼 (∆𝑡)−𝛼
+ +  0.5𝑣 ((2 − 2 cos𝜙)𝛽 ) + 𝛽
(∆𝑥)2 𝑥 Γ(2 − 𝛼) 𝑚 Γ(2 − 𝛼) 𝑛
< 1 
(∆𝑡)−𝛼  𝛼 (∆𝑥)−𝛼 (∆𝑥)−𝛼
(5-175) 
𝛿  𝛼  𝛼
Γ(2 − 𝛼) 𝑛,𝑛−1
+ 0.5𝑣𝑥 𝛿Γ(2 − 𝛼) 𝑛,𝑚
+ 0.5𝑣𝑥  𝛿Γ(2 − 𝛼) 𝑛,𝑚
(
(∆𝑥)−𝛼
)
+𝑣𝑥 (2 − 2 cos𝜙)𝛽Γ(2 − 𝛼) 𝑚
Simplifying, 
4𝐷𝐿 (∆𝑡)
−𝛼 2(∆𝑡)−𝛼 (∆𝑥)−𝛼
+ 𝛽𝑛 < + 0.5𝑣𝑥 (2 − 2 cos𝜙)𝛽  (∆𝑥)2 Γ(2 − 𝛼) Γ(2 − 𝛼) Γ(2 − 𝛼) 𝑚 (5-176) 
Under this assumption (Equation (5-174)), the upwind advection Crank-Nicolson scheme for the 
fractional advection-dispersion equation with Caputo fractional derivative is conditionally stable. 
The stability criterion for the upwind advection Crank-Nicolson finite difference scheme for the 
fractional advection-dispersion equation (Caputo) has been determined by the induction method. The 
stability criteria are thus as follows, 
2𝐷 −𝛼𝐿 (∆𝑡)
< , 
(∆𝑥)2 Γ(2 − 𝛼)
(∆𝑡)−𝛼 (∆𝑥)−𝛼
𝛽 < 𝑣 (𝛿  𝛼  + 𝛽 (1 − cos𝜙)), 
Γ(2 − 𝛼) 𝑛 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 𝑚
and, 
129 
 
4𝐷 (∆𝑡)−𝛼 2(∆𝑡)−𝛼 (∆𝑥)−𝛼𝐿
+ 𝛽 < + 𝑣 (1 − cos𝜙)𝛽  
(∆𝑥)2 Γ(2 − 𝛼) 𝑛 Γ(2 − 𝛼) 𝑥 Γ(𝛼) 𝑚
Under these conditions, the error of the approximation is not propagated throughout the solution, 
but rather decreases with each time step, where for all values of n, |?̂?𝑛+1| < |?̂?𝑜|. 
5.2.4 Explicit upwind-downwind weighted scheme  
To perform the stability analysis, induction method terms are substituted into the explicit upwind-
downwind weighted numerical scheme presented in Section 5.1.4, 
𝑎 ?̂? 𝑒𝑗𝑘𝑖𝑥 = 𝑠 ?̂? 𝑒𝑗𝑘𝑖𝑥 + 𝑢 ?̂? 𝑒𝑗𝑘𝑖(𝑥−∆𝑥)3 𝑛 3 𝑛−1 3 𝑛−1 + 𝑣3?̂? 𝑒
𝑗𝑘𝑖(𝑥+∆𝑥)
𝑛−1  
𝑛−2
−ℎ (∑(?̂? 𝑒𝑗𝑘𝑚𝑥 − ?̂? 𝑒𝑗𝑘𝑚𝑥)𝛿  𝛼3 𝑘+1 𝑘 𝑛,𝑘) 
𝑘=0 (5-177) 
𝑚−1
− 𝑙 (∑ (𝜃(?̂? 𝑒𝑗𝑘𝑖𝑥 − ?̂? 𝑒𝑗𝑘𝑖(𝑥−∆𝑥)) + (1 − 𝜃)(?̂? 𝑒𝑗𝑘𝑖(𝑥+∆𝑥) − ?̂? 𝑒𝑗𝑘𝑖𝑥  𝛼3 𝑛−1 𝑛−1 𝑛−1 𝑛−1 )) 𝛿𝑚,𝑖) 
𝑖=1
Multiple out, 
𝑎 ?̂? 𝑒𝑗𝑘𝑖𝑥3 𝑛 = 𝑠 ?̂? 𝑒
𝑗𝑘𝑖𝑥 + 𝑢 ?̂? 𝑒𝑗𝑘𝑖𝑥𝑒−𝑗𝑘𝑖∆𝑥 + 𝑣 ?̂? 𝑒𝑗𝑘𝑖𝑥𝑒𝑗𝑘𝑖∆𝑥3 𝑛−1 3 𝑛−1 3 𝑛−1  
𝑛−2
−ℎ3 (∑(?̂?
𝑗𝑘𝑚𝑥
𝑘+1𝑒 − ?̂? 𝑒
𝑗𝑘𝑚𝑥)𝛿  𝛼𝑘 𝑛,𝑘) (5-178) 
𝑘=0
𝑚−1
− 𝑙3 (∑ (𝜃(?̂?
𝑗𝑘𝑖𝑥 𝑗𝑘𝑖𝑥 −𝑗𝑘𝑖∆𝑥
𝑛−1𝑒 − ?̂?𝑛−1𝑒 𝑒 ) + (1 − 𝜃)(?̂? 𝑒
𝑗𝑘𝑖𝑥𝑒𝑗𝑘𝑖∆𝑥 − ?̂? 𝑒𝑗𝑘𝑖𝑥  𝛼𝑛−1 𝑛−1 )) 𝛿𝑚,𝑖) 
𝑖=1
 
Divide by 𝑒𝑗𝑘𝑖𝑥, 
𝑛−2
𝑎3?̂? = 𝑠 ?̂? + 𝑢 ?̂? 𝑒
−𝑗𝑘𝑖∆𝑥 + 𝑣 ?̂? 𝑒𝑗𝑘𝑖∆𝑥𝑛 3 𝑛−1 3 𝑛−1 3 𝑛−1 − ℎ3 (∑(?̂?
 𝛼
𝑘+1 − ?̂?𝑘)𝛿𝑛,𝑘) 
𝑘=0
𝑚−1 (5-179) 
− 𝑙3 (∑ (𝜃(?̂?𝑛−1 − ?̂?
−𝑗𝑘𝑖∆𝑥
𝑛−1𝑒 ) + (1 − 𝜃)(?̂? 𝑒
𝑗𝑘𝑖∆𝑥
𝑛−1 − ?̂?
 𝛼
𝑛−1)) 𝛿𝑚,𝑖) 
𝑖=1
The induction numerical stability analysis is performed in two parts; firstly it is proved for a set ∀ 𝑛 > 1,  
|?̂?𝑛| < |?̂?  𝑜| 
If 𝑛 = 1, then 
𝑚−1
𝑎 −𝑗𝑘𝑖∆𝑥 𝑗𝑘𝑖∆𝑥3?̂?1 = 𝑠3?̂?0 + 𝑢3?̂?0𝑒 + 𝑣3?̂?0𝑒 − 𝑙3 (∑ (𝜃(?̂? − ?̂? 𝑒
−𝑗𝑘𝑖∆𝑥
0 0 ) + (1 − 𝜃)(?̂? 𝑒
𝑗𝑘𝑖∆𝑥
0 − ?̂?0)) 𝛿
 𝛼
𝑚,𝑖) (5-180) 
𝑖=1
A subset for 𝑚 is now considered, where 𝑚 = 1, 
𝑎3?̂?1 = 𝑠3?̂? + 𝑢
−𝑗𝑘𝑖∆𝑥
0 3?̂?0𝑒 + 𝑣3?̂? 𝑒
𝑗𝑘𝑖∆𝑥 (5-181) 0
Simplifying, 
130 
 
𝑎 ?̂? = (𝑠 + 𝑢 𝑒−𝑗𝑘𝑖∆𝑥 + 𝑣 𝑒𝑗𝑘𝑖∆𝑥)?̂?  (5-182) 3 1 3 3 3 0
Rearranging, 
?̂? −𝑗𝑘𝑖∆𝑥 𝑗𝑘𝑖∆𝑥1 𝑠3 + 𝑢3𝑒 + 𝑣3𝑒
=  (5-183) 
?̂?0 𝑎3
Applying the norm on both sides, 
|?̂?1| |𝑠3| + |𝑢3𝑒
−𝑗𝑘𝑖∆𝑥| + |𝑣 −𝑗𝑘𝑖∆𝑥3𝑒 |
=  
|?̂? (5-184) 0| |𝑎3|
The condition can thus be represented as, 
|?̂?1|
< 1  
|?̂?0|
And, remembering |𝑒𝑛| = 1, the condition is, 
|𝑠3| + |𝑢3| + |𝑣3|
< 1  
|𝑎3|
The term is expanded using the simplification terms associated with Equation (5-33), 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷 (∆𝑥)−𝛼 𝐷
| 𝛿  𝛼𝑛,𝑛−1 + 𝑣𝑥(2𝜃 − 1) 𝛿
 𝛼 𝐿
𝑛,𝑚 − 2| + |𝑣𝑥𝜃 𝛿
 𝛼 + 𝐿 |
Γ(2 − 𝛼) Γ(2 − 𝛼) (∆𝑥) Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
( )
(∆𝑥)−𝛼 𝐷
+ |𝑣  𝛼 𝐿𝑥(1 − 𝜃) 𝛿𝑛,𝑚 + |Γ(2 − 𝛼) (∆𝑥)2 (5-185) 
< 1 
(∆𝑡)−𝛼
| 𝛿  𝛼𝑛,𝑛−1|Γ(2 − 𝛼)
The assumption is made where, 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷
𝛿  𝛼
𝐿
𝑛,𝑛−1 + 𝑣𝑥(2𝜃 − 1) 𝛿
 𝛼 >  (5-186) 
Γ(2 − 𝛼) Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
Then, 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 𝛼  𝛼 2𝐷
−𝛼
𝛿 + 𝑣 (2𝜃 − 1) 𝛿 − 𝐿
(∆𝑥)  𝛼 𝐷𝐿
Γ(2 − 𝛼) 𝑛,𝑛−1 𝑥 Γ(2 − 𝛼) 𝑛,𝑚
+ 𝑣
(∆𝑥)2 𝑥
𝜃 𝛿
Γ(2 − 𝛼) 𝑛,𝑚
+
(∆𝑥)2
(
(∆𝑥)−𝛼
)
𝐷
+𝑣 (1 − 𝜃) 𝛿  𝛼 + 𝐿𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2 (5-187) 
(∆𝑡)−𝛼
< 1 
𝛿  𝛼
Γ(2 − 𝛼) 𝑛,𝑛−1
Under this assumption (Equation (5-186)), the explicit upwind-downwind weighted scheme for the 
fractional advection-dispersion equation with Caputo fractional derivative is unstable. 
When the opposite assumption is made where, 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷
𝛿  𝛼
𝐿
𝑛,𝑛−1 + 𝑣𝑥(2𝜃 − 1) 𝛿
 𝛼 <  (5-188) 
Γ(2 − 𝛼) Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
Then, 
131 
 
(∆𝑡)−𝛼  𝛼 (∆𝑥)
−𝛼 2𝐷 (∆𝑥)−𝛼 𝐷
− 𝛿𝑛,𝑛−1 − 𝑣𝑥(2𝜃 − 1) 𝛿
 𝛼 + 𝐿 + 𝑣 𝜃 𝛿  𝛼 + 𝐿
Γ(2 − 𝛼) Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
( −𝛼 )(∆𝑥) 𝐷
+𝑣𝑥(1 − 𝜃) 𝛿
 𝛼 𝐿 (5-189) 
Γ(2 − 𝛼) 𝑛,𝑚
+
(∆𝑥)2
(∆𝑡)−𝛼
< 1 
𝛿  𝛼
Γ(2 − 𝛼) 𝑛,𝑛−1
Simplifying, 
(∆𝑥)−𝛼 2𝐷𝐿 (∆𝑡)
−𝛼
𝑣𝑥(1 − 2𝜃) 𝛿
 𝛼
Γ(2 − 𝛼) 𝑛,𝑚
+ <  
(∆𝑥)2 Γ(2 − 𝛼) (5-190) 
Under this assumption (Equation (5-188)), the explicit upwind-downwind weighted scheme for the 
fractional advection-dispersion equation with Caputo fractional derivative is conditionally stable. 
A subset for 𝑚 is now considered for all 𝑚 > 1, 
𝑎3?̂?1 = 𝑠3?̂?0 + 𝑢 ?̂? 𝑒
−𝑗𝑘𝑖∆𝑥 + 𝑣 ?̂? 𝑒𝑗𝑘𝑖∆𝑥3 0 3 0  
𝑚−1 (5-191) 
− 𝑙3 (∑ (𝜃(?̂?0 − ?̂? 𝑒
−𝑗𝑘𝑖∆𝑥) + (1 − 𝜃)(?̂? 𝑒𝑗𝑘𝑖∆𝑥  𝛼0 0 − ?̂?0)) 𝛿𝑚,𝑖) 
𝑖=1
Simplifying, 
𝑚−1
𝑠 + 𝑢 𝑒−𝑗𝑘𝑖∆𝑥 + 𝑣 𝑒𝑗𝑘𝑖∆𝑥 − 𝑙 𝜃(1 − 𝑒−𝑗𝑘𝑖∆𝑥3 3 3 3 ) (∑ 𝛿
 𝛼
 𝑚,𝑖
)
 
𝑎 ?̂? =  𝑖=13 1 𝑚−1  ?̂?0 (5-192)   
− 𝑙 (1 − 𝜃)(𝑒𝑗𝑘𝑖∆𝑥 − 1)(∑ 𝛿  𝛼3 𝑚,𝑖)
( 𝑖=1 )
Considering, 
‖𝑒𝑗𝑘𝑖∆𝑥 − 1‖ = (cos𝜙 − 1)2 + 𝑠𝑖𝑛2 𝜙 
                                       = cos2𝜙 + 𝑠𝑖𝑛2 𝜙 − 2 cos𝜙 + 1 
    = 2 − 2 cos𝜙 
Thus, 
|𝑎3||?̂?1| = (|𝑠3| + |𝑢3| + |𝑣3| + |𝑙3|𝜃(2 − 2 cos𝜙)𝛽𝑚 + |𝑙3|(1 − 𝜃)(2 − 2 cos𝜙)𝛽𝑚)|?̂?0| (5-193) 
Rearranging, 
|?̂?1| |𝑠3| + |𝑢3| + |𝑣3| + |𝑙3|𝜃(2 − 2 cos𝜙)𝛽𝑚 + |𝑙3|(1 − 𝜃)(2 − 2 cos𝜙)𝛽𝑚
=  
|?̂? | |𝑎 | (5-194) 0 3
The stability condition required transforms to, 
|?̂?1|
< 1  
|?̂?0|
The condition becomes, 
|𝑠3| + |𝑢3| + |𝑣3| + |𝑙3|𝜃(2 − 2 cos𝜙)𝛽𝑚 + |𝑙3|(1 − 𝜃)(2 − 2 cos𝜙)𝛽𝑚
< 1 (5-195) 
|𝑎3|
The term is expanded using the simplification terms associated with Equation (5-33), 
132 
 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷 (∆𝑥)−𝛼 𝐷
| + 𝑣𝑥(2𝜃 − 1) 𝛿
 𝛼 − 𝐿 | + |𝑣 𝜃 𝛿  𝛼 + 𝐿 |
Γ(2 − 𝛼) Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
  
(∆𝑥)−𝛼  𝛼 𝐷 (∆𝑥)
−𝛼
 + |𝑣𝑥(1 − 𝜃) 𝛿 +
𝐿 | + |𝑣 | 𝜃(2 − 2 cos𝜙)𝛽
Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2 𝑥 Γ(2 − 𝛼) 𝑚
 
  
−𝛼 (5-196) (∆𝑥)
+ |𝑣𝑥 | (1 − 𝜃)(2 − 2 cos𝜙)𝛽( Γ(2 − 𝛼) 𝑚 )
< 1 
(∆𝑡)−𝛼
| |
Γ(2 − 𝛼)
The assumption is made where, 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷𝐿
+ 𝑣𝑥(2𝜃 − 1) 𝛿
 𝛼
𝑛,𝑚 >  (5-197) Γ(2 − 𝛼) Γ(2 − 𝛼) (∆𝑥)2
Then, 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷 (∆𝑥)−𝛼 𝐷
+ 𝑣𝑥(2𝜃 − 1) 𝛿
 𝛼 − 𝐿 + 𝑣 𝜃 𝛿  𝛼 + 𝐿
Γ(2 − 𝛼) Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
 
(∆𝑥)−𝛼 𝐷 (∆𝑥)−𝛼
 
 +𝑣𝑥(1 − 𝜃) 𝛿
 𝛼 + 𝐿 + 𝑣
Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2 𝑥
𝜃 (2 − 2 cos𝜙)𝛽  
 Γ(2 − 𝛼)
𝑚
 
(∆𝑥)−𝛼 (5-198) 
+𝑣𝑥 (1 − 𝜃)(2 − 2 cos𝜙)𝛽( Γ(2 − 𝛼) 𝑚 )
< 1 
(∆𝑡)−𝛼
Γ(2 − 𝛼)
Simplifying, it becomes clear that the upwind-downwind weighted (explicit) numerical scheme for the 
fractional advection-dispersion equation with Caputo fractional derivative is unstable. 
Making the complementary assumption, 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷𝐿
+ 2𝑣𝑥(2𝜃 − 1) 𝛿
 𝛼
𝑛,𝑚 <  (5-199) Γ(2 − 𝛼) Γ(2 − 𝛼) (∆𝑥)2
Then, 
(∆𝑡)−𝛼 (∆𝑥)−𝛼  𝛼 2𝐷𝐿 (∆𝑥)
−𝛼 𝐷
− − 2𝑣 (2𝜃 − 1) 𝛿 + + 2𝑣 𝜃 𝛿  𝛼 + 𝐿
Γ(2 − 𝛼) 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
  
(
 ∆𝑥
)−𝛼 𝐷 (∆𝑥)−𝛼
+2𝑣𝑥(1 − 𝜃) 𝛿
 𝛼
𝑛,𝑚 +
𝐿
2 + 2𝑣𝑥𝜃 (2 − 2 cos𝜙)𝛽   Γ(2 − 𝛼) (∆𝑥) Γ(2 − 𝛼)
𝑚
 
(∆𝑥)−𝛼 (5-200) 
+2𝑣𝑥 (1 − 𝜃)(2 − 2 cos𝜙)𝛽( Γ(2 − 𝛼) 𝑚 )
−𝛼 < 1 (∆𝑡)
Γ(2 − 𝛼)
Simplifying, 
2𝐷 (∆𝑡)−𝛼𝐿 (∆𝑥)
−𝛼
< + 2𝑣𝑥 (𝛿
 𝛼
𝑛,𝑚(𝜃 + 1) + 𝛽𝑚(1 − (cos𝜙))) (∆𝑥)2 Γ(2 − 𝛼) Γ(2 − 𝛼) (5-201) 
Under this assumption (Equation (5-199)), the explicit upwind-downwind weighted scheme for the 
fractional advection-dispersion equation with Caputo fractional derivative is conditionally stable. 
133 
 
Next, making the assumption that |?̂?𝑛−1| < |?̂?𝑜| is true for all time steps, the second part of the 
numerical stability analysis is to demonstrate for a set ∀ 𝑛 ≥ 1, that 
|?̂?𝑛| < |?̂?𝑜|  
Rearranging Equation (5-179) for ?̂?𝑛, 
𝑠 + 𝑢 𝑒−𝑗𝑘𝑖∆𝑥 + 𝑣 𝑒𝑗𝑘𝑖∆𝑥3 3 3
𝑚−1
𝑎3?̂?𝑛 = (
− 𝑙 (∑ (𝜃(1 − 𝑒−𝑗𝑘𝑖∆𝑥) + (1 − 𝜃)(𝑒𝑗𝑘𝑖∆𝑥 − 1)) 𝛿  𝛼
) ?̂?𝑛−1 
3 𝑚,𝑖)
𝑖=1 (5-202) 
𝑛−2
−ℎ  𝛼3 (∑(?̂?𝑘+1 − ?̂?𝑘)𝛿𝑛,𝑘) 
𝑘=0
Applying a norm on both sides, 
𝑠 + 𝑢 𝑒−𝑗𝑘𝑖∆𝑥3 3 + 𝑣3𝑒
𝑗𝑘𝑖∆𝑥
𝑚−1
|( ) ?̂?𝑛−1|
− 𝑙3 (∑ (𝜃(1 − 𝑒
−𝑗𝑘𝑖∆𝑥) + (1 − 𝜃)(𝑒𝑗𝑘𝑖∆𝑥 − 1)) 𝛿  𝛼𝑚,𝑖)
|𝑎3?̂?𝑛| = 𝑖=1  (5-203) 
𝑛−2
| |
−ℎ (∑(?̂? − ?̂? )𝛿  𝛼3 𝑘+1 𝑘 𝑛,𝑘)
𝑘=0
Thus, 
𝑠 + 𝑢 𝑒−𝑗𝑘𝑖∆𝑥 𝑗𝑘𝑖∆𝑥3 3 + 𝑣3𝑒    
𝑚−1
|𝑎3||?̂?𝑛| < |( ) | |?̂? | 
− 𝑙 (∑ (𝜃(1 − 𝑒−𝑗𝑘𝑖∆𝑥3 ) + ( )
𝑗𝑘 ∆𝑥  𝛼 𝑛−11 − 𝜃 (𝑒 𝑖 − 1))𝛿𝑚,𝑖)
(5-204) 
𝑖=1
𝑛−2
−|ℎ3| |∑(?̂?𝑘+1 − ?̂?𝑘)𝛿
 𝛼
𝑛,𝑘| 
𝑘=0
Remembering that it has been proved that for a set ∀ 𝑛 ≥ 1,   
|?̂?𝑛−1| < |?̂?𝑜| 
Thus, 
𝑚−1
|𝑎 ||?̂? | < |(𝑠 + 𝑢 𝑒−𝑗𝑘𝑖∆𝑥 + 𝑣 𝑒𝑗𝑘𝑖∆𝑥   −  𝑙 (∑ (𝜃(1 − 𝑒−𝑗𝑘𝑖∆𝑥) + (1 − 𝜃)(𝑒𝑗𝑘𝑖∆𝑥3 𝑛 3 3 3 3 − 1)) 𝛿
 𝛼
𝑚,𝑖)) | |?̂?𝑛−1|
𝑖=1
𝑛−2
+ |ℎ  𝛼3| |∑(?̂?𝑘+1 − ?̂?𝑘)𝛿𝑛,𝑘| 
𝑘=0
(5-205) 
𝑚−1
     < |(𝑠 + 𝑢 𝑒−𝑗𝑘𝑖∆𝑥 + 𝑣 𝑒𝑗𝑘𝑖∆𝑥   − 𝑙 (∑ (𝜃(1 − 𝑒−𝑗𝑘𝑖∆𝑥3 3 3 3 ) + (1 − 𝜃)(𝑒
𝑗𝑘𝑖∆𝑥 − 1)) 𝛿  𝛼𝑚,𝑖)) | |?̂?0|
𝑖=1
𝑛−2
+ |ℎ3| |∑(?̂?
 𝛼
𝑘+1 − ?̂?𝑘)𝛿𝑛,𝑘| 
𝑘=0
and, it can be inferred that, 
134 
 
𝑚−1
|𝑎3||?̂?𝑛| < |(𝑠3 + 𝑢 𝑒
−𝑗𝑘𝑖∆𝑥
3 + 𝑣 𝑒
𝑗𝑘𝑖∆𝑥
3   − 𝑙3 (∑ (𝜃(1 − 𝑒
−𝑗𝑘𝑖∆𝑥) + (1 − 𝜃)(𝑒𝑗𝑘𝑖∆𝑥 − 1)) 𝛿  𝛼𝑚,𝑖)) | |?̂?0|
𝑖=1
(5-206) 
𝑛−2
+ |ℎ3| |∑(?̂? − ?̂? )𝛿
 𝛼
𝑘+1 𝑘 𝑛,𝑘| 
𝑘=0
Simplifying and rearranging, 
|?̂?𝑛| |𝑠3| + |𝑢3| + |𝑣3| + |𝑙3|𝜃(2 − 2 cos𝜙)𝛽 + |𝑙3|(1 − 𝜃)(2 − 2 cos𝜙)𝛽 + |ℎ |𝛽
< 𝑚 𝑚
3 𝑛 
|?̂?0| |𝑎3|
(5-207) 
Thus, the solution will be stable when, 
|𝑠3|+ |𝑢3| + |𝑣3| + |𝑙3|𝜃(2 − 2 cos𝜙)𝛽 + |𝑙3|(1 − 𝜃)(2 − 2 cos𝜙)𝛽 + |ℎ |𝛽𝑚 𝑚 3 𝑛 < 1 
|𝑎 | (5-208) 3
The term is expanded using the simplification terms associated with Equation (5-33), 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷
| 𝛿 𝛼𝑛,𝑛−1 + 𝑣
 𝛼 𝐿
Γ(2 − 𝛼) 𝑥
(2𝜃 − 1) 𝛿
Γ(2 − 𝛼) 𝑛,𝑚
− |
(∆𝑥)2
(∆𝑥)−𝛼 𝐷 (∆𝑥)−𝛼 𝐷
+ |𝑣𝑥𝜃 𝛿
 𝛼 + 𝐿𝑛,𝑚 2|+ |𝑣𝑥(1 − 𝜃) 𝛿
 𝛼
𝑛,𝑚 +
𝐿
Γ(2 − 𝛼) (∆𝑥) Γ(2 − 𝛼) (∆𝑥)2
|
(∆𝑥)−𝛼 (∆𝑥)−𝛼 (∆𝑡)−𝛼 (5-209) 
+ |𝑣𝑥 | 𝜃(2 − 2 cos𝜙)𝛽 + |𝑣𝑥 | (1 − 𝜃)(2 − 2 cos𝜙)𝛽 + 𝛽Γ(2 − 𝛼) 𝑚
| |
Γ(2 − 𝛼) 𝑚 Γ(2 − 𝛼) 𝑛
(∆𝑡)−𝛼
< 1 
| 𝛿 𝛼𝑛,𝑛−1|Γ(2 − 𝛼)
The assumption is made for the norm where, 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷
𝛿  𝛼
𝐿
𝑛,𝑛−1 + 𝑣 (2𝜃 − 1) 𝛿
 𝛼 >  (5-210) 
Γ(2 − 𝛼) 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
Then, 
(∆𝑡)−𝛼 (∆𝑥)−𝛼
𝛿 𝛼
2𝐷
+ 𝑣 (2𝜃 − 1) 𝛿 𝛼 − 𝐿
Γ(2 − 𝛼) 𝑛,𝑛−1 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
(∆𝑥)−𝛼  𝛼 𝐷𝐿 (∆𝑥)
−𝛼
 𝛼 𝐷+𝑣𝑥𝜃 𝛿𝑛,𝑚 + 2 + 𝑣𝑥(1 − 𝜃) 𝛿 +
𝐿
Γ(2 − 𝛼) (∆𝑥) Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
(∆𝑥)−𝛼 (∆𝑥)−𝛼 (∆𝑡)−𝛼 (5-211) 
+ 𝑣𝑥 𝜃(2 − 2 cos𝜙)𝛽 + 𝑣 (1 − 𝜃)(2 − 2 cos𝜙)𝛽 + 𝛽Γ(2 − 𝛼) 𝑚 𝑥 Γ(2 − 𝛼) 𝑚 Γ(2 − 𝛼) 𝑛
< 1 
(∆𝑡)−𝛼
𝛿 𝛼
Γ(2 − 𝛼) 𝑛,𝑛−1
Multiplying out and simplifying, it is found that the explicit upwind-downwind weighted scheme for 
the fractional advection-dispersion equation with Caputo fractional derivative is unstable under the 
assumption made in Equation (5-210). 
When the complementary assumption is made, 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷
𝛿  𝛼
𝐿
+ 𝑣 (2𝜃 − 1) 𝛿  𝛼 <  (5-212) 
Γ(2 − 𝛼) 𝑛,𝑛−1 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
135 
 
Then, 
(∆𝑡)−𝛼 (∆𝑥)−𝛼
− 𝛿 𝛼
2𝐷
𝑛,𝑛−1 − 𝑣𝑥(2𝜃 − 1) 𝛿
 𝛼 + 𝐿
Γ(2 − 𝛼) Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
(∆𝑥)−𝛼  𝛼 𝐷𝐿 (∆𝑥)
−𝛼 𝐷
+𝑣𝑥𝜃 𝛿𝑛,𝑚 + 2 + 𝑣𝑥(1 − 𝜃) 𝛿
 𝛼 + 𝐿
Γ(2 − 𝛼) (∆𝑥) Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2 (5-213) 
(∆𝑥)−𝛼 (∆𝑥)−𝛼 (∆𝑡)−𝛼
+𝑣𝑥 𝜃(2 − 2 cos𝜙)𝛽 + 𝑣 (1 − 𝜃)(2 − 2 cos𝜙)𝛽 + 𝛽Γ(2 − 𝛼) 𝑚 𝑥 Γ(2 − 𝛼) 𝑚 Γ(2 − 𝛼) 𝑛
−𝛼 < 1 (∆𝑡)
𝛿 𝛼
Γ(2 − 𝛼) 𝑛,𝑛−1
Simplifying, 
(∆𝑥)−𝛼 4𝐷 (∆𝑥)−𝛼 (∆𝑡)−𝛼
2𝑣 𝛿  𝛼
𝐿
𝑥 𝑛,𝑚 + +  2𝑣𝑥 𝛽𝑚(1 − cos𝜙) + (𝛽 − 2) < 0 Γ(2 − 𝛼) (∆𝑥)2 Γ(2 − 𝛼) Γ(2 − 𝛼) 𝑛 (5-214) 
Conversely, the explicit upwind-downwind weighted scheme for the fractional advection-dispersion 
equation with Caputo fractional derivative is also unstable under the complementary assumption 
made in Equation (5-212). Thus, the second assumption, |?̂?𝑛| < |?̂?𝑜|, has failed to produce a stable 
solution.  
5.2.5 Implicit upwind-downwind weighted scheme  
The developed implicit upwind-downwind weighted scheme discussed in Section 5.1.5 (Equation (5-
37)) is applied and induction method terms substituted, 
𝑎 ?̂? 𝑒𝑗𝑘𝑖𝑥 = 𝑠 ?̂? 𝑒𝑗𝑘𝑖𝑥 + 𝑢 ?̂? 𝑒𝑗𝑘𝑖(𝑥−∆𝑥) + 𝑣 ?̂? 𝑒𝑗𝑘𝑖(𝑥+∆𝑥)3 𝑛 3 𝑛−1 3 𝑛 3 𝑛  
𝑛−2
−ℎ3 (∑(?̂?𝑘+1𝑒
𝑗𝑘𝑚𝑥 − ?̂? 𝑗𝑘𝑚𝑥  𝛼𝑘𝑒 )𝛿𝑛,𝑘) 
(5-215) 
𝑘=0
𝑚−1
−𝑙 (∑ (𝜃(?̂? 𝑒𝑗𝑘𝑖𝑥 − ?̂? 𝑒𝑗𝑘𝑖(𝑥−∆𝑥) 𝑗𝑘𝑖(𝑥+∆𝑥) 𝑗𝑘𝑖𝑥  𝛼3 𝑛 𝑛 ) + (1 − 𝜃)(?̂?𝑛𝑒 − ?̂?𝑛𝑒 ))𝛿𝑚,𝑖) 
𝑖=1
Multiple out and dividing by 𝑒𝑗𝑘𝑖𝑥, 
𝑛−2
𝑎3?̂?𝑛 = 𝑠3?̂?𝑛−1 + 𝑢3?̂?𝑛𝑒
−𝑗𝑘𝑖∆𝑥 + 𝑣3?̂?
𝑗𝑘𝑖∆𝑥
𝑛𝑒 − ℎ3 (∑(?̂?𝑘+1 − ?̂?𝑘)𝛿
 𝛼
𝑛,𝑘) 
𝑘=0
𝑚−1 (5-216) 
− 𝑙3 (∑ (𝜃(?̂?𝑛 − ?̂?𝑛𝑒
−𝑗𝑘𝑖∆𝑥) + (1 − 𝜃)(?̂?𝑛𝑒
𝑗𝑘𝑖∆𝑥 − ?̂?𝑛)) 𝛿
 𝛼
𝑚,𝑖) 
𝑖=1
The induction numerical stability analysis is performed in two parts; firstly it is proved for a set ∀ 𝑛 > 1,  
|?̂? | < |?̂? |  𝑛 𝑜
If 𝑛 = 1, then 
𝑎3?̂?1 = 𝑠
−𝑗𝑘𝑖∆𝑥 𝑗𝑘𝑖∆𝑥
3?̂?0 + 𝑢3?̂?1𝑒 + 𝑣3?̂?1𝑒  
𝑚−1
(5-217) 
−𝑙3 (∑ (𝜃(?̂? − ?̂? 𝑒
−𝑗𝑘𝑖∆𝑥
1 1 ) + (1 − 𝜃)(?̂?
𝑗𝑘𝑖∆𝑥  𝛼
1𝑒 − ?̂?1)) 𝛿𝑚,𝑖) 
𝑖=1
136 
 
A subset for 𝑚 is now considered, where 𝑚 = 1, 
𝑎3?̂?1 = 𝑠3?̂?0 + 𝑢3?̂?1𝑒
−𝑗𝑘𝑖∆𝑥 + 𝑣3?̂? 𝑒
𝑗𝑘𝑖∆𝑥
1  
(5-218) 
Simplifying, 
(𝑎 − 𝑢 𝑒−𝑗𝑘𝑖∆𝑥 − 𝑣 𝑒𝑗𝑘𝑖∆𝑥)?̂? = 𝑠 ?̂?  (5-219) 3 3 3 1 3 0
Rearranging, 
?̂?1 𝑠3
=  (5-220) 
?̂?0 𝑎 − 𝑢 𝑒
−𝑗𝑘𝑖∆𝑥 − 𝑣 𝑒𝑗𝑘𝑖∆𝑥3 3 3
Taking a norm on both sides, 
|?̂?1| |𝑠3|
=  
|?̂?0| |𝑎3| + |𝑢 𝑒
−𝑗𝑘𝑖∆𝑥| + |𝑣 𝑒−𝑗𝑘𝑖∆𝑥| (5-221) 3 3
The stability condition is: 
|?̂?1|
< 1  
|?̂?0|
Remembering |𝑒𝑛| = 1, the condition becomes 
|𝑠3|
< 1  
|𝑎3| + |𝑢3| + |𝑣3|
The term is expanded using the simplification functions accompanying Equation (5-37), 
(∆𝑡)−𝛼
| 𝛿  𝛼𝑛,𝑛−1|Γ(2 − 𝛼)
< 1 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷 (∆𝑥)−𝛼 𝐷
| 𝛿  𝛼 + 𝑣 (2𝜃 − 1) 𝛿  𝛼 − 𝐿 | + |𝑣 𝜃 𝛿  𝛼 + 𝐿 |
Γ(2 − 𝛼) 𝑛,𝑛−1 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2 (5-222) 
(
(∆𝑥)−𝛼
)
 𝛼 𝐷+|𝑣 𝐿𝑥(1 − 𝜃) 𝛿 + |Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
The assumption is made where, 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷
𝛿  𝛼 + 𝑣 (2𝜃 − 1) 𝛿  𝛼
𝐿
>  (5-223) 
Γ(2 − 𝛼) 𝑛,𝑛−1 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
Then, 
(∆𝑡)−𝛼
𝛿  𝛼
Γ(2 − 𝛼) 𝑛,𝑛−1
< 1 
(∆𝑡)−𝛼  𝛼 (∆𝑥)
−𝛼
 𝛼 2𝐷𝐿 (∆𝑥)
−𝛼 𝐷
𝛿 + 𝑣 (2𝜃 − 1) 𝛿 − + 𝑣 𝜃 𝛿  𝛼 + 𝐿 (5-224) 
Γ(2 − 𝛼) 𝑛,𝑛−1 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
( )
(∆𝑥)−𝛼 𝐷
+𝑣 (1 − 𝜃) 𝛿  𝛼 𝐿𝑥 Γ(2 − 𝛼) 𝑛,𝑚
+
(∆𝑥)2
Simplifying, 
(∆𝑥)−𝛼
2𝑣𝑥𝜃 𝛿
 𝛼
Γ(2 − 𝛼) 𝑛,𝑚
> 0 (5-225) 
Under the assumption Equation (5-223), the implicit upwind-downwind weighted numerical scheme 
for the fractional advection-dispersion equation with Caputo fractional derivative is unconditionally 
stable, where 𝜃 > 0. 
137 
 
Making the complementary assumption, 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷
𝛿  𝛼
𝐿
𝑛,𝑛−1 + 𝑣𝑥(2𝜃 − 1) 𝛿
 𝛼
𝑛,𝑚 <  (5-226) Γ(2 − 𝛼) Γ(2 − 𝛼) (∆𝑥)2
Then, 
(∆𝑡)−𝛼
𝛿  𝛼
Γ(2 − 𝛼) 𝑛,𝑛−1
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷 (∆𝑥)−𝛼
< 1 
− 𝛿  𝛼
𝐷
𝑛,𝑛−1 − 𝑣𝑥(2𝜃 − 1) 𝛿
 𝛼 + 𝐿 + 𝑣 𝜃 𝛿  𝛼 + 𝐿
Γ(2 − 𝛼) Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2 (5-227) 
( )
(∆𝑥)−𝛼 𝐷
+𝑣𝑥(1 − 𝜃) 𝛿
 𝛼 𝐿
Γ(2 − 𝛼) 𝑛,𝑚
+
(∆𝑥)2
Simplifying, 
(∆𝑥)−𝛼 4𝐷𝐿 2(∆𝑡)
−𝛼
𝑣𝑥(2 − 𝜃) 𝛿
 𝛼
𝑛,𝑚 + >  Γ(2 − 𝛼) (∆𝑥)2 Γ(2 − 𝛼) (5-228) 
Under this assumption (Equation (5-226)), the implicit upwind-downwind weighted scheme for the 
fractional advection-dispersion equation with Caputo fractional derivative is conditionally stable. 
A subset for 𝑚 is now considered for all 𝑚 > 1, 
𝑎 ?̂? = 𝑠 ?̂? + 𝑢 ?̂? 𝑒−𝑗𝑘𝑖∆𝑥 + 𝑣 ?̂? 𝑒𝑗𝑘𝑖∆𝑥3 1 3 0 3 1 3 1
𝑚−1
(5-229) 
− 𝑙 (∑ (𝜃(?̂? − ?̂? 𝑒−𝑗𝑘𝑖∆𝑥3 1 1 ) + (1 − 𝜃)(?̂?1𝑒
𝑗𝑘𝑖∆𝑥 − ?̂?1)) 𝛿
 𝛼
𝑚,𝑖) 
𝑖=1
Simplifying, 
𝑚−1
𝑎 + 𝑢 𝑒−𝑗𝑘𝑖∆𝑥 + 𝑣 𝑒𝑗𝑘𝑖∆𝑥3 3 3 − 𝑙3𝜃(1 − 𝑒
−𝑗𝑘𝑖∆𝑥)(∑ 𝛿  𝛼
 𝑚,𝑖
)
 
 𝑖=1𝑚−1  ?̂?1 = 𝑠3?̂?0 (5-230)   
−𝑙 (1 − 𝜃)(𝑒𝑗𝑘𝑖∆𝑥3 − 1)(∑ 𝛿
 𝛼
𝑚,𝑖)
( 𝑖=1 )
Thus, 
(|𝑎3| + |𝑢3| + |𝑣3| + |𝑙3|𝜃(2 − 2 cos𝜙)𝛽𝑚 + |𝑙3|(1 − 𝜃)(2 − 2 cos𝜙)𝛽𝑚)|?̂?1| = |𝑠3||?̂?0| (5-231) 
Rearranging, 
|?̂?1| |𝑠3|
=  
|?̂?0| |𝑎3| + |𝑢3| + |
(5-232) 
𝑣3| + |𝑙3|𝜃(2 − 2 cos𝜙)𝛽𝑚 + |𝑙3|(1 − 𝜃)(2 − 2 cos𝜙)𝛽𝑚
The condition required |?̂?𝑛| < |?̂?𝑜|, becomes, 
|?̂?1|  
< 1 
|?̂?0|
The condition becomes, 
|𝑠3|
< 1 
|𝑎 | + |𝑢 | + |𝑣 | + |𝑙 |𝜃(2 − 2 cos𝜙)𝛽 + |𝑙 |(1 − 𝜃)(2 − 2 cos𝜙)𝛽 (5-233) 3 3 3 3 𝑚 3 𝑚
The simplification terms associated with Equation (5-37) are incorporated,  
138 
 
(∆𝑡)−𝛼
| 𝛿  𝛼 |
Γ(2 − 𝛼) 𝑛,𝑛−1
(∆𝑡)−𝛼 −𝛼 −𝛼
< 1 
| 𝛿  𝛼
(∆𝑥) 2𝐷 (∆𝑥) 𝐷
𝑛,𝑛−1 + 𝑣𝑥(2𝜃 − 1) 𝛿
 𝛼
𝑛,𝑚 −
𝐿
2| + |𝑣 𝜃 𝛿
 𝛼 𝐿
Γ(2 − 𝛼) Γ(2 − 𝛼) (∆𝑥) 𝑥 Γ(2 − 𝛼) 𝑛,𝑚
+ 2|(∆𝑥)
 −𝛼 −𝛼  (∆𝑥) 𝐷 (∆𝑥) (5-234) 
 +|𝑣𝑥(1 − 𝜃) 𝛿
 𝛼 + 𝐿 | + |𝑣 | 𝜃(2 − 2 cos𝜙)𝛽  
 Γ(2 − 𝛼)
𝑛,𝑚 (∆𝑥)2 𝑥 Γ(2 − 𝛼) 𝑚  
(∆𝑥)−𝛼
+ |𝑣𝑥 | (1 − 𝜃)(2 − 2 cos𝜙)𝛽( Γ(2 − 𝛼) 𝑚 )
An assumption for the norm is made where, 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷
 𝛼 𝐿+ 𝑣𝑥(2𝜃 − 1) 𝛿𝑛,𝑚 >  (5-235) Γ(2 − 𝛼) Γ(2 − 𝛼) (∆𝑥)2
Then, 
(∆𝑡)−𝛼
𝛿  𝛼
Γ(2 − 𝛼) 𝑛,𝑛−1
< 1 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 𝛼  𝛼 2𝐷 (∆𝑥)
−𝛼
𝛿 𝐿  𝛼
𝐷
𝑛,𝑛−1 + 𝑣𝑥(2𝜃 − 1) 𝛿 − + 𝑣 𝜃 𝛿 +
𝐿
Γ(2 − 𝛼) Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
  
(∆𝑥)−𝛼  𝛼 𝐷𝐿 (∆𝑥)
−𝛼 (5-236) 
 +𝑣𝑥(1 − 𝜃) 𝛿 + + 𝑣 𝜃 (2 − 2 cos𝜙)𝛽  
 Γ(2 − 𝛼)
𝑛,𝑚 (∆𝑥)2 𝑥 Γ(2 − 𝛼) 𝑚  
(∆𝑥)−𝛼
+𝑣𝑥 (1 − 𝜃)(2 − 2 cos𝜙)𝛽( Γ(2 − 𝛼) 𝑚 )
Simplifying, 
(∆𝑥)−𝛼 (∆𝑥)−𝛼
2𝑣𝜃 𝛿  𝛼𝑛,𝑚  + 2𝑣𝑥 𝛽 (1 − 3𝜃(cos𝜙)) > 0 Γ(2 − 𝛼) Γ(2 − 𝛼) 𝑚 (5-237) 
Under this assumption (Equation (5-235)), the finite difference first-order upwind-downwind 
weighted (implicit) numerical scheme for the fractional advection-dispersion equation with Caputo 
fractional derivative is unconditionally stable. 
Making the complementary assumption, 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷𝐿
+ 𝑣𝑥(2𝜃 − 1) 𝛿
 𝛼 <  (5-238) 
Γ(2 − 𝛼) Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
Then, 
(∆𝑡)−𝛼
𝛿  𝛼
Γ(2 − 𝛼) 𝑛,𝑛−1
(∆𝑡)−𝛼
< 1 
 𝛼 (∆𝑥)
−𝛼 −𝛼
− 𝛿 − 𝑣 (2𝜃 − 1) 𝛿  𝛼
2𝐷𝐿 (∆𝑥)+ + 𝑣 𝜃 𝛿  𝛼
𝐷
+ 𝐿
Γ(2 − 𝛼) 𝑛,𝑛−1 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
 
(∆𝑥)−𝛼 𝐷 (∆𝑥)−𝛼
 (5-239) 
 +𝑣𝑥(1 − 𝜃) 𝛿
 𝛼 + 𝐿
 Γ(2 − 𝛼)
𝑛,𝑚 (∆𝑥)2
+ 𝑣𝑥𝜃 (2 − 2 cos𝜙)𝛽Γ(2 − 𝛼) 𝑚
 
 
(∆𝑥)−𝛼
+𝑣𝑥 (1 − 𝜃)(2 − 2 cos𝜙)𝛽( Γ(2 − 𝛼) 𝑚 )
Simplifying, 
(∆𝑥)−𝛼 2𝐷𝐿 (∆𝑡)
−𝛼
𝑣 (1 − 𝜃) (𝛿  𝛼𝑥 𝑛,𝑚 + 𝛽𝑚(3(cos𝜙))) + >  Γ(2 − 𝛼) (∆𝑥)2 Γ(2 − 𝛼) (5-240) 
139 
 
Under the assumption made in Equation (5-238), the implicit upwind-downwind weighted numerical 
scheme for the fractional advection-dispersion equation with Caputo fractional derivative is 
conditionally stable. 
Secondly, making the assumption that |?̂?𝑛−1| < |?̂?𝑜| is true for all time steps, the second part of the 
numerical stability analysis is to demonstrate for a set ∀ 𝑛 ≥ 1, that 
|?̂?𝑛| < |?̂?𝑜|  
Rearranging Equation (5-216) for ?̂?𝑛, 
𝑚−1
(𝑎 − 𝑢 −𝑗𝑘𝑖∆𝑥 𝑗𝑘𝑖∆𝑥 −𝑗𝑘𝑖∆𝑥3 3𝑒 − 𝑣3𝑒 − 𝑙3(∑ (𝜃(1 − 𝑒 )+ (1 − 𝜃)(𝑒
𝑗𝑘𝑖∆𝑥 − 1) 𝛿 𝛼) 𝑚,𝑖)) ?̂?𝑛
𝑖=1
𝑛−2
= 𝑠3?̂?𝑛−1 − ℎ3 (∑(?̂?𝑘+1 − ?̂?𝑘)𝛿
 𝛼
𝑛,𝑘) 
𝑘=0
Taking the norm on both sides, 
𝑚−1
|(𝑎 − 𝑢 𝑒−𝑗𝑘𝑖∆𝑥 − 𝑣 𝑒𝑗𝑘𝑖∆𝑥 − 𝑙 (∑ (𝜃(1 − 𝑒−𝑗𝑘𝑖∆𝑥3 3 3 3 )+ (1 − 𝜃)(𝑒
𝑗𝑘𝑖∆𝑥 − 1)  𝛼)𝛿𝑚,𝑖)) ?̂?𝑛|
𝑖=1 (5-241) 
𝑛−2
= |𝑠3?̂?𝑛−1 − ℎ3 (∑(?̂?𝑘+1 − ?̂?
 𝛼
𝑘)𝛿𝑛,𝑘)| 
𝑘=0
Thus, 
𝑚−1
|(𝑎3 − 𝑢3𝑒
−𝑗𝑘𝑖∆𝑥 − 𝑣3𝑒
𝑗𝑘𝑖∆𝑥 − 𝑙 (∑ (𝜃(1 − 𝑒−𝑗𝑘𝑖∆𝑥)+ (1 − 𝜃)(𝑒𝑗𝑘𝑖∆𝑥3 − 1) 𝛿
 𝛼
) 𝑚,𝑖))| |?̂?𝑛| 
𝑖=1 (5-242) 
𝑛−2
< |𝑠3||?̂?𝑛−1| + |ℎ3| |∑(?̂?
 𝛼
𝑘+1 − ?̂?𝑘)𝛿𝑛,𝑘| 
𝑘=0
Remembering that it has been proved that for a set ∀ 𝑛 ≥ 1,   
|?̂?𝑛−1| < |?̂?𝑜| 
Hence, 
𝑚−1
|(𝑎 − 𝑢 𝑒−𝑗𝑘𝑖∆𝑥 − 𝑣 𝑒𝑗𝑘𝑖∆𝑥 − 𝑙 (∑ (𝜃(1 − 𝑒−𝑗𝑘𝑖∆𝑥)+ (1 − 𝜃)(𝑒𝑗𝑘𝑖∆𝑥3 3 3 3 − 1) 𝛿
 𝛼
) 𝑚,𝑖))| |?̂?𝑛| 
𝑖=1
𝑛−2 (5-243) 
< |𝑠3||?̂?𝑛−1| + |ℎ3| |∑(?̂? − ?̂? )𝛿
 𝛼
𝑘+1 𝑘 𝑛,𝑘| 
𝑘=0
𝑛−2
< |𝑠3||?̂?0| + |ℎ3| |∑(?̂?𝑘+1 − ?̂?𝑘)𝛿
 𝛼
𝑛,𝑘| 
𝑘=0
and, it can be inferred that, 
140 
 
𝑚−1
|(𝑎 − 𝑢 𝑒−𝑗𝑘𝑖∆𝑥 − 𝑣 𝑒𝑗𝑘𝑖∆𝑥3 3 3 − 𝑙3(∑ (𝜃(1 − 𝑒
−𝑗𝑘𝑖∆𝑥)+ (1 − 𝜃) 𝑒𝑗𝑘𝑖∆𝑥 − 1 𝛿 𝛼( )) 𝑚,𝑖))| |?̂?𝑛| 
𝑖=1 (5-244) 
𝑛−2
< |𝑠3||?̂?0| + |ℎ | |∑(?̂? − ?̂? )𝛿
 𝛼
3 𝑘+1 𝑘 𝑛,𝑘| 
𝑘=0
Simplifying and rearranging, 
𝑚−1
|(𝑎3 − 𝑢
−𝑗𝑘𝑖∆𝑥 𝑗𝑘𝑖∆𝑥 −𝑗𝑘𝑖∆𝑥 𝑗𝑘𝑖∆𝑥  𝛼
3𝑒 − 𝑣3𝑒 − 𝑙3 (∑ (𝜃(1 − 𝑒 ) + (1 − 𝜃)(𝑒 − 1)) 𝛿𝑚,𝑖))| |?̂?𝑛| 
𝑖=1
< (|𝑠3| + |ℎ3|𝛽𝑛)|?̂?0| (5-245) 
|?̂?𝑛| |𝑠3| + |ℎ3|𝛽
< 𝑛  
|?̂?0| |𝑎3| + |𝑢3| + |𝑣3| + |𝑙3|𝜃(2 − 2 cos𝜙)𝛽 + |𝑙 |(1 − 𝜃)(2 − 2 cos𝜙)𝛽𝑚 3 𝑚
Thus, the solution will be stable when, 
|𝑠3| + |ℎ3|𝛽𝑛 < 1 
|𝑎3| + |𝑢3| + |𝑣3| + |𝑙3|𝜃(2 − 2 cos𝜙)𝛽 + |𝑙3|(1 − 𝜃)(2 − 2 cos𝜙)𝛽
(5-246) 
𝑚 𝑚
The term is expanded using the simplification terms associated with Equation (5-37), 
(∆𝑡)−𝛼 −𝛼
| 𝛿 𝛼
(∆𝑡)
Γ(2 − 𝛼) 𝑛,𝑛−1
| + | |𝛽
Γ(2 − 𝛼) 𝑛
(∆𝑡)−𝛼
< 1 
 𝛼 (∆𝑥)−𝛼 2𝐷 (∆𝑥)−𝛼 𝐷| 𝛿𝑛,𝑛−1 + 𝑣𝑥(2𝜃 − 1) 𝛿
 𝛼 − 𝐿  𝛼 𝐿
Γ(2 − 𝛼) Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
|+ |𝑣𝑥𝜃 𝛿 + |Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
−𝛼 (5-247) (∆𝑥) 𝐷 (∆𝑥)−𝛼
+ |𝑣 (1 − 𝜃) 𝛿 𝛼𝑥 +
𝐿 | + |𝑣 | 𝜃(2 − 2 cos𝜙)𝛽
Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2 𝑥 Γ(2 − 𝛼) 𝑚
(∆𝑥)−𝛼
+ |𝑣𝑥 | (1 − 𝜃)(2 − 2 cos𝜙)𝛽Γ(2 − 𝛼) 𝑚
The assumption is made where, 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷𝐿
𝛿  𝛼𝑛,𝑛−1 + 𝑣𝑥(2𝜃 − 1) 𝛿
 𝛼
𝑛,𝑚 >  (5-248) Γ(2 − 𝛼) Γ(2 − 𝛼) (∆𝑥)2
Then, 
(∆𝑡)−𝛼 (∆𝑡)−𝛼
𝛿 𝛼 + 𝛽
Γ(2 − 𝛼) 𝑛,𝑛−1 Γ(2 − 𝛼) 𝑛
(∆𝑡)−𝛼
< 1 
 𝛼 (∆𝑥)−𝛼  𝛼 2𝐷 (∆𝑥)−𝛼𝛿 + 𝑣 (2𝜃 − 1) 𝛿 − 𝐿 + 𝑣 𝜃 𝛿 𝛼
𝐷
+ 𝐿
Γ(2 − 𝛼) 𝑛,𝑛−1 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
(∆𝑥)−𝛼  𝛼 𝐷𝐿 (∆𝑥)
−𝛼 (5-249) 
+𝑣𝑥(1 − 𝜃) 𝛿 + + 𝑣 𝜃 (2 − 2 cos𝜙)𝛽Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2 𝑥 Γ(2 − 𝛼) 𝑚
(∆𝑥)−𝛼
+𝑣𝑥 (1 − 𝜃)(2 − 2 cos𝜙)𝛽Γ(2 − 𝛼) 𝑚
Multiplying out and simplifying, 
(∆𝑥)−𝛼 (∆𝑥)−𝛼 (∆𝑡)−𝛼
𝑣𝑥(𝜃 + 1) 𝛿
 𝛼
𝑛,𝑚 +  2𝑣𝑥(1 − cos𝜙) 𝛽 > 𝛽  Γ(2 − 𝛼) Γ(2 − 𝛼) 𝑚 Γ(2 − 𝛼) 𝑛 (5-250) 
141 
 
Thus, the implicit upwind-downwind weighted scheme for the fractional advection-dispersion 
equation with Caputo fractional derivative is conditionally stable under the assumption made in 
Equation (5-248). 
When the corresponding assumption is made, 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷
𝛿  𝛼
𝐿
𝑛,𝑛−1 + 𝑣𝑥(2𝜃 − 1) 𝛿
 𝛼 <  (5-251) 
Γ(2 − 𝛼) Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
Then, 
(∆𝑡)−𝛼  𝛼 (∆𝑡)
−𝛼
𝛿𝑛,𝑛−1 + 𝛽Γ(2 − 𝛼) Γ(2 − 𝛼) 𝑛
< 1 
(∆𝑡)−𝛼 −𝛼 𝛼 (∆𝑥) 2𝐷 (∆𝑥)−𝛼 𝐷− 𝛿𝑛,𝑛−1 − 𝑣𝑥(2𝜃 − 1) 𝛿
 𝛼 + 𝐿 + 𝑣 𝜃 𝛿 𝛼 + 𝐿
Γ(2 − 𝛼) Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
(∆𝑥)−𝛼 𝐷 (∆𝑥)−𝛼 (5-252) 
+𝑣𝑥(1 − 𝜃) 𝛿
 𝛼
𝑛,𝑚 +
𝐿
2 + 𝑣𝑥𝜃 (2 − 2 cos𝜙)𝛽Γ(2 − 𝛼) (∆𝑥) Γ(2 − 𝛼) 𝑚
(∆𝑥)−𝛼
+𝑣𝑥 (1 − 𝜃)(2 − 2 cos𝜙)𝛽Γ(2 − 𝛼) 𝑚
Simplifying, 
(∆𝑥)−𝛼 (∆𝑥)−𝛼 4𝐷𝐿 (∆𝑡)
−𝛼
2𝑣𝑥(1 − 𝜃) 𝛿
 𝛼 + 2𝑣 (1 − cos𝜙) 𝛽 + > (2 + 𝛽
Γ(2 − 𝛼) 𝑛,𝑚 𝑥 Γ(2 − 𝛼) 𝑚 (∆𝑥)2 𝑛
)  
Γ(2 − 𝛼) (5-253) 
Under this assumption (Equation (5-251)), the implicit upwind-downwind weighted scheme for the 
fractional advection-dispersion equation with Caputo fractional derivative is conditionally stable. 
The implicit upwind-downwind weighted finite difference scheme for the fractional advection-
dispersion equation (Caputo) has the following stability criterion, as proven by the induction method, 
(∆𝑥)−𝛼 4𝐷 2(∆𝑡)−𝛼𝐿
𝑣𝑥(2 − 𝜃) 𝛿
 𝛼 + > , 
Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2 Γ(2 − 𝛼)
(∆𝑥)−𝛼 (∆𝑥)−𝛼 2𝐷 (∆𝑡)−𝛼𝐿
𝑣𝑥(1 − 𝜃) 𝛿
 𝛼  + 𝑣 (1 − 3𝜃(cos𝜙)) 𝛽 + > , 
Γ(2 − 𝛼) 𝑛,𝑚 𝑥 Γ(2 − 𝛼) 𝑚 (∆𝑥)2 Γ(2 − 𝛼)
(∆𝑥)−𝛼 (∆𝑥)−𝛼 (∆𝑡)−𝛼
𝑣𝑥(𝜃 + 1) 𝛿
 𝛼
𝑛,𝑚 +  2𝑣𝑥(1 − cos𝜙) 𝛽Γ(2 − 𝛼) Γ(2 − 𝛼) 𝑚
> 𝛽 , 
Γ(2 − 𝛼) 𝑛
and, 
(∆𝑥)−𝛼 (∆𝑥)−𝛼 4𝐷𝐿 (∆𝑡)
−𝛼
2𝑣𝑥(1 − 𝜃) 𝛿
 𝛼
𝑛,𝑚 + 2𝑣𝑥(1 − cos𝜙) 𝛽𝑚 + > (2 + 𝛽 )  Γ(2 − 𝛼) Γ(2 − 𝛼) (∆𝑥)2 𝑛 Γ(2 − 𝛼)
Because under these conditions, the error of the approximation is not propagated throughout the 
solution, but rather decreases with each time step, as according to the induction method, where for 
all values of n, |?̂?𝑛+1| < |?̂?𝑜|. 
142 
 
5.2.6 Evaluation of numerical stability results 
Upwind numerical schemes are developed for the one-dimensional, non-reactive space-time 
fractional advection-dispersion equation (Caputo), including the traditional upwind as well as the 
newly proposed upwind Crank-Nicolson and a weighted upwind-downwind scheme. The numerical 
schemes were subjected to numerical stability analysis using the recursive method, with a summary 
of these results tabulated in Table 5-1.  
The traditional upwind (explicit) numerical scheme for the space-time fractional advection-dispersion 
equation (Caputo) is found to be unstable under certain assumptions, while the traditional upwind 
(implicit) numerical scheme is conditionally stable under all assumptions. The upwind Crank-Nicolson 
scheme, which has a half weighted approach to implicit and explicit for the advection term, is found 
to be conditionally stable under all assumptions. The weighted upwind-downwind (explicit) scheme is 
found to be unstable under certain assumptions, similarly to the traditional upwind (explicit) 
numerical scheme. The weighted upwind-downwind (implicit) scheme is conditionally stable under all 
assumptions, again similar to the traditional upwind (implicit) scheme. This trend of implicit numerical 
scheme formulations being more stable than the corresponding explicit formulation has been found 
by other researchers (Lynch et al., 2003; Meerschaert and Tadjeran, 2004; Liu et al., 2007).  
The new weighted upwind-downwind (explicit) scheme is not more applicable than the traditional 
upwind (explicit) scheme, because the new weighted (explicit) scheme tends to be unstable under 
more assumptions. On the other hand, the weighted (implicit) scheme does provide an improvement 
on the traditional upwind (implicit) scheme in terms of stability, where the inclusion of the weighting 
factor (𝜃) provides a means to improve the likelihood of upholding the stability condition. Thus, the 
upwind Crank-Nicolson and weighted upwind-downwind (implicit) schemes are applicable for solution 
of the space-time fractional advection-dispersion equation (Caputo), if the stability criterion are 
upheld. 
5.3 Chapter summary 
The space-time fractional advection-dispersion equation with Caputo fractional derivatives is defined, 
and the upwind numerical schemes developed in Chapter 2 are applied to numerically approximate 
the solution. Each scheme is analysed for stability, where it is found that the implicit weighted scheme 
is an improvement on the traditional implicit upwind scheme in terms of stability, where the inclusion 
of the weighting factor (𝜃) provides a means to improve the likelihood of upholding the stability 
condition. It was concluded that the upwind advection Crank-Nicolson and implicit weighted upwind-
downwind schemes are applicable for solution of the space-time fractional advection-dispersion 
equation (Caputo), when the stability criterion are upheld.  
143 
 
Table 5-1 Summary of the assumptions made and corresponding stability condition for each numerical 
approximation scheme for the fractional advection-dispersion equation with Caputo fractional derivative 
Scheme Assumptions Stability condition 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷
𝛿  𝛼
𝐿
𝑛,𝑛−1 + 𝑣 𝛿
 𝛼
𝑥 >  Unstable Γ(2 − 𝛼) Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
Explicit 
Upwind  
−𝛼 −𝛼 4𝐷𝐿 (∆𝑥)
−𝛼 (∆𝑡)−𝛼 2(∆𝑡)−𝛼
(∆𝑡) (∆𝑥) 2𝐷
 𝛼  𝛼 𝐿  + 𝑣𝑥 (2 − 2 cos𝜙)𝛽𝑚 + 𝛽 <  𝛿 + 𝑣 2𝑛,𝑛−1 𝑥 𝛿𝑛,𝑚 <  2 (∆𝑥) Γ(2 − 𝛼) Γ(2 − 𝛼)
𝑛 Γ(2 − 𝛼)
Γ(2 − 𝛼) Γ(2 − 𝛼) (∆𝑥)
Conditionally stable 
(∆𝑡)−𝛼−𝛼 −𝛼 (∆𝑥)
−𝛼
(∆𝑡) (∆𝑥) 2𝐷  𝛼
𝛿  𝛼 + 𝑣 𝛿  𝛼
𝐿 𝛽
>  Γ(2 − 𝛼) 𝑛
< 𝑣𝑥 (2𝛿  + (2 − 2 cos𝜙)𝛽 ) 
𝑛,𝑛−1 𝑥 𝑛,𝑚 2 Γ(2 − 𝛼)
𝑛,𝑚 𝑚
Γ(2 − 𝛼) Γ(2 − 𝛼) (∆𝑥)
Unconditionally stable / Conditionally stable 
Implicit 
Upwind  
(∆𝑡)−𝛼 (∆𝑥)−𝛼 4𝐷
(∆𝑡)−𝛼 (∆𝑥)−𝛼 2𝐷 𝐿
 𝛼 𝐿 (2 + 𝛽 ) < 𝑣 (2 − 2 cos𝜙)𝛽 +   𝛿𝑛,𝑛−1 + 𝑣𝑥 𝛿
 𝛼 <  Γ(2 − 𝛼) 𝑛 𝑥 Γ(2 − 𝛼) 𝑚 (∆𝑥)2
Γ(2 − 𝛼) Γ(2 − 𝛼) 𝑛,𝑚 (∆𝑥)2
Conditionally stable 
(∆𝑡)−𝛼 (∆𝑥)−𝛼 −𝛼
𝛿  𝛼 > 𝑣 0.5  𝛿  𝛼 (∆𝑡) (∆𝑥)
−𝛼
Γ(2 − 𝛼) 𝑛,𝑛−1 𝑥 Γ(2 − 𝛼) 𝑛,𝑚 𝛽𝑛 < 𝑣 (𝛿
 𝛼
𝑥  + 0.5𝛽 (2 − 2 cos𝜙)) 
2𝐷 Γ(2 − 𝛼) Γ(2 − 𝛼)
𝑛,𝑚 𝑚
𝐿
+  Unconditionally stable / Conditionally stable 
(∆𝑥)2
Upwind       2𝐷𝐿 (∆𝑡)
−𝛼
Crank- <  
−𝛼 −𝛼 (∆𝑥)2 Γ(2 − 𝛼)
Nicolson (∆𝑡) (∆𝑥) 𝛼  𝛼 (∆𝑡)−𝛼 (∆𝑥)−𝛼𝛿
Γ(2 − 𝛼) 𝑛,𝑛−1
< 𝑣𝑥0.5  𝛿Γ(2 − 𝛼) 𝑛,𝑚 𝛽𝑛 < 𝑣𝑥 (𝛿
 𝛼
𝑛,𝑚  + 0.5𝛽𝑚(2 − 2 cos𝜙)) 
2𝐷 Γ(2 − 𝛼) Γ(2 − 𝛼)𝐿
+  4𝐷 −𝛼 −𝛼 −𝛼
2 𝐿
(∆𝑡) 2(∆𝑡) (∆𝑥)
(∆𝑥) + 𝛽𝑛 < + 0.5𝑣 (2 − 2 cos 𝜙)𝛽  (∆𝑥)2 Γ(2 − 𝛼) Γ(2 − 𝛼) 𝑥 Γ(2 − 𝛼) 𝑚
Conditionally stable 
(∆𝑡)−𝛼 (∆𝑥)−𝛼
𝛿  𝛼  𝛼
Γ(2 − 𝛼) 𝑛,𝑛−1
+ 𝑣𝑥(2𝜃 − 1) 𝛿Γ(2 − 𝛼) 𝑛,𝑚
2𝐷 Unstable 
Explicit 𝐿>  
( )2
weighted ∆𝑥
upwind- (∆𝑡)−𝛼 (∆𝑥)−𝛼
downwind  𝛿  𝛼𝑛,𝑛−1 + 𝑣𝑥(2𝜃 − 1) 𝛿
 𝛼
Γ(2 − 𝛼) Γ(2 − 𝛼) 𝑛,𝑚
2𝐷 Conditionally stable / Unstable 𝐿
<  
(∆𝑥)2
(∆𝑡)−𝛼 (∆𝑥)−𝛼
(∆𝑡)−𝛼 (∆𝑥)−𝛼
 𝛼  𝛼 𝛽𝑛 < 𝑣𝑥 𝛿
 𝛼 (𝜃 + 1)
𝛿𝑛,𝑛−1 + 𝑣𝑥(2𝜃 − 1) 𝛿𝑛,𝑚 Γ(2 − 𝛼) Γ(2 − 𝛼)
𝑛,𝑚
Γ(2 − 𝛼) Γ(2 − 𝛼) (∆𝑥)−𝛼
2𝐷𝐿 +  2𝑣𝑥 𝛽𝑚(1 − cos𝜙) >  
2 Γ(2 − 𝛼)
Implicit (∆𝑥) Unconditionally stable / conditionally stable 
weighted 
upwind-
downwind  −𝛼 −𝛼 (∆𝑡)
−𝛼 (∆𝑥)−𝛼
(∆𝑡) (∆𝑥) (2 + 𝛽 ) < 2𝑣 𝛿  𝛼 𝛼 (1 − 𝜃)𝛿𝑛,𝑛−1 + 𝑣
 𝛼
𝑥(2𝜃 − 1) 𝛿𝑛,𝑚 Γ(2 − 𝛼)
𝑛 𝑥 Γ(2 − 𝛼) 𝑛,𝑚
Γ(2 − 𝛼) Γ(2 − 𝛼) (∆𝑥)−𝛼 4𝐷
2𝐷 𝐿𝐿 + 2𝑣𝑥 𝛽 (1 − cos𝜙) +  <  
2 Γ(2 − 𝛼)
𝑚 (∆𝑥)2
(∆𝑥)
Conditionally stable 
 
144 
 
 
6  FRACTIONAL ADVECTION-    
 DISPERSION EQUATION WITH    
 ATANGANA-BALEANU IN CAPUTO  
 SENSE (ABC) DERIVATIVE 
 
It has been discussed that modelling groundwater transport in fractured aquifer systems is 
complicated due to the uncertainty associated with defining preferential pathways along which water 
and potential contaminants can be transported along. Misrepresenting an expected movement of a 
potential contaminant in a groundwater system can lead to environmental implications due to 
inadequate mitigation or remediation measures (Schmelling and Ross, 1990; Zimmerman et al., 1998; 
Fomin et al., 2005; Goode et al., 2007; Cello et al., 2009; Shapiro, 2011; Masciopinto and Palmiotta, 
2013). To minimise misrepresentations in fractured systems, the newest fractional derivative 
definition, Atangana-Baleanu, is used to develop an advection-focused fractional advection-dispersion 
equation. The Atangana-Baleanu fractional derivative definition is presented in Chapter 4, along with 
the potential advantages of using this definition, where it was concluded that the new fractional 
derivatives definitions have the potential to incorporate different memory effects, which could change 
the way anomalous diffusion is modelled. The applicability of this particular formulation and fractional 
derivative definition is investigated for groundwater transport within fractured aquifers.   
145 
 
The Caputo fractional derivative definition was applied in Chapter 5, and a similar process is followed 
for the application of the Atangana-Baleanu in Caputo sense (ABC) fractional derivative definition. 
Firstly, the numerical approximation schemes are developed using the augmented upwind schemes, 
as well as the traditional upwind schemes, presented in Chapter 2. A numerical stability analysis is 
performed for each scheme, where the traditional upwind schemes serve to form a base of 
comparison in terms of numerical stability for the developed schemes. 
6.1 Advection-focused transport model with Atangana-Baleanu in Caputo sense 
(ABC) derivative 
The advection-focused fractional advection-dispersion equation with the Atangana-Baleanu in Caputo 
sense (ABC) fractional derivative definition is, 
𝜕2
𝐴𝐵𝐶 𝛼
0𝐷𝑡  (𝑐(𝑥, 𝑡)) = −𝑣  
𝐴𝐵𝐶 𝛼
𝑥 0𝐷𝑥 (𝑐(𝑥, 𝑡)) + 𝐷𝐿 (𝑐(𝑥, 𝑡)) (6-1) 𝜕𝑥2
To investigate the qualitative properties of the formulated equation, the boundedness, existence and 
uniqueness of the fractional advection-dispersion equation with ABC fractional derivative is 
determined using the Picard-Lindelöf theorem. Additionally, the stability of the defined fractional 
advection-dispersion equation is evaluated in time.  
6.1.1 Picard-Lindelöf theorem for existence and uniqueness 
Applying the AB integral to both sides of the fractional advection-dispersion equation with Atangana-
Baleanu in Caputo sense (ABC) derivative, to obtain: 
𝜕2
𝑐(𝑥, 𝑡) − 𝑐(𝑥, 0) = 𝐴𝐵0𝐼
𝛼
𝑥 {−𝑣
𝐴𝐵𝐶 𝛼
𝑥 0𝐷𝑥 (𝑐(𝑥, 𝜏)) + 𝐷𝐿 (𝑐(𝑥, 𝜏))} 𝑑𝜏  𝜕𝑥2
1 − 𝛼 𝜕2
                                = {−𝑣  𝐴𝐵𝐶 𝛼𝑥 0𝐷𝑥 (𝑐(𝑥, 𝜏)) + 𝐷𝐿 (𝑐(𝑥, 𝜏)2 )} 𝐴𝐵(𝛼) 𝜕𝑥 (6-2) 
𝛼 𝑡 𝜕2
+ ∫ {−𝑣 𝐴𝐵𝐶𝐷𝛼(𝑐(𝑥, 𝜏)) + 𝐷 𝑐(𝑥, 𝜏) (𝑡 − 𝜏)𝛼−1𝑑𝜏 
𝐴𝐵(𝛼)Γ(𝛼) 𝑥 0 𝑥 𝐿 𝜕𝑥2
( )}
0
Consider a new function 𝐹(𝑥, 𝑡, 𝑐) to simplify: 
𝜕2
𝐹(𝑥, 𝑡, 𝑐) = −𝑣 𝐴𝐵𝐶 𝛼𝑥 0𝐷𝑥 (𝑐(𝑥, 𝑡)) + 𝐷𝐿 (𝑐(𝑥, 𝑡)) (6-3) 𝜕𝑥2
Thus, 
1 − 𝛼 𝛼 𝑡
𝑐(𝑥, 𝑡) − 𝑐(𝑥, 0) = 𝐹(𝑥, 𝑡, 𝑐) + ∫ 𝐹(𝑥, 𝑡, 𝑐)(𝑡 − 𝜏)𝛼−1𝑑𝜏 
𝐴𝐵(𝛼) 𝐴𝐵(𝛼)Γ(𝛼) (6-4) 0
Let 
𝐶 = 𝐼̅̅ ̅̅𝜆,𝛽 𝜆(𝑡̅̅ ̅0) × 𝐵̅̅ ̅̅ ̅̅𝛽(𝑥̅̅ ̅0) 
146 
 
where, 
𝐼̅̅ ̅̅𝜆(𝑡̅̅ ̅0) = [𝑡0 − 𝜆, 𝑡0 + 𝜆] 
̅̅𝐵̅̅ ̅̅̅̅𝛽(
̅̅̅̅𝑥̅̅̅̅ ̅̅0) = [𝑥0 − 𝛽, 𝑥0 + 𝛽] 
The Banach fixed-point theorem is applied by introducing the norm of the supremum (statistical limit 
of a set) for 𝐼𝜆, 
  𝑠𝑢𝑝|𝜑(𝑡)|
𝑀 = ‖𝜑‖∞ =   𝑡 ∈ 𝐼𝜆(𝑡0)
(6-5) 
Considering the practical meaning of 𝑐(𝑥, 𝑡), it can be assumed that the initial concentration (𝑐0) will 
always be greater than subsequent concentrations (𝑐𝑛) due to advection, dispersion and diffusion 
processes which reduce the concentration over time and space, 
‖𝑐‖∞ < 𝑐0 (6-6) 
Considering the max norm for the function 𝐹(𝑥, 𝑡, 𝑐), 
𝐴𝐵(𝛼) 𝑥 𝑑 𝛼 𝜕2
‖𝐹‖∞ = ‖−𝑣𝑥   ∫ 𝑐(𝜏, 𝑥)𝐸𝛼 [− (𝑥 − 𝜏)
𝛼]𝑑𝜏+ 𝐷𝐿 2 (𝑐(𝑥, 𝑡))‖  (6-7) (1 − 𝛼) 0 𝑑𝜏 1 − 𝛼 𝜕𝑥 ∞
Thus, 
𝐴𝐵(𝛼) 𝑥 𝑑 𝛼 𝜕2
‖𝐹‖∞ ≤ 𝑣𝑥  ‖ ∫ 𝑐(𝜏, 𝑥)𝐸𝛼 [− (𝑥 − 𝜏)
𝛼]𝑑𝜏‖ + 𝐷𝐿 ‖ 2 (𝑐(𝑥, 𝑡))‖  (6-8) (1 − 𝛼) 0 𝑑𝜏 1 − 𝛼 𝜕𝑥∞ ∞
Applying the proven theorem in Section 3.2.2, the second order derivative is bounded (𝑀1), thus 
𝐴𝐵(𝛼) 𝑥 𝑑 𝛼 
‖𝐹‖∞ ≤ 𝑣𝑥 ∫ ‖ 𝐶( ) 𝜏, 𝑥 ‖ ‖𝐸𝛼 [− (𝑥 − 𝜏)
𝛼]‖ 𝑑𝜏+ 𝐷 𝑀  
(1 − 𝛼) 𝑑𝜏 1 − 𝛼 𝐿 1 (6-9) 0 ∞ ∞
The Mittag-Leffler function is bounded because 1 > 𝛼 > 0, and the first order derivative is bounded 
to the physical meaning of the derivative of the spread of a particle in terms of its concentration (𝑀2). 
Thus, the derivative is considered at the maximum physical time that is applicable to the existence of 
the concentration (𝑇𝑚𝑎𝑥), 
𝐴𝐵(𝛼)
‖𝐹‖∞ ≤ 𝑣𝑥 𝑀 𝑇 + 𝐷 𝑀 < ∞ (6-10) (1 − 𝛼) 2 𝑚𝑎𝑥 𝐿 1
Therefore, the solution is bounded because we obtain a positive constant, such that 
  𝑠𝑢𝑝|𝐹(𝑥, 𝑡, 𝑐)|
𝐴 = ‖𝐹‖∞ =  (6-11) 𝑡 ∈ 𝐶𝜆,𝛽
Let 𝐶𝜆,𝛽 be a set where, 𝐹: 𝐶𝜆,𝛽  → 𝐶𝜆,𝛽, such that 
1 − 𝛼 𝛼 𝑡
Γ𝜙(𝑥, 𝑡) = 𝑐(𝑥, 0) + 𝐹(𝑥, 𝑡, 𝜙) + ∫ 𝐹(𝑥, 𝜏, 𝜙)(𝑡 − 𝜏)𝛼−1𝑑𝜏 (6-12) 
𝐴𝐵(𝛼) 𝐴𝐵(𝛼)Γ(𝛼) 0
Thus, 
147 
 
1 − 𝛼 𝛼 𝑡
‖Γ𝜙(𝑥, 𝑡) − 𝑐(𝑥, 0)‖∞ = ‖ 𝐹(𝑥, 𝑡, 𝜙) + ∫ 𝐹(𝑥, 𝜏, 𝜙)(𝑡 − 𝜏)
𝛼−1𝑑𝜏‖  
𝐴𝐵(𝛼) 𝐴𝐵(𝛼)Γ(𝛼) 0 ∞ (6-13) 
1 − 𝛼 𝛼 𝑡
‖Γ𝜙(𝑥, 𝑡) − 𝑐(𝑥, 0)‖∞ ≤ ‖𝐹(𝑥, 𝑡, 𝜙)‖ + ∫ (𝑡 − 𝜏)
𝛼−1
∞ ‖𝐹(𝑥, 𝜏, 𝜙)‖ 𝑑𝜏 𝐴𝐵(𝛼) 𝐴𝐵(𝛼)Γ(𝛼) ∞0
The function 𝐹(𝑥, 𝑡, 𝑐) has been shown to be bounded (Equation (6-7) - (6-11)), 
1 − 𝛼 𝛼 𝑡
‖Γ𝜙(𝑥, 𝑡) − 𝑐(𝑥, 0)‖∞ ≤ 𝐴 + 𝐴∫ (𝑡 − 𝜏)
𝛼−1𝑑𝜏 
𝐴𝐵(𝛼) 𝐴𝐵(𝛼)Γ(𝛼) 0
The integral is considered at the maximum physical time that is applicable to the existence of the 
concentration (𝑇𝑚𝑎𝑥), 
1 − 𝛼 𝛼 𝑇𝛼𝑚𝑎𝑥
‖Γ𝜙(𝑥, 𝑡) − 𝑐(𝑥, 0)‖∞ ≤ 𝐴 + 𝐴  𝐴𝐵(𝛼) 𝐴𝐵(𝛼)Γ(𝛼) 𝛼
1 − 𝛼 𝐴 𝑇𝛼𝑚𝑎𝑥
                                         ≤ 𝐴 + < ∞ 
𝐴𝐵(𝛼) 𝐴𝐵(𝛼)Γ(𝛼)
Therefore, Γ is well-posed because we obtain a positive constant.  
To prove that Γ is Lipschitz, 
1 − 𝛼
‖Γ𝜙1 − Γ𝜙2‖∞ = ‖ (𝐹(𝑥, 𝑡, 𝜙1) − 𝐹(𝑥, 𝑡, 𝜙2))𝐴𝐵(𝛼)
𝛼 𝑡
+ ∫ (𝐹(𝑥, 𝜏, 𝜙 ) − 𝐹(𝑥, 𝜏, 𝜙 ))(𝑡 − 𝜏)𝛼−1𝑑𝜏‖  
𝐴𝐵(𝛼)Γ(𝛼) 1 20 ∞
(6-14) 
1 − 𝛼
‖Γ𝜙1 − Γ𝜙2‖∞ ≤ ‖(𝐹(𝑥, 𝑡, 𝜙1) − 𝐹(𝑥, 𝑡, 𝜙𝐴𝐵(𝛼) 2
))‖
∞
𝛼 𝑡
+ ∫ (𝑡 − 𝜏)𝛼−1‖𝐹(𝑥, 𝜏, 𝜙 ) − 𝐹(𝑥, 𝜏, 𝜙 )‖ 𝑑𝜏 
𝐴𝐵(𝛼)Γ(𝛼) 1 2 ∞0
To achieve this, first evaluate 
𝜕2
‖𝐹(𝑥, 𝑡, 𝜙1) − 𝐹(𝑥, 𝑡, 𝜙
𝐴𝐵𝐶 𝛼
2)‖∞ = ‖−𝑣𝑥 0𝐷𝑥 (𝜙1 − 𝜙2) + 𝐷𝐿 (𝜙1 − 𝜙2)‖  𝜕𝑥2
∞
(6-15) 
𝜕2
‖𝐹(𝑥, 𝑡, 𝜙1) − 𝐹(𝑥, 𝑡, 𝜙2)‖∞ ≤ 𝑣𝑥  ‖
𝐴𝐵𝐶 𝛼
0𝐷𝑥 (𝜙1 − 𝜙2)‖ + 𝐷𝐿 ‖ (𝜙 − 𝜙 )‖  ∞ 𝜕𝑥2 1 2
∞
Applying the Atangana-Baleanu fractional derivative, and applying the proven theorem in Section 
3.2.2 for the second order derivative is bounded (𝜉22 ), 
‖𝐹(𝑥, 𝑡, 𝜙1) − 𝐹(𝑥, 𝑡, 𝜙2)‖∞ ≤ 
𝐴𝐵(𝛼) 𝑥 𝑑 𝛼 (6-16) 
𝑣𝑥 ∫ ‖ (𝜙1 − 𝜙2)‖ ‖𝐸𝛼 [− (𝑥 − 𝜏)
𝛼
 ]‖ 𝑑𝜏 + 𝐷 𝜉
2
𝐿 ‖(𝜙1 − 𝜙2)‖  (1 − 𝛼) 𝑑𝜏 1 − 𝛼 ∞0 ∞ ∞
The Mittag-Leffler function is bounded because 1 > 𝛼 > 0, and the first order derivative is bounded 
to the physical meaning of the derivative of the spread of a particle in terms of its concentration (𝜉1). 
148 
 
Thus, the derivative is considered at the maximum physical space that is applicable to the existence 
of the concentration (𝑋𝑚𝑎𝑥), 
𝐴𝐵(𝛼)
‖𝐹(𝑥, 𝑡, 𝜙1) − 𝐹(𝑥, 𝑡, 𝜙2)‖∞ ≤ 𝑣𝑥  𝑋𝑚𝑎𝑥𝜉 ‖( )‖1 𝜙1 − 𝜙2 ∞ + 𝐷 𝜉
2
𝐿 2‖(𝜙1 − 𝜙 )‖(1 − 𝛼) 2 ∞
 (6-17) 
Simplifying, 
𝐴𝐵(𝛼)
‖𝐹(𝑥, 𝑡, 𝜙1) − 𝐹(𝑥, 𝑡, 𝜙
2
2)‖∞ ≤ (𝑣𝑥  𝑋𝑚𝑎𝑥𝜉1 + 𝐷𝐿𝜉2) ‖(𝜙1 − 𝜙2)‖∞ < Κ𝛼‖(𝜙1 − 𝜙2)‖∞ (6-18) (1 − 𝛼)
Applying to Equation (6-14), 
1 − 𝛼 𝛼 𝑡
‖Γ𝜙1 − Γ𝜙2‖∞ ≤ Κ𝛼‖(𝜙1 −𝜙2)‖∞ + Κ𝛼‖(𝜙1 −𝜙2)‖∞∫ (𝑡 − 𝜏)
𝛼−1𝑑𝜏 (6-19) 
𝐴𝐵(𝛼) 𝐴𝐵(𝛼)Γ(𝛼) 0
Applying a similar process as previously, 
1 − 𝛼 𝛼 𝑇𝛼𝑚𝑎𝑥
‖Γ𝜙1 − Γ𝜙2‖∞ ≤ Κ𝛼‖(𝜙1 − 𝜙2)‖∞ + Κ𝛼‖(𝜙 − 𝜙 )‖  𝐴𝐵(𝛼) 𝐴𝐵(𝛼)Γ(𝛼) 1 2 ∞ 𝛼
1 − 𝛼 𝑇𝛼𝑚𝑎𝑥
≤ ( Κ𝛼 + Κ𝛼)‖(𝜙1 − 𝜙 )‖  
(6-20) 
𝐴𝐵(𝛼) 𝐴𝐵(𝛼)Γ(𝛼) 2 ∞
                                           ≤ 𝑉‖(𝜙1 − 𝜙2)‖∞ 
Therefore, Γ is a contraction when 𝑉 < 1, which translates to a condition 
1
Κ𝛼 < 𝛼  1 − 𝛼 𝑇
+ 𝑚𝑎𝑥 (6-21) 
𝐴𝐵(𝛼) 𝐴𝐵(𝛼)Γ(𝛼)
Then 𝐹(𝑥, 𝑡, 𝑐) has a fixed point using the Banach fixed-point theorem and the fractional advection-
dispersion equation with ABC fractional derivative is bounded and has a unique solution under this 
condition.  
6.1.2 Semi-discretisation stability 
The defined fractional advection-dispersion equation with ABC fractional derivative is discretised in 
time while the concentration in space is considered constant to evaluate the stability of the equation 
in time. The forward difference scheme in time is applied to the Atangana-Baleanu in Caputo sense 
(ABC) fractional derivative, considered for a specific time (𝑡𝑛), and the numerical integration of the 
Mittag-Leffler function as performed by Alkahtani et al. (2017) is applied, 
𝑛−1
𝐴𝐵(𝛼)
 𝐴𝐵𝐶𝐷𝛼0 𝑡 (𝑐(𝑥, 𝑡𝑛)) = ∑(𝑐
𝑘+1
𝑥 − 𝑐
𝑘
𝑥 )𝛿
𝛼
𝑛,𝑘   (1 − 𝛼) 
𝑘=0
where, 
𝛼 ∆𝑡 𝛼 ∆𝑡
𝛿𝛼𝑛,𝑘 = (𝑛 − 𝑘)𝐸𝛼,2 [− (𝑛 − 𝑘)] − (𝑛 − 𝑘 − 1)𝐸𝛼,2 [− (𝑛 − 𝑘 − 1)] (6-22) 1 − 𝛼 1 − 𝛼 
149 
 
Substituting back into the fractional advection-dispersion equation with the Atangana-Baleanu in 
Caputo sense (ABC) derivative, and applying the assumption of discretisation in time only 
𝑛−1
𝐴𝐵(𝛼) 𝜕2
∑(𝑐𝑘+1𝑥 − 𝑐
𝑘 𝛼
𝑥 )𝛿𝑛,𝑘 = −𝑣𝑥 
𝐴𝐵𝐶 𝛼 𝑛 𝑛+1
 0
𝐷𝑥 (𝑐𝑥 ) + 𝐷𝐿 (𝑐 ) 2 𝑥 (6-23) (1 − 𝛼) 𝜕𝑥
𝑘=0
A function 𝐴𝛼𝑛,𝑘 is applied to simplify, 
𝑛−1
𝜕2
∑(𝑐𝑘+1 − 𝑐𝑘)𝐴𝛼 = −𝑣 𝐴𝐵𝐶 𝛼 𝑛 𝑛+1𝑥 𝑥 𝑛,𝑘 𝑥 0𝐷𝑥 (𝑐𝑥 ) + 𝐷𝐿 (𝑐𝑥 ) 2 (6-24) 𝜕𝑥
𝑘=0
Reformulating to obtain, 
𝑛−1
𝜕2
( 𝑛+1 𝑛) 𝛼 𝑘+1 𝑘 𝛼 𝐴𝐵𝐶 𝛼( 𝑛) ( 𝑛+1) (6-25) 𝑐𝑥 − 𝑐𝑥 𝐴𝑛 +∑(𝑐𝑥 − 𝑐𝑥 )𝐴𝑛,𝑘 = −𝑣𝑥 0𝐷𝑥 𝑐𝑥 + 𝐷𝐿 𝑐  𝜕𝑥2 𝑥
𝑘=0
Rearranging, 
𝑣 2
𝑛−1
𝑛+1 𝑛 𝑥
𝐷
𝐴𝐵𝐶 𝛼 𝑛 𝐿
𝜕 1
𝑐𝑥 = 𝑐𝑥 − 𝛼  0𝐷𝑥 (𝑐𝑥 ) +
𝑛+1
𝛼 (𝑐 ) − ∑(𝑐
𝑘+1 − 𝑐𝑘)𝐴𝛼  (6-26) 
𝐴𝑛 𝐴𝑛 𝜕𝑥
2 𝑥 𝐴𝛼 𝑥 𝑥 𝑛,𝑘𝑛
𝑘=0
Equation (6-26) is the numerical approximation of the fractional advection-equation (ABC) with 
respect to time. Now, the semi-stability can be evaluated defining the following norms, 
(𝑓, 𝑔) = ∫ (𝑓 ∙ 𝑔)(𝑥)𝑑𝑥 
Ω
where, 
‖𝑔‖0 = √(𝑔 ∙ 𝑔) 
𝑑2
‖𝑔‖1 = √‖𝑔‖0 + 𝜀 ‖ 𝑔‖  𝑑𝑥2
0
When 𝑛 = 0, Equation (6-26) becomes 
𝑣 𝐷 𝜕2
𝑐1 = 𝑐0
𝑥
− 𝐴𝐵𝐶 𝛼
𝐿
𝑥 𝑥 𝐷 (𝑐
0) + (𝑐1) (6-27) 
𝐴𝛼 0 𝑥 𝑥 𝛼 2 𝑥𝑛 𝐴𝑛 𝜕𝑥
Simplifying using functions 𝜆1 and 𝜆2, 
𝜕2
𝑐1𝑥 = 𝑐
0
𝑥 − 𝜆  
𝐴𝐵𝐶𝐷𝛼(𝑐0) + 𝜆 (𝑐1) (6-28) 1 0 𝑥 𝑥 2 𝜕𝑥2 𝑥
Applying the norm with respect to 𝑔, 
𝜕2 𝜕2
(𝑐1, 𝑔) = (𝑐0𝑥 𝑥 , 𝑔) − 𝜆1(
𝐴𝐵𝐶
0𝐷
𝛼
𝑥 𝑐
0
𝑥 ,  
𝐴𝐵𝐶
0𝐷
𝛼
𝑥 𝑔) + 𝜆2 ( 𝑐
1
𝑥 , 𝑔) (6-29) 𝜕𝑥2 𝜕𝑥2
Let ∀𝑔 ∈ 𝐻1(Ω), 𝑔 = 𝑐1𝑥  
𝜕2 𝜕2
(𝑐1, 𝑐1) = (𝑐0, 𝑐1) − 𝜆 (𝐴𝐵𝐶𝐷𝛼𝑐0,  𝐴𝐵𝐶𝐷𝛼𝑐1) + 𝜆 ( 𝑐1, 𝑐1) (6-30) 𝑥 𝑥 𝑥 𝑥 1 0 𝑥 𝑥 0 𝑥 𝑥 2 𝜕𝑥2 𝑥 𝜕𝑥2 𝑥
From the defined norms, the following statement is to be proven, 
‖𝑐1𝑥‖1 ≤ ‖𝑐
0
𝑥‖0 
150 
 
Reformulating in terms of the defined norms, 
𝜕2𝑐1 𝜕2𝑐1
(𝑐1 1
𝑥 𝑥
𝑥 , 𝑐𝑥) − 𝜆2 ( , = (𝑐
0, 𝑐1) − 𝜆 𝐴𝐵𝐶𝐷𝛼𝑐0,  𝐴𝐵𝐶𝐷𝛼𝑐1  
𝜕𝑥2 𝜕𝑥2
) 𝑥 𝑥 1( 0 𝑥 𝑥 0 𝑥 𝑥)
(6-31) 
‖𝑐1‖2𝑥 1 = ‖𝑐
0 1
𝑥‖0‖𝑐𝑥‖0 − 𝜆 ‖
𝐴𝐵𝐶𝐷𝛼𝑐0 𝐴𝐵𝐶1 0 𝑥 𝑥‖ ‖ 0𝐷
𝛼 1
𝑥 𝑐𝑥‖  0 0
where, 
𝐴𝐵(𝛼) 𝑥 𝑑𝑐0
𝐴𝐵𝐶 𝛼 0 𝜏
𝛼 
‖ 0𝐷𝑥 𝑐𝑥‖0 = ‖  ∫ 𝐸𝛼 [− (𝑥 − 𝜏)
𝛼] 𝑑𝜏‖  
(1 − 𝛼) 0 𝑑𝜏 1 − 𝛼 0
Thus, 
𝐴𝐵(𝛼) 𝑥 𝑑𝑐0 𝛼 
‖𝐴𝐵𝐶𝐷𝛼𝑐0
𝜏
‖ ≤  ∫ ‖ ‖ ‖𝐸 [− (𝑥 − 𝜏)𝛼0 𝑥 𝑥 0 𝛼 ] ‖ 𝑑𝜏 (6-32) (1 − 𝛼) 0 𝑑𝜏 1 − 𝛼 0 0
The Mittag-Leffler function is bounded because 1 > 𝛼 > 0, and the first order derivative is bounded 
to the physical meaning of the derivative of the spread of a particle in terms of its concentration. Thus, 
the derivative is considered at the maximum physical space that is applicable to the existence of the 
concentration (𝑋𝑚𝑎𝑥), 
𝐴𝐵(𝛼) 𝑥 𝑑𝑐0
‖𝐴𝐵𝐶
𝜏
0𝐷
𝛼
𝑥 𝑐
0
𝑥‖0 ≤ ∫ ‖ ‖ 𝑑𝜏 (1 − 𝛼) 0 𝑑𝜏 0
𝐴𝐵(𝛼) 𝑥𝑚𝑎𝑥
                           ≤  𝜃‖𝑐0‖ ∫ 𝑑𝜏 (6-33) 
(1 − 𝛼) 𝑥 0 0
𝐴𝐵(𝛼)
                    ≤  𝜃‖𝑐0𝑥‖0𝑋  (1 − 𝛼) 𝑚𝑎𝑥
Substituting back into Equation (6-31), 
𝐴𝐵(𝛼) 𝐴𝐵(𝛼)
‖𝑐1‖2 0 1 0 1𝑥 1 < ‖𝑐𝑥‖0‖𝑐𝑥‖0 − 𝜆1 { 𝜃‖𝑐 ‖ 𝑋 } { 𝜃‖𝑐 ‖ 𝑋 } (6-34) (1 − 𝛼) 𝑥 0 𝑚𝑎𝑥 (1 − 𝛼) 𝑥 0 𝑚𝑎𝑥
Rearranging, 
2
𝐴𝐵(𝛼)𝜃𝑋
1 2 0 1 𝑚𝑎𝑥  ‖𝑐𝑥‖
0 1
1 < ‖𝑐𝑥‖0‖𝑐𝑥‖0 − 𝜆1 { (1 − 𝛼) 
} ‖𝑐𝑥‖ ‖𝑐𝑥‖  0 0
(6-35) 
2
𝐴𝐵(𝛼)𝜃𝑋𝑚𝑎𝑥
< (1 − 𝜆 { } )‖𝑐01  𝑥‖0‖𝑐
1
𝑥‖0 (1 − 𝛼)
Applying the assumption that  
‖𝑐1𝑥‖0 ≤ ‖𝑐
1
𝑥‖1 
Simplifying the stability conditions becomes, 
2
𝐴𝐵(𝛼)𝜃𝑋𝑚𝑎𝑥
           ‖𝑐1‖2𝑥 1 < (1 − 𝜆
0 1
1 { } )‖𝑐𝑥‖ ‖ 0 𝑐𝑥‖1 (1 − 𝛼) (6-36) 
2
𝐴𝐵(𝛼)𝜃𝑋
‖𝑐1
𝑚𝑎𝑥
𝑥‖1 < (1 − 𝜆 { } )‖𝑐
0
1  𝑥
‖0 (1 − 𝛼)
151 
 
‖𝑐1
2
𝑥‖1 𝐴𝐵(𝛼)𝜃𝑋𝑚𝑎𝑥
< 1 − 𝜆1 { }  
‖𝑐0𝑥‖ (1 − 𝛼)
 
0
where, 
2
𝐴𝐵(𝛼)𝜃𝑋𝑚𝑎𝑥
1 − 𝜆1 { } < 1 (1 − 𝛼) 
2
𝐴𝐵(𝛼)𝜃𝑋𝑚𝑎𝑥
𝜆1 { (1 − 𝛼) 
} > 0 
The first condition is thus upheld and unconditionally stable.  
Secondly, Let ∀𝑔 ∈ 𝐻1(Ω), 𝑔 = 𝑐𝑛+1𝑥  
𝜕2 𝜕2
(𝑐𝑛+1𝑥 , 𝑐
𝑛+1
𝑥 ) = (𝑐
𝑛, 𝑐𝑛+1) − 𝜆 (𝐴𝐵𝐶 𝛼 𝑛 𝐴𝐵𝐶 𝛼 𝑛+1 𝑛+1 𝑛+1𝑥 𝑥 1 0𝐷𝑥 𝑐𝑥 ,  0𝐷𝑥 𝑐𝑥 ) + 𝜆2 ( 𝑐 , 𝑐 ) 𝜕𝑥2 𝑥 𝜕𝑥2 𝑥
(6-37) 
𝑛−1
−𝜆 ∑((𝑐𝑘+1, 𝑐𝑛+1 𝑘 𝑛+13 𝑥 𝑥 ) − (𝑐𝑥 , 𝑐𝑥 )) 
𝑘=0
where, 
1
𝜆 = 𝐴𝛼3  𝐴𝛼 𝑛,𝑘𝑛
From the defined norms, the following statement is to be proven, 
‖𝑐𝑛+1 0𝑥 ‖1 ≤ ‖𝑐𝑥‖0 
Reformulating Equation (6-37) in terms of the defined norms, 
𝜕2𝑐𝑛+1 𝜕2𝑐𝑛+1
(𝑐𝑛+1
𝑥 𝑥
𝑥 , 𝑐
𝑛+1
𝑥 ) − 𝜆2 ( , ) = 𝜕𝑥2 𝜕𝑥2
𝑛−1
(𝑐𝑛, 𝑐𝑛+1) − 𝜆 (𝐴𝐵𝐶 𝛼 𝑛 𝐴𝐵𝐶 𝛼 𝑛+1𝑥 𝑥 1 0𝐷𝑥 𝑐𝑥 ,  0𝐷𝑥 𝑐𝑥 ) − 𝜆3∑((𝑐
𝑘+1
𝑥 , 𝑐
𝑛+1
𝑥 ) − (𝑐
𝑘 𝑛+1
𝑥 , 𝑐𝑥 )) 
𝑘=0 (6-38) 
‖𝑐𝑛+1‖2 = ‖𝑐𝑛‖ ‖𝑐𝑛+1‖ 𝐴𝐵𝐶 𝛼 𝑛 𝐴𝐵𝐶 𝛼 𝑛+1𝑥 1 𝑥 0 𝑥 0 − 𝜆1‖ 0𝐷𝑥 𝑐𝑥 ‖ ‖ 0𝐷𝑥 𝑐𝑥 ‖  0 0
𝑛−1
−𝜆3∑((‖𝑐
𝑘+1‖ ‖𝑐𝑛+1𝑥 𝑥 ‖0) − (‖𝑐
𝑘‖ ‖𝑐𝑛+1𝑥 𝑥 ‖0)) 0 0
𝑘=0
Applying Equation (6-33), 
𝑛−1
‖𝑐𝑛+1‖2 ≤ ‖𝑐𝑛‖ ‖𝑐𝑛+1‖ − 𝜆 𝐴‖𝑐𝑛‖ ‖𝑐𝑛+1𝑥 1 𝑥 0 𝑥 0 1 𝑥 0 𝑥 ‖0 − 𝜆3∑((‖𝑐
𝑘+1‖ ‖𝑐𝑛+1 𝑘 𝑛+1 (6-39) 𝑥 0 𝑥 ‖0) − (‖𝑐𝑥‖ ‖𝑐 ‖ )) 0 𝑥 0
𝑘=0
where, 
2
𝐴𝐵(𝛼)𝜃𝑋𝑚𝑎𝑥
𝐴 = { }  
(1 − 𝛼) 
Using the inductive method for 
‖𝑐𝑛𝑥 ‖0 ≤ ‖𝑐
0
𝑥‖0 
152 
 
Equation (6-39) becomes, 
𝑛−1 (6-40) 
‖𝑐𝑛+1‖2 ≤ ‖𝑐0‖ ‖𝑐𝑛+1𝑥 1 𝑥 𝑥 ‖0 − 𝜆1𝐴‖𝑐
0
𝑥‖ ‖𝑐
𝑛+1
𝑥 ‖0 − 𝜆3∑((‖𝑐
0
𝑥‖ ‖𝑐
𝑛+1
𝑥 ‖0) − (‖𝑐
0
𝑥‖ ‖𝑐
𝑛+1‖ )) 
0 0 0 0 𝑥 0
𝑘=0
Reformulating in terms of the defined norms, 
‖𝑐𝑛+1 2 0 𝑛+1 0𝑥 ‖1 ≤ ‖𝑐𝑥‖0‖𝑐𝑥 ‖1 − 𝜆1𝐴‖𝑐𝑥‖0‖𝑐
𝑛+1
𝑥 ‖1 (6-41) 
Rearranging and simplifying, 
‖𝑐𝑛+1𝑥 ‖
2 0 𝑛+1
1 ≤ (1 − 𝜆1𝐴)‖𝑐𝑥‖0‖𝑐𝑥 ‖1 (6-42) 
‖𝑐𝑛+1𝑥 ‖1 ≤ (1 − 𝜆 𝐴)‖𝑐
0
1 𝑥‖0 
where, 
1 − 𝜆1𝐴 < 1 
𝜆1𝐴 > 0 
Thus, the second condition is supported and unconditionally stable. This concludes the semi-
discretisation analysis for an evolution equation, and it can be concluded that the advection-
dominated fractional transport model with ABC fractional derivative is stable in time.  
6.2 Upwind numerical approximation schemes 
The modified upwind schemes are to be applied to the advection-focused transport model with 
Atangana-Baleanu in Caputo sense (ABC) derivative for numerical approximation (Equation (4-93)). 
Applying the Atangana-Baleanu in Caputo sense (ABC) fractional derivative definition to the developed 
advection-dispersion equation, 
𝜕2
 𝐴𝐵𝐶𝐷𝛼
(6-43)  
𝑎 𝑡  (𝑐(𝑥, 𝑡)) = −𝑣  
𝐴𝐵𝐶𝐷𝛼𝑥 𝑎 𝑥 (𝑐(𝑥, 𝑡)) + 𝐷𝐿 (𝑐(𝑥, 𝑡)) 
𝜕𝑥2
where, 
 𝐴𝐵(𝛼) 𝑡 𝑑 𝛼 
 𝐴𝐵𝐶𝑎 𝐷
𝛼
𝑡 𝑓(𝑥) =  ∫ 𝑓(𝜏) 𝐸𝛼 [− (𝑡 − 𝜏)
𝛼] 𝑑𝜏  
(1 − 𝛼) 0 𝑑𝜏 1 − 𝛼 
A forward finite difference scheme in time is applied to the Atangana-Baleanu in Caputo sense (ABC) 
fractional derivative to investigate the numerical approximation method. The Atangana-Baleanu (ABC) 
fractional derivative is considered for a specific time, 𝑡𝑛: 
 𝐴𝐵(𝛼) 𝑡𝑛 𝑑 𝛼 
 𝐴𝐵𝐶 𝛼𝑎 𝐷𝑡 𝑓(𝑡𝑛) =  ∫ 𝑓(𝜏) 𝐸𝛼 [− (𝑡𝑛 − 𝜏)
𝛼] 𝑑𝜏 
 (6-44) (1 − 𝛼) 0 𝑑𝜏 1 − 𝛼 
The time integer-order derivative 𝜏 is replaced with the forward finite difference approximation at 
specific points in time (𝑡), and a summation is used to express the integral performed for each time 
step (time domain discretisation): 
153 
 
𝑛−1
𝐴𝐵(𝛼) 𝑡𝑘+1 𝑓𝑘+1 − 𝑓𝑘 𝛼 
 𝐴𝐵𝐶 𝛼
𝑖 𝑖
𝑎 𝐷𝑡 𝑓(𝑡𝑛) = ∑  ∫ ( ) 𝐸
𝛼
(1 − 𝛼) ∆𝑡 𝛼
[− (𝑡𝑛 − 𝜏) ] 𝑑𝜏  (6-45) 
𝑘=0 𝑡
1 − 𝛼 
𝑘
The approximation of the continuous 𝜏 function, results in two specific points of the function with 
respect to time (𝑡), which allows the approximated derivative to be taken out of the integral: 
𝑛−1
𝐴𝐵(𝛼) 𝑓𝑘+1 − 𝑓𝑘 𝑡𝑘+1𝑖 𝑖 𝛼 
 𝐴𝐵𝐶𝐷𝛼𝑓(𝑡 ) = ∑  ( )∫ 𝐸 [− (𝑡 − 𝜏)𝛼𝑎 𝑡 𝑛  𝛼 𝑛 ] 𝑑𝜏  (6-46) (1 − 𝛼) ∆𝑡 𝑡 1 − 𝛼 𝑘=0 𝑘
The numerical approximation by to this point has been similar to the approach followed in the Caputo 
definition of the fractional derivative, but now a different approach is required to take into account 
the Mittag-Leffler function (𝐸𝛼). The Mittag-Leffler function is numerically integrated using the known 
properties of the function (Alkahtani et al., 2017): 
𝑡𝑘+1 𝛼 𝛼 
∫ 𝐸𝛼 [− (𝑡
𝛼
𝑛 − 𝜏) ] 𝑑𝜏 = (𝑡𝑛 − 𝑡𝑘)𝐸1 − 𝛼 𝛼,2
[− (𝑡 − 𝑡 )] 
𝑡 1 − 𝛼 
𝑛 𝑘
𝑘 (6-47) 
𝛼 
                                                      −(𝑡𝑛 − 𝑡𝑘+1)𝐸𝛼,2 [− (𝑡𝑛 − 𝑡𝑘+1)]  1 − 𝛼 
Considering the specific time can be represented as the number of time steps required to reach that 
time (𝑡𝑛 = ∆𝑡 ∙ 𝑛) and similarly, 𝑡𝑘 = ∆𝑡 ∙ 𝑘. Thus, the same can be applied to achieve 𝑡𝑛 − 𝑡𝑘 =
∆𝑡(𝑛 − 𝑘) and 𝑡𝑛 − 𝑡𝑘+1 = ∆𝑡(𝑛 − 𝑘 − 1): 
𝑡𝑘+1 𝛼 𝛼 
∫ 𝐸𝛼 [− (𝑡𝑛 − 𝜏)
𝛼] 𝑑𝜏 = (∆𝑡(𝑛 − 𝑘))𝐸𝛼,2 [− (∆𝑡(𝑛 − 𝑘))] 
𝑡 1 − 𝛼 1 − 𝛼 (6-48) 𝑘
𝛼 
                                                      −(∆𝑡(𝑛 − 𝑘 − 1))𝐸𝛼,2 [− (∆𝑡(𝑛 − 𝑘 − 1))]  1 − 𝛼 
Simplifying, 
𝛼 ∆𝑡 
𝑡𝑘+1 𝛼 (𝑛 − 𝑘)𝐸𝛼,2 [− (𝑛 − 𝑘)]
∫ 𝐸 [− (𝑡 − 𝜏)𝛼] 𝑑𝜏 = { 1 − 𝛼 𝛼 ∆𝑡 }  (6-49) 
𝑡 1 − 𝛼 
𝑛 𝛼 ∆𝑡
𝑘 −(𝑛 − 𝑘 − 1)𝐸𝛼,2 [− (𝑛 − 𝑘 − 1)]1 − 𝛼 
Substituting the numerically integrated Mittag-Leffler function into Equation (6-46):  
𝛼 ∆𝑡 
𝑛−1
𝐴𝐵(𝛼) (𝑛 − 𝑘)𝐸𝛼,2 [− (𝑛 − 𝑘)]
 𝐴𝐵𝐶 𝛼𝑎 𝐷𝑡 𝑓(𝑡𝑛) = ∑(𝑓
𝑘+1 − 𝑓𝑘){ 1 − 𝛼 }  (6-50) 
(1 − 𝛼) 𝑖 𝑖 𝛼 ∆𝑡
𝑘=0 −(𝑛 − 𝑘 − 1)𝐸𝛼,2 [− (𝑛 − 𝑘 − 1)]1 − 𝛼 
Equation (6-50) is the numerically approximated Atangana-Baleanu in Caputo sense (ABC) fractional 
derivative, where the function is considered at two discrete points in time, and additionally the 
fractional components of the Mittag-Leffler function are included to account for changes in-between 
those two discrete points in time.  
154 
 
6.2.1 Explicit upwind  
The numerical approximation of the Atangana-Baleanu in Caputo sense (ABC) fractional derivative 
with respect to time has been considered in Equation (6-50), where, a function 𝛿𝛼𝑛,𝑘 is applied to 
simplify, 
𝑛−1
𝐴𝐵(𝛼)
𝐴𝐵𝐶 𝛼 (6-51)  0 𝐷𝑡 (𝑐(𝑥𝑚, 𝑡𝑘)) = ∑(𝑐
𝑘+1 − 𝑐𝑘 )𝛿𝛼
(1 − 𝛼) 𝑚 𝑚 𝑛,𝑘
 
𝑘=0
where, 
𝛼 ∆𝑡 𝛼 ∆𝑡
𝛼  𝛿𝑛,𝑘 = (𝑛 − 𝑘)𝐸𝛼,2 [− (𝑛 − 𝑘)] − (𝑛 − 𝑘 − 1)𝐸𝛼,2 [− (𝑛 − 𝑘 − 1)] 1 − 𝛼 1 − 𝛼 
Similarly, the ABC fractional derivative with respect to space (explicit) and the upwind scheme is 
(Alkahtani et al., 2017), 
𝛼 ∆𝑥 
𝑚
𝐴𝐵(𝛼) (𝑚 − 𝑖)𝐸𝛼,2 [− (𝑚 − 𝑖)] 
 𝐴𝐵𝐶𝐷𝛼(𝑐(𝑥 , 𝑡 )) = ∑(𝑐𝑛−10 𝑥 𝑚 𝑘 𝑖 − 𝑐
𝑛−1 1 − 𝛼 (6-52) 
(1 − 𝛼) 𝑖−1
) { } 
𝛼 ∆𝑥
𝑖=0 −(𝑚 − 𝑖 − 1)𝐸𝛼,2 [− (𝑚 − 𝑖 − 1)]1 − 𝛼 
where, a function 𝛿𝛼𝑚,𝑖 is applied to simplify, 
𝑚
𝐴𝐵(𝛼)
 𝐴𝐵𝐶𝐷𝛼 𝑛−1 𝑛−1 𝛼0 𝑥 (𝑐(𝑥𝑚, 𝑡𝑘)) = ∑(𝑐 𝑖 − 𝑐𝑖−1 )𝛿𝑚,𝑖 (6-53) (1 − 𝛼)
𝑖=0
Substituting this into the advection-dispersion equation, and using the traditional finite difference 
approach for the local second order derivative, 
𝑛−1 𝑚
𝐴𝐵(𝛼) 𝐴𝐵(𝛼)
∑(𝑐𝑘+1 − 𝑐𝑘 )𝛿𝛼 + 𝑣 ∑(𝑐𝑛−1 − 𝑐𝑛−1 𝛼𝑚 𝑚 𝑛,𝑘 𝑥 𝑖 𝑖−1 )𝛿𝑚,𝑖 (1 − 𝛼) (1 − 𝛼) (6-54) 
𝑘=0 𝑖=0
𝑐𝑛−1𝑚+1 − 2𝑐
𝑛−1
𝑚 + 𝑐
𝑛−1
𝑚−1
−𝐷𝐿 ( ) = 0 (∆𝑥)2
Reformulating the following can be obtained, 
𝑛−2
𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 𝐴𝐵(𝛼)
(𝑐𝑛 𝑛−1 𝛼 𝑘+1 𝑘 𝛼 𝑛−1 𝑛−1 𝛼
(1 − 𝛼) 𝑚
− 𝑐𝑚 )𝛿𝑛,𝑛−1 + ∑ 𝑐 − 𝑐 𝛿 + 𝑣 (𝑐 − 𝑐 )𝛿(1 − 𝛼) 
( 𝑚 𝑚) 𝑛,𝑘 𝑥 (1 − 𝛼) 𝑚 𝑚−1 𝑚,𝑖
𝑘=0 (6-55) 
𝑚
𝐴𝐵(𝛼) 𝑛−1𝑛−1 𝑛−1 𝛼 𝑐𝑚+1 − 2𝑐
𝑛−1
𝑚 + 𝑐
𝑛−1
𝑚−1
+ 𝑣𝑥 ∑ 𝑐(1 − 𝛼) 
( 𝑖 − 𝑐𝑖−1 )𝛿𝑚,𝑖 − 𝐷𝐿 ( ) = 0 (∆𝑥)2
𝑖=0
Rearranging, 
𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 2𝐷
𝑛 𝐿𝑐𝑚 ( 𝛿
𝛼 𝑛−1 𝛼 𝛼
(1 − 𝛼) 𝑛,𝑛−1
) = 𝑐𝑚 ( 𝛿(1 − 𝛼) 𝑛,𝑛−1
− 𝑣𝑥 𝛿(1 − 𝛼) 𝑚,𝑖
− ) 
(∆𝑥)2
𝑛−2
𝐴𝐵(𝛼) 𝛼 𝐷𝐿 𝐷𝐿 𝐴𝐵(𝛼)+𝑐𝑛−1𝑚−1 (𝑣𝑥 𝛿𝑚,𝑖 + )+ 𝑐
𝑛−1
𝑚+1 ( ) − ∑(𝑐
𝑘+1
𝑚 − 𝑐
𝑘 )𝛿𝛼
(1 − 𝛼) (∆𝑥)2 (∆𝑥)2 (1 − 𝛼) 𝑚 𝑛,𝑘 (6-56) 
𝑘=0
𝑚
𝐴𝐵(𝛼)
− 𝑣 ∑(𝑐𝑛−1 − 𝑐𝑛−1𝑥  𝑖 𝑖−1 )𝛿
𝛼
(1 − 𝛼) 𝑚,𝑖
 
𝑖=0
155 
 
This is the first-order explicit upwind finite difference scheme for the one-dimensional, non-reactive 
fractional advection-dispersion equation (ABC). The numerical scheme can be simplified as follows,  
𝑛−2 𝑚
𝑎 𝑐𝑛 = 𝑏 𝑐𝑛−1 + 𝑑 𝑐𝑛−1 + 𝑓 𝑐𝑛−1 𝑘+1 𝑘 𝛼 𝑛−1 𝑛−1 𝛼
(6-57) 
4 𝑚 4 𝑚 4 𝑚−1 4 𝑚+1 − 𝑔4∑(𝑐𝑚 − 𝑐𝑚)𝛿𝑛,𝑘 − 𝑣𝑥𝑔4∑(𝑐𝑖 − 𝑐𝑖−1 )𝛿𝑚,𝑖  
𝑘=0 𝑖=0
where, 
𝐴𝐵(𝛼)
𝑎 = 𝛿𝛼4  (1 − 𝛼) 𝑛,𝑛−1
𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 2𝐷𝐿
𝑏 = 𝛿𝛼4 − 𝑣
𝛼
(1 − 𝛼) 𝑛,𝑛−1 𝑥
𝛿 −  
(1 − 𝛼) 𝑚,𝑖 (∆𝑥)2
𝐴𝐵(𝛼) 𝐷𝐿
𝑑4 = 𝑣𝑥 𝛿
𝛼 +   
(1 − 𝛼) 𝑚,𝑖 (∆𝑥)2
𝐷𝐿
𝑓4 =  (∆𝑥)2
𝐴𝐵(𝛼)
 𝑔4 =  (1 − 𝛼) 
6.2.2 Implicit upwind  
Applying the same methodology as the explicit upwind numerical approximation, the following is 
obtained, 
𝑛−1 𝑚
𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 𝑐𝑛 − 2𝑐𝑛 + 𝑐𝑛𝑚+1 𝑚 𝑚−1
∑(𝑐𝑘+1𝑚 − 𝑐
𝑘
𝑚)𝛿
𝛼 𝑛 𝑛 𝛼
 𝑛,𝑘
+ 𝑣𝑥 ∑(𝑐 𝑖 − 𝑐𝑖−1)𝛿𝑚,𝑖 − 𝐷𝐿 ( 2 ) = 0 
(6-58) 
(1 − 𝛼) (1 − 𝛼) (∆𝑥)
𝑘=0 𝑖=0
Reformulating and rearranging the following can be obtained, 
𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 2𝐷
𝑐𝑛 ( 𝛿𝛼
𝐿
𝑚 𝑛,𝑛−1 + 𝑣 𝛿
𝛼
(1 − 𝛼) 𝑥 (1 − 𝛼) 𝑚,𝑖
+ )
(∆𝑥)2
𝐴𝐵(𝛼) 𝐷 𝐷
= 𝑐𝑛 (𝑣 𝛿𝛼
𝐿 𝑛 𝐿
𝑚−1 𝑥 (1 − 𝛼) 𝑚,𝑖
− + 𝑐 ( ) 
(∆𝑥)2
) 𝑚+1 (∆𝑥)2 (6-59) 
𝑛−2 𝑚
𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 𝑛 𝑛 𝛼 𝐴𝐵(𝛼)− ∑(𝑐𝑘+1 − 𝑐𝑘 𝛼 𝑛−1 𝛼
(1 − 𝛼) 𝑚 𝑚
)𝛿𝑛,𝑘 − 𝑣𝑥 ∑(𝑐 − 𝑐(1 − 𝛼) 𝑖 𝑖−1
)𝛿𝑚,𝑖 + 𝑐𝑚 ( 𝛿 ) (1 − 𝛼) 𝑛,𝑛−1
𝑘=0 𝑖=0
Equation (6-59) is the implicit upwind finite difference scheme for the one-dimensional fractional 
advection-dispersion equation (ABC). The numerical scheme can be further simplified by substituting 
functions,  
𝑛−2 𝑚
ℎ4𝑐
𝑛
𝑚 − 𝑗
𝑛
4𝑐𝑚−1 − 𝑓
𝑛 𝑘+1 𝑘 𝛼 𝑛 𝑛 𝛼 𝑛−1
4𝑐𝑚+1 + 𝑔4∑(𝑐𝑚 − 𝑐𝑚)𝛿𝑛,𝑘 + 𝑣𝑥𝑔4∑(𝑐𝑖 − 𝑐𝑖−1)𝛿𝑚,𝑖 = 𝑎4𝑐𝑚  (6-60) 
𝑘=0 𝑖=0
where, 
𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 2𝐷𝐿
ℎ 𝛼4 = 𝛿𝑛,𝑛−1 + 𝑣𝑥 𝛿
𝛼
(1 − 𝛼) (1 − 𝛼) 𝑚,𝑖
+  
(∆𝑥)2
𝐴𝐵(𝛼) 𝐷𝐿
 𝑗4 = 𝑣𝑥 𝛿
𝛼
(1 − 𝛼) 𝑚,𝑖
−  
(∆𝑥)2
156 
 
6.2.3 Upwind advection Crank-Nicolson scheme 
For the upwind advection Crank-Nicolson finite difference scheme, the time component remains the 
same as with the first-order implicit and explicit schemes, and the space components change to, 
𝑚
𝐴𝐵(𝛼)
 𝐴𝐵𝐶𝐷𝛼(𝑐(𝑥 , 𝑡 )) =  ∑[0.5(𝑐𝑛−1 − 𝑐𝑛−1 𝑛0 𝑥 𝑚 𝑛 𝑖 𝑖−1 ) + 0.5(𝑐𝑖 − 𝑐
𝑛
𝑖−1)] 𝛿
𝛼    (6-61) 
(1 − 𝛼) 𝑚,𝑖
𝑖=0
Substituting this back into the advection-dispersion equation, 
𝑛−1 𝑚
𝐴𝐵(𝛼) 𝐴𝐵(𝛼)
∑(𝑐𝑘+1 − 𝑐𝑘 )𝛿𝛼 + 𝑣  ∑[0.5(𝑐𝑛−1 𝑛−1 𝑛 𝑛 𝛼
(1 − 𝛼) 𝑚 𝑚 𝑛,𝑘 𝑥 (1 − 𝛼) 𝑖
− 𝑐𝑖−1 ) + 0.5(𝑐𝑖 − 𝑐𝑖−1)] 𝛿𝑚,𝑖 
𝑘=0 𝑖=0
𝑐𝑛 𝑛 𝑛𝑚+1 − 2𝑐𝑚 + 𝑐𝑚−1
−𝐷𝐿 ( ) = 0 (∆𝑥)2
Reformulating and rearranging, the following can be obtained 
𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 2𝐷 𝐴𝐵(𝛼) 𝐴𝐵(𝛼)
𝑐𝑛 ( 𝛿𝛼
𝐿
𝑚 + 0.5𝑣 𝛿
𝛼 + ) = 𝑐𝑛−1 ( 𝛿𝛼 + 0.5𝑣 𝛿𝛼 ) 
(1 − 𝛼) 𝑛,𝑛−1 𝑥 (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2 𝑚 (1 − 𝛼) 𝑛,𝑛−1 𝑥 (1 − 𝛼) 𝑚,𝑖
𝐴𝐵(𝛼) 𝐷 𝐷 𝐴𝐵(𝛼)
+𝑐𝑛 𝛼
𝐿 𝑛 𝐿 𝑛−1 𝛼
𝑚−1 (0.5𝑣𝑥 𝛿𝑚,𝑖 − )+ 𝑐 ( ) + 𝑐 (0.5𝑣 𝛿 ) (1 − 𝛼) (∆𝑥)2 𝑚+1 (∆𝑥)2 𝑚−1 𝑥 (1 − 𝛼) 𝑚,𝑖 (6-62) 
𝑛−2 𝑚
𝐴𝐵(𝛼) 𝐴𝐵(𝛼)
− ∑(𝑐𝑘+1 − 𝑐𝑘 )𝛿𝛼 − 𝑣  ∑[0.5(𝑐𝑛−1 − 𝑐𝑛−1 𝑛 𝑛 𝛼
(1 − 𝛼) 𝑚 𝑚 𝑛,𝑘 𝑥 (1 − 𝛼) 𝑖 𝑖−1
) + 0.5(𝑐𝑖 − 𝑐𝑖−1)] 𝛿𝑚,𝑖 
𝑘=0 𝑖=0
Equation (6-62) is the upwind advection Crank-Nicolson finite difference scheme for the fractional 
advection-dispersion equation (ABC). The numerical scheme can be simplified with functions,  
𝑛−2
𝑙4𝑐
𝑛
𝑚 = 𝑚 𝑐
𝑛−1
4 𝑚 + 𝑜4𝑐
𝑛
𝑚−1 + 𝑓4𝑐
𝑛
𝑚+1 + 𝑝4𝑐
𝑛−1
𝑚−1 − 𝑔4∑(𝑐
𝑘+1
𝑚 − 𝑐
𝑘
𝑚)𝛿
𝛼
𝑛,𝑘 
𝑘=0 (6-63) 
𝑚
−𝑣𝑔  ∑[0.5(𝑐𝑛−1 − 𝑐𝑛−14 𝑖 𝑖−1 ) + 0.5(𝑐
𝑛 𝑛 𝛼
𝑖 − 𝑐𝑖−1)] 𝛿𝑚,𝑖 
𝑖=0
where, 
𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 2𝐷
𝑙 = 𝛿𝛼 𝛼
𝐿
4 𝑛,𝑛−1 + 0.5𝑣𝑥 𝛿𝑚,𝑖 +  (1 − 𝛼) (1 − 𝛼) (∆𝑥)2
𝐴𝐵(𝛼) 𝐴𝐵(𝛼)
 𝑚4 = 𝛿
𝛼 𝛼
(1 − 𝛼) 𝑛,𝑛−1
+ 0.5𝑣𝑥 𝛿(1 − 𝛼) 𝑚,𝑖
  
𝐴𝐵(𝛼) 𝐷𝐿
𝑜4 = 0.5𝑣𝑥 𝛿
𝛼
(1 − 𝛼) 𝑚,𝑖
−  
(∆𝑥)2
𝐴𝐵(𝛼)
𝑝4 = 0.5𝑣
𝛼
𝑥 𝛿𝑚,𝑖  (1 − 𝛼) 
6.2.4 Explicit upwind-downwind weighted scheme 
The upwind and downwind weighted scheme for the advection term is controlled by a ratio of upwind 
to downwind (𝜃), where 0 ≤ 𝜃 ≤ 1. Thus, the space advection component is altered to, 
157 
 
𝑚
𝐴𝐵(𝛼)
 𝐴𝐵𝐶𝐷𝛼(𝑐(𝑥 , 𝑡 )) =  ∑[𝜃(𝑐𝑛−1 − 𝑐𝑛−1 𝑛−10 𝑥 𝑚 𝑛 𝑖 𝑖−1 ) + (1 − 𝜃)(𝑐𝑖+1 − 𝑐
𝑛−1)] 𝛿𝛼   (6-64) 
(1 − 𝛼) 𝑖 𝑚,𝑖
 
𝑖=0
Substituting this back into the advection-dispersion equation, 
𝑛−1 𝑚
𝐴𝐵(𝛼) 𝐴𝐵(𝛼)
∑(𝑐𝑘+1 − 𝑐𝑘 )𝛿𝛼 + 𝑣  ∑[𝜃(𝑐𝑛−1 𝑛−1 𝑛−1 𝑛−1 𝛼
(1 − 𝛼) 𝑚 𝑚 𝑛,𝑘 𝑥 (1 − 𝛼) 𝑖
− 𝑐𝑖−1 ) + (1 − 𝜃)(𝑐𝑖+1 − 𝑐𝑖 )] 𝛿𝑚,𝑖 
𝑘=0 𝑖=0   (6-65) 
𝑐𝑛−1𝑚+1 − 2𝑐
𝑛−1 + 𝑐𝑛−1𝑚 𝑚−1
−𝐷𝐿 ( ) = 0 (∆𝑥)2
Reformulating and rearranging, the following can be obtained 
𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 2𝐷
𝑐𝑛 𝛿𝛼 = 𝑐𝑛−1 ( 𝛿𝛼
𝐿
𝑚 − 𝑣 𝜃 𝛿
𝛼 + 𝑣 (1 − 𝜃) 𝛿𝛼 − ) 
(1 − 𝛼) 𝑛,𝑛−1 𝑚 (1 − 𝛼) 𝑛,𝑛−1 𝑥 (1 − 𝛼) 𝑚,𝑖 𝑥 (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2
𝐴𝐵(𝛼) 𝐷𝐿 𝐴𝐵(𝛼) 𝐷
+𝑐𝑛−1 𝛼 𝑛−1
𝐿
𝑚−1 (𝑣𝑥𝜃 𝛿𝑚,𝑖 + )− 𝑐𝑚+1 (𝑣𝑥(1 − 𝜃) 𝛿
𝛼 + ) 
(1 − 𝛼) (∆𝑥)2 (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2 (6-66) 
𝑛−2 𝑚
𝐴𝐵(𝛼) 𝐴𝐵(𝛼)
− ∑(𝑐𝑘+1 − 𝑐𝑘 )𝛿𝛼  − 𝑣  ∑[𝜃(𝑐𝑛−1 − 𝑐𝑛−1𝑚 𝑚 𝑛,𝑘 𝑥 𝑖 𝑖−1 ) + (1 − 𝜃)(𝑐
𝑛−1 𝑛−1
𝑖+1 − 𝑐𝑖 )] 𝛿
𝛼
(1 − 𝛼) (1 − 𝛼) 𝑚,𝑖
 
𝑘=0 𝑖=0
Equation (6-66) is the explicit upwind-downwind weighted finite difference scheme for the one-
dimensional, non-reactive fractional advection-dispersion equation (ABC). Simplifying, the following 
functions as substituted,  
𝑎 𝑐𝑛 = 𝑞 𝑐𝑛−1 + 𝑟 𝑐𝑛−1 𝑛−14 𝑚 4 𝑚 4 𝑚−1 − 𝑠4𝑐𝑚+1 
𝑛−2 𝑚 (6-67) 
−𝑔4∑(𝑐
𝑘+1 − 𝑐𝑘 𝛼 𝑛−1 𝑛−1 𝑛−1 𝑛−1 𝛼𝑚 𝑚)𝛿𝑛,𝑘 − 𝑣𝑥𝑔4∑[𝜃(𝑐𝑖 − 𝑐𝑖−1 ) + (1 − 𝜃)(𝑐𝑖+1 − 𝑐𝑖 )] 𝛿𝑚,𝑖 
𝑘=0 𝑖=0
where, 
𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 2𝐷𝐿
𝑞4 = 𝛿
𝛼 𝛼 𝛼
(1 − 𝛼) 𝑛,𝑛−1
− 𝑣𝑥𝜃 𝛿(1 − 𝛼) 𝑚,𝑖
+ 𝑣𝑥(1 − 𝜃) 𝛿(1 − 𝛼) 𝑚,𝑖
−  
(∆𝑥)2
𝐴𝐵(𝛼) 𝐷𝐿
𝑟4 = 𝑣𝑥𝜃 𝛿
𝛼
𝑚,𝑖 +   (1 − 𝛼) (∆𝑥)2
𝐴𝐵(𝛼) 𝐷𝐿
𝑠4 = 𝑣𝑥(1 − 𝜃) 𝛿
𝛼 +  
(1 − 𝛼) 𝑚,𝑖 (∆𝑥)2
6.2.5 Implicit upwind-downwind weighted scheme  
Similarly, considering both the upwind and downwind direction for the advection term for the first-
order upwind finite difference scheme approximation (implicit), a ratio of upwind to downwind is 
applied (𝜃), where 0 ≤ 𝜃 ≤ 1. Then, the space advection component becomes, 
𝑚
𝐴𝐵(𝛼)
 𝐴𝐵𝐶𝐷𝛼(𝑐(𝑥 , 𝑡 )) =  ∑[𝜃(𝑐𝑛 − 𝑐𝑛 𝑛 𝑛 𝛼   (6-68) 0 𝑥 𝑚 𝑛 (1 − 𝛼) 𝑖 𝑖−1
) + (1 − 𝜃)(𝑐𝑖+1 − 𝑐𝑖 )] 𝛿𝑚,𝑖  
𝑖=0
Substituting this back into the advection-dispersion equation, 
158 
 
𝑛−1 𝑚
𝐴𝐵(𝛼) 𝐴𝐵(𝛼)
∑(𝑐𝑘+1 𝑘𝑚 − 𝑐𝑚)𝛿
𝛼 + 𝑣 𝑛 𝑛 𝑛 𝑛 𝛼
(1 − 𝛼) 𝑛,𝑘 𝑥
∑[𝜃(𝑐
(1 − 𝛼) 𝑖
− 𝑐𝑖−1) + (1 − 𝜃)(𝑐𝑖+1 − 𝑐𝑖 )] 𝛿𝑚,𝑖 
𝑘=0 𝑖=0   (6-69) 
𝑐𝑛𝑚+1 − 2𝑐
𝑛
𝑚 + 𝑐
𝑛
𝑚−1
−𝐷𝐿 ( ) = 0 (∆𝑥)2
Reformulating and rearranging, the following can be obtained 
𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 2𝐷𝐿
𝑐𝑛𝑚 ( 𝛿
𝛼
𝑛,𝑛−1 + 𝑣
𝛼 𝛼
𝑥𝜃 𝛿𝑚,𝑖 − 𝑣𝑥(1 − 𝜃) 𝛿𝑚,𝑖 + )(1 − 𝛼) (1 − 𝛼) (1 − 𝛼) (∆𝑥)2
𝐷 𝐴𝐵(𝛼) 𝐷 𝐴𝐵(𝛼)
= 𝑐𝑛
𝐿 𝛼 𝑛 𝐿 𝛼
𝑚+1 ( − 𝑣 (1 − 𝜃) 𝛿 ) + 𝑐 ( + 𝑣 𝜃 𝛿 )(∆𝑥)2 𝑥 (1 − 𝛼) 𝑚,𝑖 𝑚−1 (∆𝑥)2 𝑥 (1 − 𝛼) 𝑚,𝑖
𝑛−2
𝐴𝐵(𝛼) 𝐴𝐵(𝛼) (6-70) 
+ 𝑐𝑛−1 𝛿𝛼 − ∑(𝑐𝑘+1𝑚 𝑛,𝑛−1 𝑚 − 𝑐
𝑘
𝑚)𝛿
𝛼
(1 − 𝛼) (1 − 𝛼) 𝑛,𝑘
𝑘=0
𝑚
𝐴𝐵(𝛼)
− 𝑣  ∑[𝜃(𝑐𝑛𝑥 − 𝑐
𝑛 ) + (1 − 𝜃)(𝑐𝑛 − 𝑐𝑛)] 𝛿𝛼  
(1 − 𝛼) 𝑖 𝑖−1 𝑖+1 𝑖 𝑚,𝑖
𝑖=0
Equation (6-70) is the implicit upwind-downwind weighted finite difference scheme for the fractional 
advection-dispersion equation (ABC). The numerical scheme is simplified substituting terms as 
followings,  
𝑢 𝑐𝑛4 𝑚 = 𝑣4𝑐
𝑛
𝑚+1 + 𝑟 𝑐
𝑛 𝑛−1
4 𝑚−1 + 𝑎4𝑐𝑚  
𝑛−2 𝑚 (6-71) 
−𝑔4∑(𝑐
𝑘+1
𝑚 − 𝑐
𝑘
𝑚)𝛿
𝛼
𝑛,𝑘 − 𝑣𝑥𝑔4  ∑[𝜃(𝑐
𝑛 𝑛 𝑛 𝑛 𝛼
𝑖 − 𝑐𝑖−1) + (1 − 𝜃)(𝑐𝑖+1 − 𝑐𝑖 )] 𝛿𝑚,𝑖  
𝑘=0 𝑖=0
where, 
𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 2𝐷
𝑢 = 𝛿𝛼 + 𝑣 𝜃 𝛿𝛼 − 𝑣 (1 − 𝜃) 𝛿𝛼
𝐿
4 +  (1 − 𝛼) 𝑛,𝑛−1 𝑥 (1 − 𝛼) 𝑚,𝑖 𝑥 (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2
𝐷𝐿 𝐴𝐵(𝛼)
𝑣 = − 𝑣 (1 − 𝜃) 𝛿𝛼4  (∆𝑥)2 𝑥 (1 − 𝛼) 𝑚,𝑖
This concludes the formulation of the numerical approximations schemes to be investigated for the 
fractional advection-dispersion equation with ABC fractional derivative. In the following section, the 
numerical stability of each scheme will be analysed. 
6.3 Numerical stability analysis 
The numerical stability method used in Chapter 2 for the local operator numerical approximation 
schemes, and Chapter 5 for the fractional advection-dispersion equation with Caputo fractional 
derivative, will be applied to the developed numerical approximation schemes for the fractional 
advection-dispersion equation with ABC fractional derivative. The numerical stability for the upwind 
schemes are evaluated to validate their use in solving the fractional advection-dispersion equation 
with the ABC fractional definition.  
159 
 
6.3.1 Explicit upwind  
Substituting the induction method terms for the developed explicit upwind numerical scheme 
discussed in Section 6.2.1, 
𝑎 ?̂? 𝑒𝑗𝑘𝑖𝑚 = 𝑏 ?̂? 𝑒𝑗𝑘𝑖𝑚 + 𝑑 ?̂? 𝑒𝑗𝑘𝑖𝑥(𝑚−∆𝑚) 𝑗𝑘𝑖(𝑚+∆𝑚)4 𝑛 4 𝑛−1 4 𝑛−1 + 𝑓4?̂?𝑛−1𝑒  
𝑛−2 𝑚
(6-72) 
−𝑔4∑(?̂?𝑘+1𝑒
𝑗𝑘𝑖𝑚 − ?̂? 𝑒𝑗𝑘𝑖𝑚)𝛿𝛼 − 𝑣 𝑔 ∑(?̂? 𝑒𝑗𝑘𝑖𝑚 − ?̂? 𝑒𝑗𝑘𝑖(𝑚−∆𝑚) 𝛼𝑘 𝑛,𝑘 𝑥 4 𝑛−1 𝑛−1 )𝛿𝑚,𝑖 
𝑘=0 𝑖=0
Multiple out and dividing by 𝑒𝑗𝑘𝑖𝑚, 
𝑛−2
𝑎 ?̂? = 𝑏 ?̂? + 𝑑 ?̂? 𝑒−𝑗𝑘𝑖∆𝑚 + 𝑓 ?̂? 𝑒𝑗𝑘𝑖∆𝑚 𝛼4 𝑛 4 𝑛−1 4 𝑛−1 4 𝑛−1 − 𝑔4∑(?̂?𝑘+1 − ?̂?𝑘)𝛿𝑛,𝑘 
𝑘=0
𝑚 (6-73) 
−𝑣 𝑔 ∑(?̂? − ?̂? 𝑒−𝑗𝑘𝑖∆𝑚𝑥 4 𝑛−1 𝑛−1 )𝛿
𝛼
𝑚,𝑖 
𝑖=0
The first procedure for the induction numerical stability analysis entails proving for a set ∀ 𝑛 > 1, 
|?̂?𝑛| < |?̂?𝑜|  
If 𝑛 = 1, then 
𝑚
𝑎4?̂?
−𝑗𝑘𝑖∆𝑚 𝑗𝑘𝑖∆𝑚
1 = 𝑏4?̂?0 + 𝑑4?̂?0𝑒 + 𝑓4?̂?0𝑒 − 𝑣𝑥𝑔4∑(?̂? − ?̂? 𝑒
−𝑗𝑘𝑖∆𝑚
0 0 )𝛿
𝛼 (6-74) 
𝑚,𝑖 
𝑖=0
A subset for 𝑚 is considered, where 𝑚 = 0, 
𝑎4?̂?1 = 𝑏4?̂?0 + 𝑑
−𝑗𝑘𝑖∆𝑚 𝑗𝑘𝑖∆𝑚
4?̂?0𝑒 + 𝑓4?̂?0𝑒  (6-75) 
Simplifying and rearranging, 
?̂? −𝑗𝑘𝑖∆𝑚 𝑗𝑘𝑖∆𝑚1 𝑏4 + 𝑑4𝑒 + 𝑓4𝑒
=  (6-76) 
?̂?0 𝑎4
Applying a norm, the condition for the first induction requirement becomes, 
|𝑏4| + |𝑑4| + |𝑓4|
< 1 (6-77) 
|𝑎4|
The term is expanded using the simplification terms associated with Equation (6-57), 
𝐴𝐵(𝛼)
| 𝛿𝛼
𝐴𝐵(𝛼) 2𝐷 𝐴𝐵(𝛼) 𝐷 𝐷
− 𝑣 𝛿𝛼 𝐿 𝛼 𝐿 𝐿
(1 − 𝛼) 𝑛,𝑛−1 𝑥 (1 − 𝛼) 𝑚,𝑖
− | + |𝑣 𝛿 + | + | |
(∆𝑥)2 𝑥 (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2 (∆𝑥)2
< 1 (6-78) 
𝐴𝐵(𝛼)
| 𝛼
(1 − 𝛼) 
𝛿𝑛,𝑛−1|
The assumption is made where, 
𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 2𝐷𝐿
𝛿𝛼 𝛼𝑛,𝑛−1 < 𝑣𝑥 𝛿𝑚,𝑖 +  (6-79) (1 − 𝛼) (1 − 𝛼) (∆𝑥)2
Then, the condition is 
160 
 
𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 𝛼 2𝐷𝐿 𝐴𝐵(𝛼) 𝛼 𝐷𝐿 𝐷− 𝐿
(1 − 𝛼) 
𝛿𝑛,𝑛−1 + 𝑣𝑥 𝛿 + + 𝑣(1 − 𝛼) 𝑚,𝑖 (∆𝑥)2 𝑥 (1 − 𝛼) 
𝛿𝑚,𝑖 + +(∆𝑥)2 (∆𝑥)2
< 1 (6-80) 
𝐴𝐵(𝛼)
 𝛿
𝛼
(1 − 𝛼) 𝑛,𝑛−1
Simplifying, 
𝐴𝐵(𝛼) 𝛼 2𝐷𝐿 𝐴𝐵(𝛼)𝑣𝑥 𝛿𝑚,𝑖 + < 𝛿
𝛼
(1 − 𝛼) (∆𝑥)2 (1 − 𝛼) 𝑛,𝑛−1
 (6-81) 
Thus, under the assumption in Equation (6-79), the first inductive stability condition for this subset is 
upheld and conditionally stable. 
A subset for 𝑚 is now considered for all 𝑚 ≥ 1, 
𝑚
𝑎4?̂?1 = 𝑏 ?̂? + 𝑑 ?̂? 𝑒
−𝑗𝑘𝑖∆𝑚
4 0 4 0 + 𝑓4?̂? 𝑒
𝑗𝑘𝑖∆𝑚
0 − 𝑣𝑥𝑔4∑(?̂?0 − ?̂? 𝑒
−𝑗𝑘𝑖∆𝑚
0 )𝛿
𝛼 (6-82) 
𝑚,𝑖 
𝑖=0
Simplifying, 
𝑚
𝑎4?̂? = (𝑏 + 𝑑 𝑒
−𝑗𝑘𝑖∆𝑚 + 𝑓 𝑗𝑘𝑖∆𝑚1 4 4 4𝑒 − 𝑣𝑥𝑔4(1 − 𝑒
−𝑗𝑘𝑖∆𝑚)∑𝛿𝛼𝑚,𝑖) ?̂?0 (6-83) 
𝑖=0
Expanding the summation, and simplifying 
𝑚 𝑚
𝛼 𝛼 ∆𝑥 𝛼 ∆𝑥∑𝛿𝑚,𝑖 =∑(𝑚− 𝑖)𝐸𝛼,2 [− (𝑚− 𝑖)] − (𝑚− 𝑖 − 1)𝐸𝛼,2 [− (𝑚− 𝑖 − 1)] 1 − 𝛼 1 − 𝛼 
𝑖=0 𝑖=0
𝛼 ∆𝑥 𝛼 ∆𝑥
                    = {(𝑚)𝐸𝛼,2 [− (𝑚)] − (𝑚− 1)𝐸𝛼,2 [− (𝑚− 1)]}1 − 𝛼 1 − 𝛼 
𝛼 ∆𝑥 𝛼 ∆𝑥
+ {(𝑚− 1)𝐸𝛼,2 [− (𝑚− 1)] − (𝑚− 2)𝐸 [− (𝑚− 2)]}1 − 𝛼 𝛼,2 1 − 𝛼 
(6-84) 
𝛼 ∆𝑥 𝛼 ∆𝑥
+ {(𝑚− 2)𝐸𝛼,2 [− (𝑚− 2)] − (𝑚− 3)𝐸𝛼,2 [− (𝑚− 3)]}+⋯1 − 𝛼 1 − 𝛼 
𝛼 ∆𝑥 𝛼 ∆𝑥
+ {𝐸𝛼,2 [− ]− (−1)𝐸 [− (−1)]} 1 − 𝛼 𝛼,2 1 − 𝛼 
𝛼 ∆𝑥 𝛼 ∆𝑥 𝛼 ∆𝑥
         = {(𝑚)𝐸𝛼,2 [− (𝑚)]}+ {𝐸1 − 𝛼 𝛼,2
[− ] − (−1)𝐸 [− (−1)]} 
1 − 𝛼 𝛼,2 1 − 𝛼 
Substituting the expanded summation and applying Euler’s formula, 
𝑎 −𝑗𝑘𝑖∆𝑚 𝑗𝑘𝑖∆𝑚4?̂?1 = (𝑏4 + 𝑑4𝑒 + 𝑓4𝑒 − 𝑣𝑥𝑔4(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜃)𝛽𝑚,𝐸 )?̂?  𝛼,2 0 (6-85) 
Applying a norm and simplifying, 
|𝑎4||?̂?|1 = (|𝑏4| + |𝑑4| + |𝑓4| + 𝑣𝑥|𝑔4|(2 − 2 cos𝜙)|𝛽𝑚,𝐸 |)|?̂?0| (6-86) 𝛼,2
Rearranging, 
|?̂?1| |𝑏4| + |𝑑4| + |𝑓4| + 𝑣𝑥|𝑔4|(2 − 2 cos𝜙)|𝛽𝑚,𝐸 |𝛼,2
=  (6-87) 
|?̂?0| |𝑎4|
161 
 
Thus, the condition becomes, 
|𝑏4| + |𝑑4| + |𝑓4| + 𝑣𝑥|𝑔4|(2 − 2 cos𝜙)|𝛽𝑚,𝐸 | (6-88) 𝛼,2
< 1 
|𝑎4|
The term is expanded using the simplification terms associated with Equation (6-57), 
𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 2𝐷 𝐴𝐵(𝛼) 𝐷 (6-89) |  𝛿𝑛,𝑛−1 − 𝑣𝑥  𝛿
𝛼
𝑚,𝑖 −
𝐿
2| + |𝑣 𝛿
𝛼 + 𝐿 |
(1 − 𝛼) (1 − 𝛼) (∆𝑥) 𝑥 (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2
( )
𝐷𝐿 𝐴𝐵(𝛼)+ | 2| + 𝑣𝑥 |  | (2 − 2 cos𝜙)|𝛽(∆𝑥) (1 − 𝛼) 𝑚,𝐸
|
𝛼,2
< 1 
𝐴𝐵(𝛼)
| 𝛿𝛼 |
(1 − 𝛼) 𝑛,𝑛−1
The same assumption is made (Equation (6-79)), then the condition is, 
𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 𝛼 2𝐷𝐿 𝐴𝐵(𝛼) 𝛼 𝐷−  𝛿𝑛,𝑛−1 + 𝑣𝑥  𝛿𝑚,𝑖 + 2 + 𝑣𝑥  𝛿𝑚,𝑖 +
𝐿
(1 − 𝛼) (1 − 𝛼) (∆𝑥) (1 − 𝛼) (∆𝑥)2
( )
𝐷𝐿 𝐴𝐵(𝛼)+ 2 + 𝑣𝑥  (2 − 2 cos𝜙)𝛽( ) ( ) 𝑚,𝐸 (6-90) ∆𝑥 1 − 𝛼 𝛼,2
< 1 
𝐴𝐵(𝛼)
𝛿𝛼
(1 − 𝛼) 𝑛,𝑛−1
Simplifying, 
𝐴𝐵(𝛼) 𝛼 4𝐷𝐿 2𝐴𝐵(𝛼)2𝑣𝑥 (𝛿𝑚,𝑖  + (1 − cos𝜙)𝛽 ) + < 𝛿
𝛼
(1 − 𝛼) 𝑚,𝐸𝛼,2 (∆𝑥)2 (1 − 𝛼) 𝑛,𝑛−1
 (6-91) 
Therefore, the first inductive stability condition for the second subset of m is conditionally stable 
under this assumption. 
The induction numerical stability analysis has a second process that requires proving for a set ∀ 𝑛 ≥ 1, 
|?̂?𝑛| < |?̂?𝑜|  
Rearranging Equation (6-73) for ?̂?𝑛, 
𝑚
𝑎4?̂?𝑛 = (𝑏 + 𝑑 𝑒
−𝑗𝑘𝑖∆𝑚
4 4 + 𝑓
𝑗𝑘𝑖∆𝑚 −𝑗𝑘𝑖∆𝑚 𝛼
4𝑒 − 𝑣𝑥𝑔4(1 − 𝑒 )∑𝛿𝑚,𝑖) ?̂?𝑛−1 
𝑖=0
𝑛−2 (6-92) 
−𝑔4∑(?̂? − ?̂? )𝛿
𝛼
𝑘+1 𝑘 𝑛,𝑘 
𝑘=0
Following a similar simplification process of applying Euler and expanding the summation as previously 
performed, 
𝑎4?̂?𝑛 = (𝑏 + 𝑑 𝑒
−𝑗𝑘𝑖∆𝑚 + 𝑓 𝑒𝑗𝑘𝑖∆𝑚4 4 4 − 𝑣𝑥𝑔4(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)𝛽𝑚,𝐸 )?̂?𝛼,2 𝑛−1 
𝑛−2 (6-93) 
−𝑔4∑(?̂? − ?̂? )𝛿
𝛼
𝑘+1 𝑘 𝑛,𝑘 
𝑘=0
Taking the norm on both sides, 
162 
 
|𝑎4?̂?𝑛| = |(𝑏4 + 𝑑 𝑒
−𝑗𝑘𝑖∆𝑚
4 + 𝑓
𝑗𝑘𝑖∆𝑚
4𝑒 − 𝑣𝑥𝑔 (4 1 − cos𝜙+ 𝑖 𝑠𝑖𝑛 𝜙)𝛽𝑚,𝐸 )?̂?𝑛−1   𝛼,2
𝑛−2 (6-94) 
− 𝑔 ∑(?̂? − ?̂? )𝛿𝛼4 𝑘+1 𝑘 𝑛,𝑘| 
𝑘=0
Therefore,  
|𝑎4||?̂?𝑛| < |𝑏 + 𝑑 𝑒
−𝑗𝑘𝑖∆𝑚
4 4 + 𝑓4𝑒
𝑗𝑘𝑖∆𝑚 − 𝑣𝑥𝑔 (4 1 − cos𝜙+ 𝑖 𝑠𝑖𝑛 𝜙)𝛽𝑚,𝐸 ||?̂? |𝛼,2 𝑛−1
𝑛−2
(6-95) 
+ |𝑔 ( ) 𝛼4| | ∑ ?̂?𝑘+1 − ?̂?𝑘 𝛿𝑛,𝑘| 
𝑘=0
Remembering that it has been proved that for a set ∀ 𝑛 > 1,   
|?̂?𝑛−1| < |?̂?𝑜| 
Thus, 
|𝑎 ||?̂? | < |𝑏 + 𝑑 𝑒−𝑗𝑘𝑖∆𝑚4 𝑛 4 4 + 𝑓 𝑒
𝑗𝑘𝑖∆𝑚
4 − 𝑣𝑥𝑔 (4 1 − cos𝜙+ 𝑖 𝑠𝑖𝑛 𝜙)𝛽𝑚,𝐸 ||?̂? |𝛼,2 𝑛−1
𝑛−2
+ |𝑔4| | ∑(?̂?
𝛼
𝑘+1 − ?̂?𝑘)𝛿𝑛,𝑘| 
𝑘=0
(6-96) 
          < |𝑏 + 𝑑 𝑒−𝑗𝑘𝑖∆𝑚 + 𝑓 𝑒𝑗𝑘𝑖∆𝑚4 4 4 − 𝑣𝑥𝑔 (4 1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)𝛽𝑚,𝐸 ||?̂? |𝛼,2 𝑜
𝑛−2
+ |𝑔4| | ∑(?̂?𝑘+1 − ?̂?𝑘)𝛿
𝛼
𝑛,𝑘| 
𝑘=0
Therefore, it can be inferred that, 
|𝑎 −𝑗𝑘𝑖∆𝑚 𝑗𝑘𝑖∆𝑚4||?̂?𝑛| < |𝑏4 + 𝑑4𝑒 + 𝑓 𝑒 − 𝑣𝑥𝑔 (1 − cos𝜙+ 𝑖 𝑠𝑖𝑛 𝜙)𝛽𝑚,𝐸 ||?̂?𝑜|4 4 𝛼,2
𝑛−2
(6-97) 
+ |𝑔 ( 𝛼4| | ∑ ?̂?𝑘+1 − ?̂?𝑘)𝛿𝑛,𝑘| 
𝑘=0
The remaining summation is considered at the upper limit, 
𝑛−2 𝑛−2
?̂?𝑘
|∑(?̂?  𝛼  𝛼𝑘+1 − ?̂?𝑘)𝛿𝑛,𝑘| < ∑|?̂?𝑘+1| (|1 − |) 𝛿?̂? 𝑛,𝑘
  
𝑘+1
𝑘=0 𝑘=0
Subset 𝑘 will follow the same assumption made for a set ∀ 𝑛 ≥ 1, where 
𝑛−2 𝑛−2
?̂?𝑘
∑?̂?𝑘+1 (1 − )𝛿
 𝛼
𝑛,𝑘 < |?̂?0|∑ 𝛿
 𝛼
?̂? 𝑛,𝑘
  
𝑘+1
𝑘=0 𝑘=0
Substituting back into Equation (6-97), 
|𝑎4||?̂?𝑛| < |𝑏4 + 𝑑4𝑒
−𝑗𝑘𝑖∆𝑚 + 𝑓 𝑒𝑗𝑘𝑖∆𝑚4 − 𝑣𝑥𝑔4(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)𝛽𝑚,𝐸 ||?̂?𝛼,2 𝑜|
𝑛−2
(6-98) 
+ |𝑔 ||?̂? |∑ 𝛿  𝛼4 0 𝑛,𝑘 
𝑘=0
Expanding the summation, 
163 
 
𝑛−2
𝛼 ∆𝑡 𝛼 ∆𝑡
∑ 𝛿  𝛼𝑛,𝑘 = (𝑛 − 𝑘)𝐸𝛼,2 [− (𝑛 − 𝑘)] − (𝑛 − 𝑘 − 1)𝐸𝛼,2 [− (𝑛 − 𝑘 − 1)] 1 − 𝛼 1 − 𝛼 
𝑘=0
𝛼 ∆𝑡 𝛼 ∆𝑡
                    = {(𝑛)𝐸𝛼,2 [− (𝑛)] − (𝑛 − 1)𝐸𝛼,2 [− (𝑛 − 1)]}1 − 𝛼 1 − 𝛼 
𝛼 ∆𝑡 𝛼 ∆𝑡
+ {(𝑛 − 1)𝐸𝛼,2 [− (𝑛 − 1)] − (𝑛 − 2)𝐸1 − 𝛼 𝛼,2
[− (𝑛 − 2)]}
1 − 𝛼 
𝛼 ∆𝑡 𝛼 ∆𝑡  
+ {(𝑛 − 2)𝐸𝛼,2 [− (𝑛 − 2)] − (𝑛 − 3)𝐸1 − 𝛼 𝛼,2
[− (𝑛 − 3)]} + ⋯
1 − 𝛼 
𝛼 ∆𝑡 𝛼 ∆𝑡
+ {2𝐸𝛼,2 [−2 ] − 𝐸𝛼,2 [− ]} 1 − 𝛼 1 − 𝛼 
𝛼 ∆𝑡 𝛼 ∆𝑡 𝛼 ∆𝑡
 = {(𝑛)𝐸𝛼,2 [− (𝑛)]} + {2𝐸𝛼,2 [−2 ] − 𝐸𝛼,2 [− ]} 1 − 𝛼 1 − 𝛼 1 − 𝛼 
Substituting the summation and rearranging, 
|𝑎4||?̂?𝑛| < (|𝑏
−𝑗𝑘𝑖∆𝑚
4 + 𝑑4𝑒 + 𝑓4𝑒
𝑗𝑘𝑖∆𝑚 − 𝑣𝑥𝑔 (4 1 − cos𝜙+ 𝑖 𝑠𝑖𝑛 𝜙)𝛽𝑚,𝐸 | + |𝑔 | |𝛽 |) |?̂? | 𝛼,2 4 𝑛,𝐸 0 (6-99) 𝛼,2
where, 
𝛼 ∆𝑡 𝑛 2𝛼 ∆𝑡 𝛼 ∆𝑡
𝛽𝑛,𝐸 = 𝑛𝐸𝛼,2 𝛼,2 [− ] + 2𝐸 [− ] − 𝐸 [− ] 1 − 𝛼 𝛼,2 1 − 𝛼 𝛼,2 1 − 𝛼 
Simplifying and rearranging, 
|?̂? | |𝑏4| + |𝑑4| + |𝑓 | − 𝑣𝑥|𝑔 |(2 − 2cos𝜙)4 4 |𝛽𝑚,𝐸 | + |𝑔4| |𝛽 |𝑛 𝛼,2 𝑛,𝐸𝛼,2
<  (6-100) 
|?̂?0| |𝑎4|
Thus, the condition becomes, 
|𝑏4| + |𝑑4| + |𝑓4| − 𝑣𝑥|𝑔4|(2 − 2cos𝜙)|𝛽𝑚,𝐸 | + |𝑔𝛼,2 4| |𝛽𝑛,𝐸 |𝛼,2 < 1 (6-101) 
|𝑎4|
The condition is expanded using the simplification terms associated with Equation (6-57), 
𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 2𝐷 𝐴𝐵(𝛼) 𝐷 𝐷|  𝛿𝑛,𝑛−1 − 𝑣 𝛿
𝛼
𝑥  𝑚,𝑖 −
𝐿 | + |𝑣 𝛿𝛼 𝐿2 𝑥  𝑚,𝑖 + 2| + |
𝐿 |
(1 − 𝛼) (1 − 𝛼) (∆𝑥) (1 − 𝛼) (∆𝑥) (∆𝑥)2
𝐴𝐵(𝛼) 𝐴𝐵(𝛼)
−𝑣𝑥 |  | (2 − 2 cos𝜙)|𝛽𝑚,𝐸 | + |  | |𝛽 |(1 − 𝛼) 𝛼,2 (1 − 𝛼) 𝑛,𝐸
(6-102) 
𝛼,2
< 1 
𝐴𝐵(𝛼)
|  𝛿
𝛼
(1 − 𝛼) 𝑛,𝑛−1
|
The assumption in Equation (6-79) is made, and the conditions is 
𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 2𝐷 𝐴𝐵(𝛼) 𝐷 𝐷−  𝛿𝑛,𝑛−1 + 𝑣𝑥  𝛿
𝛼
𝑚,𝑖 +
𝐿 𝛼 𝐿 𝐿
(1 − 𝛼) (1 − 𝛼) (∆𝑥)2
+ 𝑣𝑥 𝛿 + +(1 − 𝛼) 𝑚,𝑖 (∆𝑥)2 (∆𝑥)2
𝐴𝐵(𝛼) 𝐴𝐵(𝛼)
−𝑣𝑥 (2 − 2 cos𝜙)𝛽 + 𝛽 (6-103) (1 − 𝛼) 𝑚,𝐸𝛼,2 (1 − 𝛼) 𝑛,𝐸𝛼,2
< 1 
𝐴𝐵(𝛼)
 𝛿
𝛼
(1 − 𝛼) 𝑛,𝑛−1
Simplifying, 
𝐴𝐵(𝛼) 4𝐷𝐿 2𝐴𝐵(𝛼)
( 2𝑣𝛿𝛼  − 𝑣 (2 − 2 cos𝜙)𝛽 + 𝛽 ) + < 𝛿𝛼  
(1 − 𝛼) 𝑚,𝑖 𝑥 𝑚,𝐸𝛼,2 𝑛,𝐸𝛼,2 (∆𝑥)2 (1 − 𝛼) 𝑛,𝑛−1 (6-104) 
164 
 
Thus, under the assumption in (6-79), the second inductive stability condition for is sustained and 
conditionally stable under this condition.  
This completes the stability analysis for the explicit upwind scheme for the advection-dispersion 
equation with ABC fractional derivative, where the scheme is found to be unstable under the 
assumption not presented, and conditionally stable under the assumption made of, 
𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 2𝐷
𝛿𝛼 𝛼
𝐿
(1 − 𝛼) 𝑛,𝑛−1
< 𝑣𝑥 𝛿 +  (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2
With the following conditions, 
𝐴𝐵(𝛼) 4𝐷𝐿 2𝐴𝐵(𝛼)
2𝑣𝑥 𝛿
𝛼
𝑚,𝑖 + < 𝛿
𝛼 , 
(1 − 𝛼) (∆𝑥)2 (1 − 𝛼) 𝑛,𝑛−1
𝐴𝐵(𝛼) 𝛼 4𝐷𝐿 2𝐴𝐵(𝛼)2𝑣𝑥 (𝛿𝑚,𝑖  + (1 − cos𝜙)𝛽
𝛼
(1 − 𝛼) 𝑚,𝐸
) + < 𝛿
𝛼,2 (∆𝑥)2 (1 − 𝛼) 𝑛,𝑛−1
, 
𝐴𝐵(𝛼) 𝛼 4𝐷𝐿 2𝐴𝐵(𝛼)( 2𝑣𝛿𝑚,𝑖  − 𝑣𝑥(2 − 2 cos𝜙)𝛽
𝛼
(1 − 𝛼) 𝑚,𝐸
+ 𝛽 ) + < 𝛿 , 
𝛼,2 𝑛,𝐸𝛼,2 (∆𝑥)2 (1 − 𝛼) 𝑛,𝑛−1
Simplified to, 
2𝐴𝐵(𝛼)
𝑚𝑎𝑥(𝜆, 𝜇, 𝜌) < 𝛿𝛼  
(1 − 𝛼) 𝑛,𝑛−1
where, 
𝐴𝐵(𝛼) 4𝐷𝐿
𝜆 = 2𝑣 𝛼𝑥 𝛿 +  (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2
𝐴𝐵(𝛼) 4𝐷𝐿
𝜇 = 2𝑣𝑥 (𝛿
𝛼
𝑚,𝑖  + (1 − cos𝜙)𝛽(1 − 𝛼) 𝑚,𝐸
) +  
𝛼,2 (∆𝑥)2
𝐴𝐵(𝛼) 4𝐷
𝛼 𝐿𝜌 =  ( 2𝑣𝑥𝛿𝑚,𝑖  − 𝑣𝑥(2 − 2 cos𝜙)𝛽 + 𝛽(1 − 𝛼) 𝑚,𝐸𝛼,2 𝑛,𝐸
) +  
𝛼,2 (∆𝑥)2
6.3.2 Implicit upwind  
Induction stability terms are substituted for the developed finite difference implicit upwind numerical 
scheme discussed in Section 6.2.2, 
𝑛−2
ℎ 𝑗𝑘𝑖𝑚4?̂?𝑛𝑒 = 𝑗4?̂? 𝑒
𝑗𝑘𝑖(𝑚−∆𝑚) + 𝑓 ?̂? 𝑒𝑗𝑘𝑖𝑥(𝑚+∆𝑚) − 𝑔 ∑(?̂? 𝑒𝑗𝑘𝑖𝑚 − ?̂? 𝑒𝑗𝑘𝑖𝑚 𝛼𝑛 4 𝑛 4 𝑘+1 𝑘 )𝛿𝑛,𝑘 
𝑘=0
𝑚 (6-105) 
−𝑣 𝑔 ∑(?̂? 𝑒𝑗𝑘𝑖𝑚 − ?̂? 𝑒𝑗𝑘𝑖(𝑚−∆𝑚)𝑥 4 𝑛 𝑛 )𝛿
𝛼
𝑚,𝑖 + 𝑎 ?̂? 𝑒
𝑗𝑘𝑖𝑚
4 𝑛−1  
𝑖=0
Multiple out and simplify, 
𝑛−2 𝑚
ℎ4?̂?𝑛 = 𝑗4?̂?
−𝑗𝑘𝑖∆𝑚
𝑛𝑒 + 𝑓 ?̂? 𝑒
𝑗𝑘𝑖∆𝑚
4 𝑛 − 𝑔4∑(?̂?𝑘+1 − ?̂?𝑘)𝛿
𝛼
𝑛,𝑘 − 𝑣𝑥𝑔4∑(?̂?
−𝑗𝑘𝑖∆𝑚 𝛼
𝑛 − ?̂?𝑛𝑒 )𝛿𝑚,𝑖 (6-106) 
𝑘=0 𝑖=0
+𝑎4?̂?𝑛−1 
165 
 
The first procedure for the induction numerical stability analysis requires proving for a set ∀ 𝑛 > 1, that 
|?̂?𝑛| < |?̂?  𝑜| 
If 𝑛 = 1, then 
𝑚
ℎ4?̂?1 = 𝑗4?̂?1𝑒
−𝑗𝑘𝑖∆𝑚 + 𝑓4?̂?1𝑒
𝑗𝑘𝑖∆𝑚 − 𝑣 𝑔 ∑(?̂? − ?̂? 𝑒−𝑗𝑘𝑖∆𝑚𝑥 4 1 1 )𝛿
𝛼
𝑚,𝑖 + 𝑎4?̂?0 (6-107) 
𝑖=0
A subset for 𝑚 is now considered, where 𝑚 = 0, 
ℎ4?̂?1 = 𝑗4?̂? 𝑒
−𝑗𝑘𝑖∆𝑚 + 𝑓 ?̂? 𝑗𝑘𝑖∆𝑚1 4 1𝑒 + 𝑎4?̂?0 (6-108) 
Simplifying and rearranging, 
?̂?1 𝑎4
=  
−𝑗𝑘 ∆𝑚 𝑗𝑘 ∆𝑚 (6-109) ?̂?0 ℎ4 − 𝑗 𝑒 𝑖 − 𝑓 𝑒 𝑖4 4
Applying a norm, the condition for the first induction requirement becomes, 
|𝑎4|
< 1 (6-110) 
|ℎ4| + |𝑗 | + |𝑓 |4 4
The term is expanded using the simplification terms associated with Equation (6-60), 
𝐴𝐵(𝛼)
|  𝛿
𝛼
(1 − 𝛼) 𝑛,𝑛−1
|
< 1 
𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 2𝐷 𝐴𝐵(𝛼) 𝐷 𝐷 (6-111) |  𝛿𝑛,𝑛−1 + 𝑣𝑥 𝛿
𝛼 𝐿 𝛼 𝐿 𝐿
(1 − 𝛼) (1 − 𝛼) 𝑚,𝑖
+ | + |𝑣
(∆𝑥)2 𝑥  
𝛿𝑚,𝑖 − | + | |(1 − 𝛼) (∆𝑥)2 (∆𝑥)2
The assumption is made where, 
𝐴𝐵(𝛼) 𝛼 𝐷𝐿𝑣𝑥 𝛿𝑚,𝑖 >  (6-112) (1 − 𝛼) (∆𝑥)2
Then, the condition is 
𝐴𝐵(𝛼)
𝛿𝛼
(1 − 𝛼) 𝑛,𝑛−1
< 1 
𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 2𝐷 𝐴𝐵(𝛼) 𝐷 𝐷 (6-113) 
 𝛿
𝛼 𝛼 𝐿 𝛼 𝐿 𝐿
(1 − 𝛼) 𝑛,𝑛−1
+ 𝑣𝑥 𝛿 + + 𝑣 𝛿 − +(1 − 𝛼) 𝑚,𝑖 (∆𝑥)2 𝑥 (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2 (∆𝑥)2
Simplifying, 
𝐴𝐵(𝛼) 𝛼 2𝐷𝐿2𝑣𝑥 𝛿𝑚,𝑖 + > 0 (1 − 𝛼) (∆𝑥)2 (6-114) 
Therefore, under the assumption in Equation (6-112), the first inductive stability condition for this 
subset is upheld and unconditionally stable. 
The complementary assumption is made, and the condition becomes, 
𝐴𝐵(𝛼)
𝛿𝛼
(1 − 𝛼) 𝑛,𝑛−1
< 1 
𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 2𝐷 𝐴𝐵(𝛼) 𝐷 𝐷 (6-115) 
 𝛿
𝛼 𝛼 𝐿 𝛼 𝐿 𝐿
(1 − 𝛼) 𝑛,𝑛−1
+ 𝑣𝑥 𝛿 + − 𝑣 𝛿 + +(1 − 𝛼) 𝑚,𝑖 (∆𝑥)2 𝑥 (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2 (∆𝑥)2
166 
 
Simplifying, 
4𝐷𝐿
> 0 
(∆𝑥)2 (6-116) 
Thus, the first inductive stability condition for this subset is upheld and unconditionally stable under 
this assumption as well. 
A subset for 𝑚 is now considered for all 𝑚 ≥ 1, 
𝑚
ℎ4?̂?1 = 𝑗4?̂? 𝑒
−𝑗𝑘𝑖∆𝑚 + 𝑓 ?̂? 𝑒𝑗𝑘𝑖∆𝑚 − 𝑣 𝑔 ∑(?̂? − ?̂? 𝑒−𝑗𝑘𝑖∆𝑚 𝛼1 4 1 𝑥 4 1 1 )𝛿𝑚,𝑖 + 𝑎4?̂?  (6-117) 0
𝑖=0
Simplifying, 
𝑚
(ℎ − 𝑗 𝑒−𝑗𝑘𝑖∆𝑚 − 𝑓 𝑒𝑗𝑘𝑖∆𝑚4 4 4 + 𝑣
−𝑗𝑘𝑖∆𝑚 𝛼
𝑥𝑔4(1 − 𝑒 )∑𝛿𝑚,𝑖) ?̂?1 = 𝑎4?̂?0 (6-118) 
𝑖=0
Expanding the summation as presented previously, 
ℎ − 𝑗 𝑒−𝑗𝑘𝑖∆𝑚 − 𝑓 𝑒𝑗𝑘𝑖∆𝑚4 4 4 + 𝑣𝑥𝑔4(1 − 𝑒
−𝑗𝑘𝑖∆𝑚) ∙
( 𝛼 ∆𝑥 𝛼 ∆𝑥 𝛼 ∆𝑥 ) ?̂?1 = 𝑎4?̂?0 (6-119) 
((𝑚)𝐸𝛼,2 [− (𝑚)]+ 𝐸𝛼,2 [− ] − (−1)𝐸𝛼,2 [− (−1)])1 − 𝛼 1 − 𝛼 1 − 𝛼 
where the function 𝛽𝑚,𝐸  is used to simplify as follows, 𝛼,2
(ℎ4 − 𝑗 𝑒
−𝑗𝑘𝑖∆𝑚 − 𝑓 𝑒𝑗𝑘𝑖∆𝑚4 4 + 𝑣 𝑔 (1 − 𝑒
−𝑗𝑘𝑖∆𝑚
𝑥 4 )𝛽𝑚,𝐸 )?̂? = 𝑎 ?̂?𝛼,2 1 4 0 (6-120) 
Let a function simplify to, 
𝜙 = 𝑘𝑖∆𝑥 
where, 
𝑒−𝑗𝜙 = 𝑒−𝑗𝑘𝑖∆𝑥 
Remembering Euler’s formula for complex numbers, and substituting back into Equation (6-120), 
(ℎ4 − 𝑗4𝑒
−𝑗𝑘𝑖∆𝑚 − 𝑓 𝑒𝑗𝑘𝑖∆𝑚4 + 𝑣𝑥𝑔4(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)𝛽𝑚,𝐸 )?̂?1 = 𝑎𝛼,2 4?̂?0 (6-121) 
Applying a norm on both sides and simplifying, 
(|ℎ4| + |𝑗4| + |𝑓4| + 𝑣𝑥|𝑔4|(2 − 2 cos𝜙)|𝛽𝑚,𝐸 |)|?̂?1| = |𝑎4||?̂?0| (6-122) 𝛼,2
Rearranging, 
|?̂?1| |𝑎4| (6-123) 
=  
|?̂?0| (|ℎ4| + |𝑗4| + |𝑓4| + 𝑣𝑥|𝑔4|(2 − 2 cos𝜙)|𝛽𝑚,𝐸 |)𝛼,2
Thus, the condition becomes, 
|𝑎4| (6-124) 
< 1 
(|ℎ4| + |𝑗4| + |𝑓4| + 𝑣𝑥|𝑔4|(2 − 2 cos𝜙)|𝛽𝑚,𝐸 |)𝛼,2
The term is expanded using the simplification terms associated with Equation (6-60), 
167 
 
𝐴𝐵(𝛼) 𝛼 (6-125) |
(1 − 𝛼) 
𝛿𝑛,𝑛−1|
< 1 
𝐴𝐵(𝛼)
| 𝛿𝛼
𝐴𝐵(𝛼) 𝛼 2𝐷𝐿 𝐴𝐵(𝛼) 𝐷 𝐷
 𝑛,𝑛−1 + 𝑣𝑥  𝛿𝑚,𝑖 + 2| + |𝑣
𝛼
𝑥  𝛿𝑚,𝑖 −
𝐿 | + | 𝐿 |
(1 − 𝛼) (1 − 𝛼) (∆𝑥) (1 − 𝛼) (∆𝑥)2 (∆𝑥)2
( )
𝐴𝐵(𝛼)
+𝑣𝑥 |  | (2 − 2 cos𝜙)|𝛽(1 − 𝛼) 𝑚,𝐸
|
𝛼,2
Considering the assumption made in Equation (6-112), the conditions becomes, 
𝐴𝐵(𝛼) (6-126) 
 𝛿
𝛼
(1 − 𝛼) 𝑛,𝑛−1
< 1 
𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 𝛼 2𝐷𝐿 𝐴𝐵(𝛼) 𝛼 𝐷𝐿 𝐷𝛿 + 𝑣 𝛿 + + 𝑣 𝛿 − + 𝐿
(1 − 𝛼) 𝑛,𝑛−1 𝑥 (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2 𝑥 (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2 (∆𝑥)2
( )
𝐴𝐵(𝛼)
+𝑣𝑥 (1 − 𝛼) 
(2 − 2 cos𝜙)𝛽𝑚,𝐸𝛼,2
Simplifying, 
𝛼 𝐴𝐵(𝛼) 2𝐷𝐿 (6-127) 2𝑣𝑥(𝛿𝑚,𝑖 + (1 − cos𝜙)𝛽𝑚,𝐸 ) +  > 0 𝛼,2 (1 − 𝛼) (∆𝑥)2
Therefore, under the assumption in Equation (6-112), the first inductive stability condition for the 
second subset of m is upheld and unconditionally stable. 
The opposite assumption is made, and the condition becomes, 
𝐴𝐵(𝛼)
𝛿𝛼
(1 − 𝛼) 𝑛,𝑛−1
< 1 
𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 𝛼 2𝐷𝐿 𝐴𝐵(𝛼) 𝐷 𝐷𝛿 + 𝑣 𝛿 + − 𝑣 𝛿𝛼 + 𝐿 + 𝐿
(1 − 𝛼) 𝑛,𝑛−1 𝑥 (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2 𝑥 (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2 (∆𝑥)2
(6-128) 
( )
𝐴𝐵(𝛼)
+𝑣𝑥  (2 − 2 cos𝜙)𝛽(1 − 𝛼) 𝑚,𝐸𝛼,2
Simplifying, 
4𝐷𝐿 𝐴𝐵(𝛼)
 + 𝑣𝑥(2 − 2 cos𝜙) 𝛽𝑚,𝐸 > 0 (∆𝑥)2 (1 − 𝛼) 𝛼,2 (6-129) 
Therefore, the first inductive stability condition for the second subset of m is upheld and 
unconditionally stable under this assumption as well. 
The second procedure for the induction numerical stability analysis requires proving for a set ∀ 𝑛 ≥ 1, 
|?̂?𝑛| < |?̂?𝑜|  
Rearranging Equation (6-106) for ?̂?𝑛, 
𝑚
(ℎ + 𝑗 𝑒−𝑗𝑘𝑖∆𝑚 − 𝑓 𝑒𝑗𝑘𝑖∆𝑚 + 𝑣 𝑔 (1 − 𝑒−𝑗𝑘𝑖∆𝑚4 4 4 𝑥 4 )∑𝛿
𝛼
𝑚,𝑖) ?̂?𝑛 
𝑖=0
𝑛−2 (6-130) 
= 𝑎4?̂?𝑛−1 − 𝑔4∑(?̂?𝑘+1 − ?̂?
𝛼
𝑘)𝛿𝑛,𝑘 
𝑘=0
Following a similar simplification process as previously performed, 
168 
 
(ℎ + 𝑗 𝑒−𝑗𝑘𝑖∆𝑚 − 𝑓 𝑗𝑘𝑖∆𝑚4 4 4𝑒 + 𝑣𝑥𝑔4(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)𝛽𝑚,𝐸 )?̂?  𝛼,2 𝑛
𝑛−2
(6-131) 
= 𝑎4?̂?𝑛−1 − 𝑔4∑(?̂?
𝛼
𝑘+1 − ?̂?𝑘)𝛿𝑛,𝑘 
𝑘=0
Applying a norm on both sides, 
|(ℎ + 𝑗 𝑒−𝑗𝑘𝑖∆𝑚 − 𝑓 𝑒𝑗𝑘𝑖∆𝑚4 4 4 + 𝑣𝑥𝑔 (4 1 − cos𝜙+ 𝑖 𝑠𝑖𝑛 𝜙)𝛽𝑚,𝐸 )?̂?𝑛| 𝛼,2
𝑛−2
(6-132) 
= |𝑎4?̂?𝑛−1 − 𝑔 ∑(?̂?𝑘+1 − ?̂?𝑘)
𝛼
4 𝛿𝑛,𝑘| 
𝑘=0
Therefore,  
|ℎ4 + 𝑗
−𝑗𝑘𝑖∆𝑚 𝑗𝑘𝑖∆𝑚 ( )
4𝑒 − 𝑓4𝑒 + 𝑣𝑥𝑔4 1 − cos𝜙+ 𝑖 𝑠𝑖𝑛 𝜙 𝛽𝑚,𝐸 ||?̂?𝛼,2 𝑛| 
𝑛−2
(6-133) 
< |𝑎4||?̂?𝑛−1| + |𝑔 | |∑(4 ?̂?
𝛼
𝑘+1 − ?̂?𝑘)𝛿𝑛,𝑘| 
𝑘=0
Remembering that it has been proved that for a set ∀ 𝑛 > 1,   
|?̂?𝑛−1| < |?̂?𝑜| 
Thus, 
|ℎ −𝑗𝑘𝑖∆𝑚 𝑗𝑘𝑖∆𝑚4 + 𝑗4𝑒 − 𝑓4𝑒 + 𝑣𝑥𝑔 (4 1 − cos𝜙+ 𝑖 𝑠𝑖𝑛 𝜙)𝛽𝑚,𝐸 ||?̂? | 𝛼,2 𝑛
𝑛−2
< |𝑎4||?̂?𝑛−1| + |𝑔4| |∑(?̂?
𝛼
𝑘+1 − ?̂?𝑘)𝛿𝑛,𝑘| 
𝑘=0 (6-134) 
𝑛−2
< |𝑎4||?̂?𝑜| + |𝑔 | |∑(?̂? − ?̂? )𝛿
𝛼
4 𝑘+1 𝑘 𝑛,𝑘| 
𝑘=0
Therefore, it can be inferred that, 
|ℎ −𝑗𝑘𝑖∆𝑚 𝑗𝑘𝑖∆𝑚4 + 𝑗4𝑒 − 𝑓4𝑒 + 𝑣𝑥𝑔 (4 1 − cos𝜙+ 𝑖 𝑠𝑖𝑛 𝜙)𝛽𝑚,𝐸 ||?̂? | 𝛼,2 𝑛
𝑛−2
(6-135) 
< |𝑎4||?̂?𝑜| + |𝑔 | |∑(?̂? − ?̂? )𝛿
𝛼
4 𝑘+1 𝑘 𝑛,𝑘| 
𝑘=0
The remaining summation is considered at the upper limit as previously, 
|ℎ + 𝑗 𝑒−𝑗𝑘𝑖∆𝑚 − 𝑓 𝑒𝑗𝑘𝑖∆𝑚4 4 4 + 𝑣𝑥𝑔4(1 − cos𝜙 + 𝑖 sin𝜙)𝛽𝑚,𝐸 ||?̂?𝛼,2 𝑛| 
𝑛−2
(6-136) 
< |𝑎4||?̂?𝑜| + |𝑔 ||?̂? |∑ 𝛿
 𝛼
4 0 𝑛,𝑘 
𝑘=0
Expanding the summation and simplifying as previously outlined, 
|ℎ4 + 𝑗 𝑒
−𝑗𝑘𝑖∆𝑚
4 − 𝑓 𝑒
𝑗𝑘𝑖∆𝑚
4 + 𝑣𝑥𝑔4(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)𝛽𝑚,𝐸 ||?̂?𝛼,2 𝑛| 
(6-137) 
< |𝑎4||?̂?𝑜| + |𝑔4||?̂?0|𝛽𝑛,𝐸  𝛼,2
Rearranging, 
|?̂?𝑛| |𝑎4| + |𝑔4|𝛽𝑛,𝐸𝛼,2
<  
| | (6-138) ?̂?𝑜 |ℎ4| + |𝑗4| + |𝑓4| + 𝑣𝑥|𝑔4|(|1 − cos𝜙| + 𝑖 |𝑠𝑖𝑛 𝜙|)|𝛽𝑚,𝐸 |𝛼,2
Thus, the condition becomes, 
169 
 
|𝑎4| + |𝑔4|𝛽𝑛,𝐸𝛼,2
< 1 (6-139) 
|ℎ4| + |𝑗4| + |𝑓4| + 𝑣𝑥|𝑔4|(|1 − cos𝜙| + 𝑖 |𝑠𝑖𝑛 𝜙|)|𝛽𝑚,𝐸 |𝛼,2
The condition is expanded using the simplification terms associated with Equation (6-60), 
𝐴𝐵(𝛼) 𝐴𝐵(𝛼)
| 𝛼 𝛿𝑛,𝑛−1| + |(1 − 𝛼) (1 − 𝛼) 
| 𝛽𝑛,𝐸𝛼,2
< 1 
𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 2𝐷| 𝛿 +  𝑣 𝛿𝛼 + 𝐿
𝐴𝐵(𝛼) 𝐷 𝐷
 𝑛,𝑛−1  𝑚,𝑖 2| + |𝑣 𝛿
𝛼 − 𝐿 𝐿
(1 − 𝛼) (1 − 𝛼) (∆𝑥) (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2
| + |
( )2
| (6-140) 
∆𝑥
𝐴𝐵(𝛼)
+𝑣 |  | (2 − 2 cos𝜙)|𝛽𝑚,𝐸 |(1 − 𝛼) 𝛼,2
The assumption in Equation (6-112) is applied, and the condition develops, 
𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼)𝛿 + 𝛽
(1 − 𝛼) 𝑛,𝑛−1 (1 − 𝛼) 𝑛,𝐸𝛼,2
< 1 
𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 𝛼 2𝐷𝐿 𝐴𝐵(𝛼) 𝛼 𝐷𝐿 𝐷𝐿
(1 − 𝛼) 
𝛿𝑛,𝑛−1 + 𝑣𝑥 𝛿 + + 𝑣 𝛿 − +(1 − 𝛼) 𝑚,𝑖 (∆𝑥)2 𝑥 (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2 (∆𝑥)2
(6-141) 
𝐴𝐵(𝛼)
+𝑣𝑥  (2 − 2 cos𝜙)𝛽(1 − 𝛼) 𝑚,𝐸𝛼,2
Simplifying, 
2𝐷𝐿
2𝑣𝑥(𝛿
𝛼
𝑚,𝑖 + (1 − cos𝜙)𝛽𝑚,𝐸 ) + > 𝛽𝛼,2 (∆𝑥)2 𝑛,𝐸
 
𝛼,2 (6-142) 
Thus, under the assumption (Equation (6-112)), the second inductive stability condition for is upheld 
and conditionally stable, under this condition.  
The opposite assumption is made, and the condition becomes, 
𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼)𝛿
(1 − 𝛼) 𝑛,𝑛−1
+ 𝛽
(1 − 𝛼) 𝑛,𝐸𝛼,2
< 1 
𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 𝛼 2𝐷𝐿 𝐴𝐵(𝛼) 𝐷 𝐷
 𝛿𝑛,𝑛−1 + 𝑣𝑥  𝛿𝑚,𝑖 + 2 − 𝑣𝑥  𝛿
𝛼 𝐿
𝑚,𝑖 + 2 +
𝐿
(1 − 𝛼) (1 − 𝛼) ( (6-143) ∆𝑥) (1 − 𝛼) (∆𝑥) (∆𝑥)2
𝐴𝐵(𝛼)
+𝑣𝑥 (2 − 2 cos𝜙)𝛽(1 − 𝛼) 𝑚,𝐸𝛼,2
Simplifying, 
4𝐷𝐿
 2𝑣𝑥(1 − cos𝜙)𝛽𝑚,𝐸 + > 𝛽𝛼,2 (∆𝑥)2 𝑛,𝐸
 
𝛼,2 (6-144) 
Thus, under this assumption, the second inductive stability condition for is also upheld and 
conditionally stable, under this condition.  
The stability analysis for the implicit upwind scheme for the advection-dispersion equation with ABC 
fractional derivative found the scheme to be unconditionally stable for the first section of the 
induction method analysis, and conditionally stable under the following conditions, 
2𝐷𝐿
2𝑣 𝛼𝑥(𝛿𝑚,𝑖 + (1 − cos𝜙)𝛽𝑚,𝐸 ) + > 𝛽 , 𝛼,2 (∆𝑥)2 𝑛,𝐸𝛼,2
and 
4𝐷𝐿
 2𝑣𝑥(1 − cos𝜙)𝛽𝑚,𝐸 + > 𝛽  𝛼,2 (∆𝑥)2 𝑛,𝐸𝛼,2
170 
 
This can be simplified to an overall condition,  
𝑚𝑖𝑛(𝛾, 𝜂) > 𝛽𝑛,𝐸  𝛼,2
where, 
2𝐷𝐿
𝛾 = 2𝑣𝑥(𝛿
𝛼
𝑚,𝑖 + (1 − cos𝜙)𝛽𝑚,𝐸 ) +  𝛼,2 (∆𝑥)2
4𝐷𝐿
𝜂 = 2𝑣𝑥(1 − cos𝜙)𝛽𝑚,𝐸 +  𝛼,2 (∆𝑥)2
The error of the approximation is thus reduced throughout the solution and decreases with each time 
step, under these conditions, where for all values of n, |?̂?𝑛+1| < |?̂?𝑜|. 
6.3.3 Upwind advection Crank-Nicolson scheme 
Substituting the induction method terms for the developed upwind advection Crank-Nicolson 
numerical scheme discussed in Section 6.2.3, 
𝑙 ?̂? 𝑒𝑗𝑘𝑖𝑚 = 𝑚 ?̂? 𝑒𝑗𝑘𝑖𝑚 + 𝑜 ?̂? 𝑒𝑗𝑘𝑖(𝑚−∆𝑚) + 𝑓 ?̂? 𝑒𝑗𝑘𝑖𝑥(𝑚+∆𝑚) + 𝑝 ?̂? 𝑒𝑗𝑘𝑖(𝑚−∆𝑚)4 𝑛 4 𝑛−1 4 𝑛 4 𝑛 4 𝑛−1  
𝑛−2 𝑚
0.5(?̂? 𝑒𝑗𝑘𝑖𝑚 − ?̂? 𝑒𝑗𝑘𝑖(𝑚−∆𝑚)) (6-145) 
−𝑔 ∑(?̂? 𝑒𝑗𝑘 𝑚
𝑛−1 𝑛−1
𝑖
4 𝑘+1 − ?̂?𝑘𝑒
𝑗𝑘𝑖𝑚)𝛿𝛼𝑛,𝑘 − 𝑣𝑥𝑔4  ∑[ ] 𝛿
𝛼
𝑚,𝑖 
+0.5(?̂? 𝑒𝑗𝑘𝑖𝑚 − ?̂? 𝑒𝑗𝑘𝑖(𝑚−∆𝑚))
𝑘=0 𝑖=0 𝑛 𝑛
Multiple out and simplify, 
𝑛−2
𝑙 ?̂? = 𝑚 ?̂? + 𝑜 ?̂? 𝑒−𝑗𝑘𝑖∆𝑚 + 𝑓 ?̂? 𝑒𝑗𝑘𝑖∆𝑚 −𝑗𝑘𝑖∆𝑚4 𝑛 4 𝑛−1 4 𝑛 4 𝑛 + 𝑝4?̂?𝑛−1𝑒 − 𝑔4∑(?̂?𝑘+1 − ?̂?𝑘)𝛿
𝛼
𝑛,𝑘 
𝑘=0
𝑚 (6-146) 
−𝑣𝑥𝑔4  ∑[0.5(?̂? − ?̂? 𝑒
−𝑗𝑘𝑖∆𝑚
𝑛−1 𝑛−1 ) + 0.5(?̂? − ?̂? 𝑒
−𝑗𝑘𝑖∆𝑚
𝑛 𝑛 )] 𝛿
𝛼
𝑚,𝑖 
𝑖=0
The first procedure for the induction numerical stability analysis requires proving for a set ∀ 𝑛 > 1, that 
|?̂?𝑛| < |?̂?𝑜|  
If 𝑛 = 1, then 
𝑙4?̂?1 = 𝑚4?̂?0 + 𝑜
−𝑗𝑘𝑖∆𝑚
4?̂?1𝑒 + 𝑓 ?̂? 𝑒
𝑗𝑘𝑖∆𝑚
4 1 + 𝑝
−𝑗𝑘𝑖∆𝑚
4?̂?0𝑒  
𝑚
(6-147) 
−𝑣 𝑔  ∑[0.5(?̂? − ?̂? 𝑒−𝑗𝑘𝑖∆𝑚)  + 0.5(?̂? − ?̂? 𝑒−𝑗𝑘𝑖∆𝑚 𝛼𝑥 4 0 0 1 1 )] 𝛿𝑚,𝑖 
𝑖=0
A subset for 𝑚 is now considered, where 𝑚 = 0, 
𝑙 ?̂? = 𝑚 ?̂? + 𝑜 ?̂? 𝑒−𝑗𝑘𝑖∆𝑚 + 𝑓 ?̂? 𝑒𝑗𝑘𝑖∆𝑚4 1 4 0 4 1 4 1 + 𝑝4?̂? 𝑒
−𝑗𝑘𝑖∆𝑚
0  (6-148) 
Simplifying and rearranging, 
?̂?1 𝑚4 + 𝑝4𝑒
−𝑗𝑘𝑖∆𝑚
=  
?̂? 𝑙 − 𝑜 𝑒−𝑗𝑘𝑖∆𝑚 − 𝑓 𝑒𝑗𝑘𝑖∆𝑚 (6-149) 0 4 4 4
Applying a norm, the condition for the first induction requirement becomes, 
171 
 
|𝑚4| + |𝑝4|
< 1 (6-150) 
|𝑙4| + |𝑜4| + |𝑓4|
The term is expanded using the simplification terms associated with Equation (6-63), 
𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 𝐴𝐵(𝛼)
| 𝛿𝛼 + 0.5𝑣 𝛿𝛼 | + |0.5𝑣 𝛿𝛼 |
(1 − 𝛼) 𝑛,𝑛−1 𝑥 (1 − 𝛼) 𝑚,𝑖 𝑥 (1 − 𝛼) 𝑚,𝑖
< 1 
𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 2𝐷 𝐴𝐵(𝛼) 𝐷 𝐷 (6-151) 
| 𝛿𝛼 𝛼 𝐿 𝑛,𝑛−1 + 0.5𝑣𝑥  𝛿𝑚,𝑖 + 2| + |0.5𝑣 𝛿
𝛼 − 𝐿 | + | 𝐿 |
(1 − 𝛼) (1 − 𝛼) (∆𝑥) 𝑥 (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2 (∆𝑥)2
The assumption is made where, 
𝐴𝐵(𝛼) 𝛼 𝐷𝐿0.5𝑣𝑥 𝛿 >  (6-152) (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2
Then, the condition is 
𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 𝐴𝐵(𝛼)
 𝛿𝑛,𝑛−1 + 0.5𝑣 𝛿
𝛼 + 0.5𝑣 𝛿𝛼
(1 − 𝛼) 𝑥 (1 − 𝛼) 𝑚,𝑖 𝑥 (1 − 𝛼) 𝑚,𝑖
< 1 
𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 2𝐷 𝐴𝐵(𝛼) 𝐷 𝐷 (6-153) 
𝛿𝛼 𝛼 𝑛,𝑛−1 + 0.5𝑣𝑥 𝛿 +
𝐿 + 0.5𝑣 𝛿𝛼 − 𝐿 + 𝐿
(1 − 𝛼) (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2 𝑥 (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2 (∆𝑥)2
Simplifying, 
2𝐷𝐿
> 0 
(∆𝑥)2 (6-154) 
Therefore, under the assumption made in Equation (6-152), the first inductive stability condition for 
this subset is upheld and unconditionally stable. 
The complementary assumption is made, and the condition becomes, 
𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 𝐴𝐵(𝛼)
𝛿𝛼 + 0.5𝑣 𝛿𝛼 + 0.5𝑣 𝛿𝛼
(1 − 𝛼) 𝑛,𝑛−1 𝑥 (1 − 𝛼) 𝑚,𝑖 𝑥 (1 − 𝛼) 𝑚,𝑖
< 1 (6-155) 
𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 𝛼 2𝐷𝐿 𝐴𝐵(𝛼) 𝛼 𝐷𝐿 𝐷𝛿 + 0.5𝑣 𝐿
(1 − 𝛼) 𝑛,𝑛−1 𝑥 (1 − 𝛼) 
𝛿𝑚,𝑖 + 2 − 0.5𝑣𝑥  𝛿𝑚,𝑖 + 2 +(∆𝑥) (1 − 𝛼) (∆𝑥) (∆𝑥)2
Simplifying, 
𝐴𝐵(𝛼) 𝛼 4𝐷𝐿𝑣𝑥 𝛿 <  (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2 (6-156) 
Thus, the first inductive stability condition for this subset is upheld and conditionally stable under this 
assumption. 
A subset for 𝑚 is now considered for all 𝑚 ≥ 1, 
𝑙4?̂?1 = 𝑚4?̂?
−𝑗𝑘𝑖∆𝑚 𝑗𝑘𝑖∆𝑚
0 + 𝑜4?̂?1𝑒 + 𝑓4?̂?1𝑒 + 𝑝4?̂? 𝑒
−𝑗𝑘𝑖∆𝑚
0
𝑚 (6-157) 
− 𝑣𝑥𝑔4  ∑[0.5(?̂? − ?̂? 𝑒
−𝑗𝑘𝑖∆𝑚
0 0 )  + 0.5(?̂?
−𝑗𝑘𝑖∆𝑚 𝛼
1 − ?̂?1𝑒 )] 𝛿𝑚,𝑖 
𝑖=0
Simplifying, 
172 
 
𝑚
(𝑙 − 𝑜 𝑒−𝑗𝑘𝑖∆𝑚 − 𝑓 𝑒𝑗𝑘𝑖∆𝑚4 4 4 − 0.5(1 − 𝑒
−𝑗𝑘𝑖∆𝑚)𝑣 𝛼𝑥𝑔4∑𝛿𝑚,𝑖) ?̂?1
𝑖=0
𝑚 (6-158) 
= (𝑚4 + 𝑝 𝑒
−𝑗𝑘𝑖∆𝑚
4 − 0.5(1 − 𝑒
−𝑗𝑘𝑖∆𝑚)𝑣 𝛼𝑥𝑔4∑𝛿𝑚,𝑖) ?̂?0 
𝑖=0
Expanding the summation, and simplifying as previously, 
(𝑙4 − 𝑜4𝑒
−𝑗𝑘𝑖∆𝑚 − 𝑓 𝑗𝑘𝑖∆𝑚4𝑒 − 0.5𝑣𝑥𝑔4(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙) 𝛽𝑚,𝐸 )?̂?𝛼,2 1
= (𝑚 + 𝑝 𝑒−𝑗𝑘𝑖∆𝑚
(6-159) 
4 4 − 0.5𝑣𝑥𝑔4(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)𝛽𝑚,𝐸 )?̂?0 𝛼,2
Taking a norm on both sides and simplifying, 
(|𝑙4| + |𝑜4| + |𝑓4| + 0.5𝑣𝑥|𝑔4|(2 − 2 cos𝜙)|𝛽𝑚,𝐸 |)|?̂?1| (6-160) 𝛼,2
= (|𝑚4| + |𝑝4| + 0.5𝑣𝑥|𝑔4|(2 − 2 cos𝜙)|𝛽𝑚,𝐸 |)|?̂? | 𝛼,2 0
Rearranging, 
|?̂?1| |𝑚4| + |𝑝4| + 𝑣𝑥|𝑔4|(1 − cos𝜙)|𝛽𝑚,𝐸 | (6-161) 𝛼,2
=  
|?̂?0| |𝑙4| + |𝑜4| + |𝑓4| − 𝑣𝑥|𝑔4|(1 − cos𝜙)|𝛽𝑚,𝐸 |𝛼,2
Thus, the condition becomes, 
|𝑚4| + |𝑝4| + 𝑣𝑥|𝑔4|(1 − cos𝜙)|𝛽𝑚,𝐸 | (6-162) 𝛼,2
< 1 
|𝑙4| + |𝑜4| + |𝑓4| − 𝑣𝑥|𝑔4|(1 − cos𝜙)|𝛽𝑚,𝐸 |𝛼,2
The term is expanded using the simplification terms associated with Equation (6-63), 
𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 𝐴𝐵(𝛼)|  𝛿𝑛,𝑛−1 + 0.5𝑣𝑥 𝛿
𝛼 | + |0.5𝑣 𝛿𝛼 | (6-163) 
(1 − 𝛼) (1 − 𝛼) 𝑚,𝑖 𝑥 (1 − 𝛼) 𝑚,𝑖
( )
𝐴𝐵(𝛼)
+𝑣𝑥 |  | (1 − cos𝜙)|𝛽 |(1 − 𝛼) 𝑚,𝐸𝛼,2
< 1 
𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 2𝐷 𝐴𝐵(𝛼) 𝐷 𝐷
|  𝛿
𝛼 𝛼 𝐿
𝑛,𝑛−1 + 0.5𝑣𝑥  𝛿𝑚,𝑖 + | + |0.5𝑣 𝛿
𝛼 − 𝐿 | + | 𝐿 |
(1 − 𝛼) (1 − 𝛼) (∆𝑥)2 𝑥 (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2 (∆𝑥)2
( )
𝐴𝐵(𝛼)
+𝑣𝑥 |  | (1 − cos𝜙)|𝛽 |(1 − 𝛼) 𝑚,𝐸𝛼,2
The same assumption is made as in Equation (6-152), then the conditions is, 
𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 𝐴𝐵(𝛼)
 𝛿𝑛,𝑛−1 + 0.5𝑣𝑥 𝛿
𝛼 + 0.5𝑣 𝛿𝛼  + 𝑣 (1 − cos𝜙)𝛽 (6-164) 
(1 − 𝛼) (1 − 𝛼) 𝑚,𝑖 𝑥 (1 − 𝛼) 𝑚,𝑖 𝑥 (1 − 𝛼) 𝑚,𝐸𝛼,2
< 1 
𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 𝛼 2𝐷𝐿 𝐴𝐵(𝛼)𝛿 𝛼
𝐷𝐿 𝐷𝐿
(1 − 𝛼) 𝑛,𝑛−1
+ 0.5𝑣𝑥 𝛿(1 − 𝛼) 𝑚,𝑖
+ 2 + 0.5𝑣𝑥  𝛿𝑚,𝑖 − +(∆𝑥) (1 − 𝛼) (∆𝑥)2 (∆𝑥)2
( )
𝐴𝐵(𝛼)
+𝑣𝑥  (1 − cos𝜙)𝛽(1 − 𝛼) 𝑚,𝐸𝛼,2
Simplifying, 
2𝐷𝐿 (6-165) 
> 0  
(∆𝑥)2
Therefore, under the assumption made in Equation (6-152), the first inductive stability condition for 
the second subset of m is upheld and unconditionally stable. 
The opposite assumption is made, and the condition becomes, 
173 
 
𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 𝐴𝐵(𝛼)𝛿 + 0.5𝑣 𝛿𝛼 + 0.5𝑣 𝛿𝛼
𝐴𝐵(𝛼)
(1 − 𝛼) 𝑛,𝑛−1 𝑥
 + 𝑣
(1 − 𝛼) 𝑚,𝑖 𝑥 (1 − 𝛼) 𝑚,𝑖 𝑥 (1 − 𝛼) 
(1 − cos𝜙)𝛽𝑚,𝐸𝛼,2
< 1 
𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 𝛼 2𝐷𝐿 𝐴𝐵(𝛼) 𝐷𝛿 + 0.5𝑣 𝛿 + − 0.5𝑣 𝛿𝛼 + 𝐿
𝐷
+ 𝐿
(1 − 𝛼) 𝑛,𝑛−1 𝑥 (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2 𝑥 (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2 (∆𝑥)2 (6-166) 
( )
𝐴𝐵(𝛼)
+𝑣𝑥  (1 − cos𝜙)𝛽(1 − 𝛼) 𝑚,𝐸𝛼,2
Simplifying, 
𝐴𝐵(𝛼) 𝛼 4𝐷𝐿𝑣𝑥 𝛿𝑚,𝑖 <   (6-167) (1 − 𝛼) (∆𝑥)2
Therefore, the first inductive stability condition for the second subset of m is upheld and conditionally 
stable under this assumption. 
The second procedure for the induction numerical stability analysis requires proving for a set ∀ 𝑛 ≥ 1, 
|?̂?𝑛| < |?̂?𝑜|  
Rearranging Equation (6-146) for ?̂?𝑛, 
𝑚
(𝑙 − 𝑜 𝑒−𝑗𝑘𝑖∆𝑚4 4 − 𝑓
𝑗𝑘𝑖∆𝑚 −𝑗𝑘𝑖∆𝑥
4𝑒 + 𝑣𝑥𝑔4 (0.5(1 − 𝑒 )∑𝛿
 𝛼
𝑚,𝑖)) ?̂?𝑛 
𝑖=0
𝑚 𝑛−2 (6-168) 
= (𝑚 + 𝑝 𝑒−𝑗𝑘𝑖∆𝑚 − 𝑣 𝑔 (0.5(1 − 𝑒−𝑗𝑘𝑖∆𝑥)∑𝛿  𝛼4 4 𝑥 4 𝑚,𝑖)) ?̂?
𝛼
𝑛−1 − 𝑔4∑(?̂?𝑘+1 − ?̂?𝑘)𝛿𝑛,𝑘 
𝑖=0 𝑘=0
Following a similar simplification process as previously performed, 
(𝑙 − 𝑜 𝑒−𝑗𝑘𝑖∆𝑚 − 𝑓 𝑒𝑗𝑘𝑖∆𝑚4 4 4 + 𝑣𝑥𝑔4(0.5(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)𝛽𝑚,𝐸 )) ?̂?𝛼,2 𝑛 
𝑛−2
(6-169) 
= (𝑚4 + 𝑝4𝑒
−𝑗𝑘𝑖∆𝑚 − 𝑣𝑥𝑔4(0.5(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)𝛽𝑚,𝐸 )) ?̂? − 𝑔
𝛼
𝛼,2 𝑛−1 4∑(?̂?𝑘+1 − ?̂?𝑘)𝛿𝑛,𝑘 
𝑘=0
Applying a norm, 
|(𝑙 − 𝑜 𝑒−𝑗𝑘𝑖∆𝑚 − 𝑓 𝑒𝑗𝑘𝑖∆𝑚4 4 4 + 𝑣𝑥𝑔4(0.5(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)𝛽𝑚,𝐸 )) ?̂? | 𝛼,2 𝑛
𝑛−2
(6-170) 
= |(𝑚 + 𝑝 𝑒−𝑗𝑘𝑖∆𝑚4 4 − 𝑣𝑥𝑔4(0.5(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)𝛽
𝛼
𝑚,𝐸 )) ?̂? − 𝑔 ∑(?̂?𝛼,2 𝑛−1 4 𝑘+1 − ?̂?𝑘)𝛿𝑛,𝑘| 
𝑘=0
Therefore,  
|(𝑙 − 𝑜 𝑒−𝑗𝑘𝑖∆𝑚 − 𝑓 𝑒𝑗𝑘𝑖∆𝑚4 4 4 + 𝑣𝑥𝑔4(0.5(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)𝛽𝑚,𝐸 ))| |?̂? | 𝛼,2 𝑛
< |(𝑚4 + 𝑝
−𝑗𝑘𝑖∆𝑚
4𝑒 − 𝑣𝑥𝑔4(0.5(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)𝛽𝑚,𝐸 ))| |?̂? | 𝛼,2 𝑛−1 (6-171) 
𝑛−2
+|𝑔4| |∑(?̂?
𝛼
𝑘+1 − ?̂?𝑘)𝛿𝑛,𝑘| 
𝑘=0
Remembering that it has been proved that for a set ∀ 𝑛 > 1,   
|?̂?𝑛−1| < |?̂?𝑜| 
Thus, 
174 
 
|(𝑙 −𝑗𝑘𝑖∆𝑚4 − 𝑜4𝑒 − 𝑓 𝑒
𝑗𝑘𝑖∆𝑚
4 + 𝑣𝑥𝑔4(0.5(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)𝛽𝑚,𝐸 ))| |?̂?𝑛| 𝛼,2
< |(𝑚4 + 𝑝4𝑒
−𝑗𝑘𝑖∆𝑚 − 𝑣𝑥𝑔4(0.5(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)𝛽𝑚,𝐸 ))| |?̂?𝛼,2 𝑛−1|
𝑛−2
+ |𝑔4| |∑(?̂?
𝛼
𝑘+1 − ?̂?𝑘)𝛿𝑛,𝑘| 
(6-172) 
𝑘=0
 < |(𝑚4 + 𝑝 𝑒
−𝑗𝑘𝑖∆𝑚
4 − 𝑣𝑥𝑔4(0.5(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)𝛽𝑚,𝐸 ))| |?̂? |𝛼,2 𝑜
𝑛−2
+ |𝑔4| |∑(?̂?𝑘+1 − ?̂?𝑘)𝛿
𝛼
𝑛,𝑘| 
𝑘=0
Therefore, it can be inferred that, 
|(𝑙4 − 𝑜4𝑒
−𝑗𝑘𝑖∆𝑚 − 𝑓 𝑒𝑗𝑘𝑖∆𝑚4 + 𝑣𝑥𝑔4(0.5(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)𝛽𝑚,𝐸 ))| |?̂? | 𝛼,2 𝑛
< |(𝑚 + 𝑝 𝑒−𝑗𝑘𝑖∆𝑚4 − 𝑣𝑥𝑔4(0.5(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)𝛽𝑚,𝐸 ))| |?̂?𝑜|𝛼,2 (6-173) 
𝑛−2
+ |𝑔4| |∑(?̂?
𝛼
𝑘+1 − ?̂?𝑘)𝛿𝑛,𝑘| 
𝑘=0
The remaining summation is considered at the upper limit as previously, 
|(𝑙 − 𝑜 𝑒−𝑗𝑘𝑖∆𝑚 − 𝑓 𝑒𝑗𝑘𝑖∆𝑚4 4 4 + 𝑣𝑥𝑔4(0.5(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)𝛽𝑚,𝐸 ))| |?̂?𝛼,2 𝑛| 
𝑛−2
(6-174) 
< |(𝑚 + 𝑝 𝑒−𝑗𝑘𝑖∆𝑚 − 𝑣 𝑔 (0.5(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)𝛽 ))| |?̂? | + |𝑔 ||?̂? |∑ 𝛿  𝛼4 4 𝑥 4 𝑚,𝐸 𝑜 4 0 𝑛,𝑘 𝛼,2
𝑘=0
Expanding the summation, 
|(𝑙 − 𝑜 𝑒−𝑗𝑘𝑖∆𝑚4 4 − 𝑓
𝑗𝑘𝑖∆𝑚
4𝑒 + 𝑣𝑥𝑔4(0.5(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)𝛽𝑚,𝐸 ))| |?̂? | 𝛼,2 𝑛
(6-175) 
< |(𝑚4 + 𝑝 𝑒
−𝑗𝑘𝑖∆𝑚
4 − 𝑣𝑥𝑔4(0.5(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)𝛽𝑚,𝐸 ))| |?̂?𝑜| + |𝑔4||?̂?0|𝛽𝛼,2 𝑛,𝐸  𝛼,2
Simplifying and rearranging, 
|?̂?𝑛| |𝑚4| + |𝑝4| + 𝑣𝑥|𝑔4|(1 − cos𝜙)|𝛽𝑚,𝐸 |𝛼,2
<  (6-176) 
|?̂?𝑜| |𝑙4| + |𝑜4| − |𝑓4| + 𝑣𝑥|𝑔4|(1 − cos𝜙)|𝛽𝑚,𝐸 |𝛼,2
Thus, the condition becomes, 
|𝑚4| + |𝑝4| + 𝑣𝑥|𝑔4|(1 − cos𝜙)|𝛽𝑚,𝐸 |𝛼,2
< 1 (6-177) 
|𝑙4| + |𝑜4| − |𝑓4| + 𝑣𝑥|𝑔4|(1 − cos𝜙)|𝛽𝑚,𝐸 |𝛼,2
The condition is expanded using the simplification terms associated with Equation (6-63), 
𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 𝐴𝐵(𝛼)|  𝛿𝑛,𝑛−1 + 0.5𝑣𝑥  𝛿𝑚,𝑖| + |0.5𝑣 𝛿
𝛼
𝑥  𝑚,𝑖| + 𝑣(1 − 𝛼) (1 − 𝛼) (1 − 𝛼) 𝑥
| | (1 − cos𝜙)|𝛽 |
(1 − 𝛼) 𝑚,𝐸𝛼,2
< 1 
𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 𝛼 2𝐷𝐿 𝐴𝐵(𝛼) 𝛼 𝐷| 𝛿 𝐿
(1 − 𝛼) 𝑛,𝑛−1
+ 0.5𝑣𝑥 𝛿(1 − 𝛼) 𝑚,𝑖
+ | + |0.5𝑣 𝛿 − |
(∆𝑥)2 𝑥 (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2 (6-178) 
( )
𝐷𝐿 𝐴𝐵(𝛼)+ | 2| + 𝑣𝑥 |  | (1 − cos𝜙)|𝛽 |(∆𝑥) (1 − 𝛼) 𝑚,𝐸𝛼,2
The same assumption is made as in Equation (6-152), and the condition then becomes, 
 
175 
 
𝐴𝐵(𝛼)
𝛿𝛼
𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 𝐴𝐵(𝛼)
 𝑛,𝑛−1 + 0.5𝑣
𝛼
𝑥  𝛿𝑚,𝑖 + 0.5𝑣 𝛿
𝛼 + 𝑣 (1 − cos𝜙)𝛽
(1 − 𝛼) (1 − 𝛼) 𝑥 (1 − 𝛼) 𝑚,𝑖 𝑥 (1 − 𝛼) 𝑚,𝐸𝛼,2
< 1 
𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 𝛼 2𝐷𝐿 𝐴𝐵(𝛼) 𝛼 𝐷
 𝛿𝑛,𝑛−1 + 0.5𝑣𝑥 𝛿 + + 0.5𝑣
𝐿
(1 − 𝛼) (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2 𝑥
𝛿
(1 − 𝛼) 𝑚,𝑖
−
( )2 (6-179) ∆𝑥
( )
𝐷𝐿 𝐴𝐵(𝛼)+ 2 + 𝑣𝑥  (1 − cos𝜙)𝛽(∆𝑥) (1 − 𝛼) 𝑚,𝐸𝛼,2
Simplifying, 
2𝐷𝐿
> 0  
2 (6-180) (∆𝑥)
Thus, under this assumption, the second inductive stability condition for is upheld and unconditionally 
stable under this condition.  
The opposite assumption is made, and the condition becomes, 
𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼)
 𝛿𝑛,𝑛−1 + 0.5𝑣𝑥  𝛿𝑚,𝑖 + 0.5𝑣𝑥  𝛿𝑚,𝑖 + 𝑣(1 − 𝛼) (1 − 𝛼) (1 − 𝛼) 𝑥
(1 − cos𝜙)𝛽
(1 − 𝛼) 𝑚,𝐸𝛼,2
< 1 
𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 𝛼 2𝐷𝐿 𝐴𝐵(𝛼) 𝛼 𝐷𝐿
(1 − 𝛼) 
𝛿𝑛,𝑛−1 + 0.5𝑣𝑥  𝛿𝑚,𝑖 + 2 − 0.5𝑣 𝛿 +(1 − 𝛼) (∆𝑥) 𝑥 (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2
(6-181) 
( )
𝐷𝐿 𝐴𝐵(𝛼)+ 2 + 𝑣𝑥  (1 − cos𝜙)𝛽(∆𝑥) (1 − 𝛼) 𝑚,𝐸𝛼,2
Simplifying, 
𝐴𝐵(𝛼) 𝛼 4𝐷𝐿 𝑣𝑥 𝛿 <   (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2 (6-182) 
Under this condition and this assumption, the second inductive stability condition for is supported and 
conditionally stable.  
This completes the stability analysis for the upwind advection Crank-Nicolson scheme for the 
advection-dispersion equation with ABC fractional derivative, where the scheme is unconditionally 
stable for assumption made in Equation (6-152), and conditionally stable under the complementary 
assumption. The determined conditions stated are given in Equation (6-156), (6-167), and (6-182). 
6.3.4 Explicit upwind-downwind weighted scheme  
Replacing the induction method terms in the developed explicit upwind-downwind weighted 
numerical scheme discussed in Section 6.2.4, 
𝑎 ?̂? 𝑒𝑗𝑘𝑖𝑚 = 𝑞 ?̂? 𝑒𝑗𝑘𝑖𝑚 + 𝑟 ?̂? 𝑒𝑗𝑘𝑖(𝑚−∆𝑚) − 𝑠 ?̂? 𝑒𝑗𝑘𝑖(𝑚+∆𝑚)4 𝑛 4 𝑛−1 4 𝑛−1 4 𝑛−1  
𝑛−2 𝑚
𝜃(?̂?𝑛−1𝑒
𝑗𝑘𝑖𝑚 − ?̂? 𝑒𝑗𝑘𝑖(𝑚−∆𝑚) (6-183) 
𝑗𝑘 𝑚 𝑗𝑘 𝑚 𝛼 𝑛−1
)
−𝑔 ∑(?̂? 𝑒 𝑖 𝑖4 𝑘+1 − ?̂?𝑘𝑒 )𝛿𝑛,𝑘 − 𝑣 𝑔 ∑[ ]𝛿
𝛼
𝑥 4
( 𝑚,𝑖
 
+ 1 − 𝜃)(?̂? 𝑒𝑗𝑘𝑖𝑥(𝑚+∆𝑚) − ?̂? 𝑗𝑘𝑖𝑚
𝑘=0 𝑖=0 𝑛−1 𝑛−1
𝑒 )
Multiple out and dividing by 𝑒𝑗𝑘𝑖𝑚, 
𝑎 −𝑗𝑘𝑖∆𝑚 𝑗𝑘𝑖∆𝑚 (6-184) 4?̂?𝑛 = 𝑞4?̂?𝑛−1 + 𝑟4?̂?𝑛−1𝑒 − 𝑠4?̂?𝑛−1𝑒  
176 
 
𝑛−2 𝑚
𝜃(?̂? −𝑗𝑘𝑖∆𝑚𝛼 𝑛−1 − ?̂?𝑛−1𝑒 )−𝑔 𝛼4∑(?̂?𝑘+1 − ?̂?𝑘)𝛿𝑛,𝑘 − 𝑣𝑥𝑔4∑[ ]𝛿  
+(1 − 𝜃)
𝑚,𝑖
(?̂?𝑛−1𝑒
𝑗𝑘𝑖∆𝑚 − ?̂?𝑛−1)𝑘=0 𝑖=0
The first procedure for the induction numerical stability analysis requires proving for a set ∀ 𝑛 > 1, that 
|?̂?𝑛| < |?̂?𝑜|  
If 𝑛 = 1, then 
𝑎4?̂?𝑛 = 𝑞4?̂?𝑛−1 + 𝑟4?̂? 𝑒
−𝑗𝑘𝑖∆𝑚
𝑛−1 − 𝑠
𝑗𝑘𝑖∆𝑚
4?̂?𝑛−1𝑒  
𝑚
𝜃(?̂? − ?̂? 𝑒−𝑗𝑘𝑖∆𝑚𝑛−1 𝑛−1 ) 𝛼 (6-185) −𝑣𝑥𝑔4∑[ ]𝛿  
+(
𝑚,𝑖
1 − 𝜃)(?̂?𝑛−1𝑒
𝑗𝑘𝑖∆𝑚 − ?̂? )
𝑖=0 𝑛−1
A subset for 𝑚 is now considered, where 𝑚 = 0, 
𝑎4?̂?1 = 𝑞4?̂?0 + 𝑟4?̂?
−𝑗𝑘𝑖∆𝑚 𝑗𝑘𝑖∆𝑚
0𝑒 − 𝑠4?̂?0𝑒  (6-186) 
Simplifying and rearranging, 
?̂?1 𝑎4
=  
−𝑗𝑘 ∆𝑚 𝑗𝑘 ∆𝑚 (6-187) ?̂?0 𝑞4 + 𝑟4𝑒 𝑖 − 𝑠4𝑒 𝑖
Applying the norm, the condition for the first induction requirement becomes, 
|𝑎4|
< 1 (6-188) 
|𝑞4| + |𝑟4| + |𝑠4|
The term is expanded using the simplification terms associated with Equation (6-67), 
𝐴𝐵(𝛼)
| 𝛿𝛼 𝑛,𝑛−1|(1 − 𝛼)
< 1 
𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼)| 𝛿 − 2𝑣 𝜃 𝛿𝛼
𝐴𝐵(𝛼) 𝛼 2𝐷+ 𝑣 𝛿 − 𝐿 |
(1 − 𝛼) 𝑛,𝑛−1 𝑥 (1 − 𝛼) 𝑚,𝑖 𝑥 (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2 (6-189) 
𝐴𝐵(𝛼) 𝐷 𝐴𝐵(𝛼) 𝐷
+ |𝑣𝑥𝜃  𝛿
𝛼 + 𝐿 | + |𝑣 (1 − 𝜃) 𝛿𝛼 + 𝐿 |
(1 − 𝛼) 𝑚,𝑖 (∆𝑥)2 𝑥 (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2
The assumption is made where, 
𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 2𝐷
𝛿𝛼
𝐿
𝑛,𝑛−1 + 𝑣 𝛿
𝛼 < 2𝑣 𝜃 𝛿𝛼𝑥 𝑚,𝑖 𝑥 𝑚,𝑖 +  (1 − 𝛼) (1 − 𝛼) (1 − 𝛼) (∆𝑥)2 (6-190) 
Under the assumption (Equation (6-190), the condition is 
𝐴𝐵(𝛼)
𝛿𝛼
(1 − 𝛼) 𝑛,𝑛−1
< 1 
𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 2𝐷
−  𝛿
𝛼 𝛼 𝛼
𝑛,𝑛−1 + 2𝑣𝑥𝜃  𝛿𝑚,𝑖 − 𝑣𝑥  𝛿𝑚,𝑖 +
𝐿
(1 − 𝛼) (1 − 𝛼) (1 − 𝛼) (∆𝑥)2 (6-191) 
𝐴𝐵(𝛼) 𝛼 𝐷𝐿 𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 𝐷+𝑣 𝜃 𝛿 + − 𝑣 𝛿 + 𝑣 𝜃 𝛿𝛼 − 𝐿𝑥 (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2 𝑥 (1 − 𝛼) 𝑚,𝑖 𝑥 (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2
Simplifying, 
𝐴𝐵(𝛼) 2𝐷𝐿 2𝐴𝐵(𝛼)
2𝑣 𝛼 𝛼𝑥(𝜃 − 1) 𝛿 + > 𝛿  (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2 (1 − 𝛼) 𝑛,𝑛−1 (6-192) 
177 
 
Thus, the first inductive stability condition for this subset is upheld and conditionally stable under this 
assumption. 
A subset for 𝑚 is now considered for all 𝑚 ≥ 1, 
𝑚
𝜃(?̂?0 − ?̂? 𝑒
−𝑗𝑘𝑖∆𝑚
0 )
𝑎4?̂?1 = 𝑞4?̂?0 + 𝑟4?̂? 𝑒
−𝑗𝑘𝑖∆𝑚
0 − 𝑠 ?̂? 𝑒
𝑗𝑘𝑖∆𝑚
4 0 − 𝑣𝑥𝑔4∑[ ]𝛿
𝛼  (6-193) 
+(1 − 𝜃)(?̂? 𝑒𝑗𝑘𝑖∆𝑚
𝑚,𝑖
− ?̂? )
𝑖=0 0 0
Simplifying, 
𝑚
𝑞 𝑒𝑗𝑘𝑖∆𝑚 + 𝑟 𝑒−𝑗𝑘𝑖∆𝑚4 4 − 𝑠 𝑒
𝑗𝑘𝑖∆𝑚 − 𝑣 𝑔 𝜃(1 − 𝑒−𝑗𝑘𝑖∆𝑚 𝛼
 4 𝑥 4
)∑𝛿𝑚,𝑖  
𝑎 𝑖=04?̂?1 =   𝑚
 ?̂?  (6-194) 
 0
−𝑣𝑔 (1 − 𝜃)(𝑒𝑗𝑘𝑖∆𝑚4 − 1)∑𝛿
𝛼
𝑚,𝑖
( 𝑖=0 )
Expanding the summation, and applying a norm as previously, 
|𝑞4| + |𝑟4| + |𝑠4| + 𝑣𝑥𝜃|𝑔4|(2 − 2 cos𝜙)|𝛽𝑚,𝐸 | (6-195) 
| || 𝛼,2𝑎4 ?̂?1| = ( ) |?̂?0| 
+𝑣𝑥|𝑔4|(1 − 𝜃)(2 − 2 cos𝜙)|𝛽𝑚,𝐸 |𝛼,2
Rearranging, 
|?̂?1| |𝑞4| + |𝑟4| + |𝑠4| + 𝑣𝑥𝜃|𝑔4|(2 − 2 cos𝜙)|𝛽𝑚,𝐸 | + 𝑣 |𝑔𝛼,2 𝑥 4|(1 − 𝜃)(2 − 2 cos𝜙)|𝛽𝑚,𝐸 | (6-196) 𝛼,2
=  
|?̂?0| |𝑎4|
Thus, the condition becomes, 
|𝑞4| + |𝑟4| + |𝑠4| + 𝑣𝑥𝜃|𝑔4|(2 − 2 cos𝜙)|𝛽𝑚,𝐸 | + 𝑣𝑥|𝑔4|(1 − 𝜃)(2 − 2 cos𝜙)|𝛽𝑚,𝐸 | (6-197) 𝛼,2 𝛼,2
< 1 
|𝑎4|
The term is expanded using the simplification terms associated with Equation (6-71), 
𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 2𝐷|  𝛿𝑛,𝑛−1 − 𝑣𝑥𝜃 𝛿
𝛼
 𝑚,𝑖 + 𝑣 𝛿
𝛼 − 𝑣 𝜃 𝛿𝛼 𝐿
(1 − 𝛼) (1 − 𝛼) 𝑥 (1 − 𝛼) 𝑚,𝑖 𝑥 (1 − 𝛼) 𝑚,𝑖
−
(∆𝑥)2
|
  
𝐴𝐵(𝛼) 𝐷+ |𝑣 𝜃 𝛿𝛼 𝐿
𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 𝛼 𝐷𝐿
𝑥
 (1 − 𝛼) 𝑚,𝑖
+ 2|  + |𝑣𝑥 𝛿 − 𝑣 𝜃 𝛿 + |  (∆𝑥) (1 − 𝛼) 𝑚,𝑖 𝑥 (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2  
(6-198) 
𝐴𝐵(𝛼) 𝐴𝐵(𝛼)
+𝑣𝑥𝜃 |  | (2 − 2 cos𝜙)|𝛽𝑚,𝐸 |  + 𝑣𝑥 |  | (1 − 𝜃)(2 − 2 cos𝜙)|𝛽( (1 − 𝛼) 𝛼,2 (1 − 𝛼) 𝑚,𝐸
|
𝛼,2 )
< 1 
𝐴𝐵(𝛼)
| 𝛼
(1 − 𝛼) 
𝛿𝑛,𝑛−1|
The same assumption is made as in Equation (6-190), and then the conditions is 
𝐴𝐵(𝛼) 𝐴𝐵(𝛼)
− 𝛿𝛼 𝛼
𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 𝛼 2𝐷𝐿
(1 − 𝛼) 𝑛,𝑛−1
+ 𝑣𝑥𝜃 𝛿(1 − 𝛼) 𝑚,𝑖
− 𝑣𝑥 𝛿 + 𝑣(1 − 𝛼) 𝑚,𝑖 𝑥
𝜃 𝛿 +
(1 − 𝛼) 𝑚,𝑖 (∆𝑥)2
  
 𝐴𝐵
(𝛼) 𝐷
+𝑣 𝜃 𝛿𝛼 + 𝐿
𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 𝐷
𝑥  𝑚,𝑖 2 − 𝑣 𝛿
𝛼 𝛼 𝐿
 (1 − 𝛼) (∆𝑥)
𝑥 (1 − 𝛼) 𝑚,𝑖
+ 𝑣𝑥𝜃 𝛿 −  (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2  
(6-199) 
𝐴𝐵(𝛼) 𝐴𝐵(𝛼)
+𝑣𝑥𝜃  (2 − 2 cos𝜙)𝛽𝑚,𝐸  + 𝑣𝑥  (1 − 𝜃)(2 − 2 cos𝜙)𝛽( (1 − 𝛼) 𝛼,2 (1 − 𝛼) 𝑚,𝐸𝛼,2 )
< 1 
𝐴𝐵(𝛼)
𝛿𝛼
(1 − 𝛼) 𝑛,𝑛−1
Simplifying, 
178 
 
2𝐴𝐵(𝛼) 2𝐴𝐵(𝛼) 4𝐷𝐿
(𝛿𝛼 𝛼𝑛,𝑛−1 + 𝑣𝑥𝛿𝑚,𝑖 + 𝑣𝑥 cos𝜙 𝛽𝑚,𝐸 ) > (2𝑣 𝜃𝛿
𝛼 + 𝑣 𝛽 ) +  
(1 − 𝛼) 𝛼,2 (1 − 𝛼) 𝑥 𝑚,𝑖 𝑥 𝑚,𝐸𝛼,2 (∆𝑥)2 (6-200) 
Therefore, the first inductive stability condition for the second subset of m is upheld and conditionally 
stable under this assumption. 
The second procedure for the induction numerical stability analysis requires proving for a set ∀ 𝑛 ≥ 1, 
|?̂?𝑛| < |?̂?𝑜|  
Rearranging Equation (6-184), 
𝑞 + 𝑟 𝑒−𝑗𝑘𝑖∆𝑚 − 𝑠 𝑒𝑗𝑘𝑖∆𝑚4 4 4 𝑛−2
𝑚
𝑎 ?̂? = ( 𝛼4 𝑛 −𝑣 𝑔 (𝜃(1 − 𝑒−𝑗𝑘 ∆𝑚
) ?̂? − 𝑔 ∑(?̂? − ?̂? )𝛿  (6-201) 
𝑖
𝑥 4 )+ (1 − 𝜃)(𝑒
𝑗𝑘𝑖∆𝑚 − 1))∑ 𝛿𝛼 𝑛−1 4 𝑘+1 𝑘 𝑛,𝑘𝑚,𝑖
𝑘=0
𝑖=0
Following a similar simplification process as previously performed, 
𝑞 + 𝑟 𝑒−𝑗𝑘𝑖∆𝑚 − 𝑠 𝑒𝑗𝑘𝑖∆𝑚4 4 4
𝑎4?̂?𝑛 = ( ) ?̂?  −𝑣𝑥𝑔4 (𝜃(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙) + (1 − 𝜃)((cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙) − 1))𝛽
𝑛−1
𝑚,𝐸𝛼,2 (6-202) 
𝑛−2
−𝑔 ( 𝛼4∑ ?̂?𝑘+1 − ?̂?𝑘)𝛿𝑛,𝑘 
𝑘=0
Taking the norm on both sides, 
𝑞 + 𝑟 𝑒−𝑗𝑘𝑖∆𝑚 − 𝑠 𝑒𝑗𝑘𝑖∆𝑚4 4 4
|𝑎4?̂?𝑛| = |( ) ?̂?𝑛−1  
−𝑣𝑥𝑔4 (𝜃(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙) + (1 − 𝜃)((cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)− 1))𝛽𝑚,𝐸𝛼,2
𝑛−2 (6-203) 
− 𝑔 ∑(?̂? − ?̂? )𝛿𝛼4 𝑘+1 𝑘 𝑛,𝑘| 
𝑘=0
Therefore,  
𝑞 + 𝑟 𝑒−𝑗𝑘𝑖∆𝑚4 4 − 𝑠 𝑒
𝑗𝑘𝑖∆𝑚
4
|𝑎4||?̂?𝑛| < |( )| |?̂?𝑛−1|
−𝑣𝑥𝑔 (4 (𝜃 1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙) + (1 − 𝜃)((cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)− 1))𝛽𝑚,𝐸𝛼,2
𝑛−2 (6-204) 
+ |𝑔 | |∑(?̂? − ?̂? )𝛿𝛼4 𝑘+1 𝑘 𝑛,𝑘| 
𝑘=0
Remembering that it has been proved that for a set ∀ 𝑛 > 1,   
|?̂?𝑛−1| < |?̂?𝑜| 
Therefore, it can be inferred that, 
𝑞 + 𝑟 𝑒−𝑗𝑘𝑖∆𝑚 − 𝑠 𝑗𝑘𝑖∆𝑚4 4 4𝑒
|𝑎4||?̂?𝑛| < |( )| |?̂?𝑜| 
−𝑣𝑥𝑔 (4 (𝜃 1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙) + (1 − 𝜃)((cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)− 1))𝛽𝑚,𝐸𝛼,2
𝑛−2 (6-205) 
+|𝑔4| |∑(?̂?𝑘+1 − ?̂?
𝛼
𝑘)𝛿𝑛,𝑘| 
𝑘=0
The remaining summation is considered at the upper limit and expanded as previously, 
179 
 
𝑞4 + 𝑟
−𝑗𝑘𝑖∆𝑚 𝑗𝑘𝑖∆𝑚
4𝑒 − 𝑠4𝑒
|𝑎4||?̂?𝑛| < (| |  −𝑣𝑥𝑔4 (𝜃(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙) + (1 − 𝜃)((cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙) − 1))𝛽𝑚,𝐸𝛼,2
(6-206) 
+ |𝑔4||𝛽𝑛,𝐸 |) |?̂?𝑜| 𝛼,2
Simplifying and rearranging, 
|𝑞4| + |𝑟4| + |𝑠4|  + 𝑣𝑥𝜃|𝑔4|(2 − 2 cos𝜙)|𝛽𝑚,𝐸 |𝛼,2
( )
|?̂?𝑛| +𝑣𝑥|𝑔4|(1 − 𝜃)(2 − 2 cos𝜙)|𝛽𝑚,𝐸 | + |𝑔||𝛽𝑛,𝐸 | (6-207) 𝛼,2 𝛼,2
<  
|?̂?𝑜| |𝑎4|
Thus, the condition becomes, 
|𝑞4| + |𝑟4| + |𝑠4|  + 𝑣𝑥𝜃|𝑔4|(2 − 2 cos𝜙)|𝛽𝑚,𝐸 |𝛼,2
( )
+𝑣𝑥|𝑔4|(1 − 𝜃)(2 − 2 cos𝜙)|𝛽𝑚,𝐸 | + |𝑔||𝛽𝑛,𝐸 | (6-208) 𝛼,2 𝛼,2
< 1 
|𝑎4|
The condition is expanded using the simplification terms associated with Equation (6-67), 
𝐴𝐵(𝛼)
| 𝛼
𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 𝛼 2𝐷𝐿
(1 − 𝛼) 
𝛿𝑛,𝑛−1 − 2𝑣𝑥𝜃 𝛿 + 𝑣 𝛿 − |(1 − 𝛼) 𝑚,𝑖 𝑥 (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2
𝐴𝐵(𝛼)
+ |𝑣 𝜃 𝛿𝛼
𝐷
𝑥  𝑚,𝑖 +
𝐿 |
(1 − 𝛼) (∆𝑥)2
𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 𝛼 𝐷 𝐴𝐵(𝛼)+ |𝑣𝑥  𝛿𝑚,𝑖 − 𝑣
𝐿
𝑥𝜃  𝛿𝑚,𝑖 + 2|  + 𝑣𝑥𝜃 |  | (2 − 2 cos𝜙)|𝛽 |(1 − 𝛼) (1 − 𝛼) (∆𝑥) (1 − 𝛼) 𝑚,𝐸𝛼,2 (6-209) 
𝐴𝐵(𝛼) 𝐴𝐵(𝛼)
+𝑣𝑥 |  | (1 − 𝜃)(2 − 2 cos𝜙)|𝛽𝑚,𝐸 | + |  | |𝛽𝑛,𝐸 |(1 − 𝛼) 𝛼,2 (1 − 𝛼) 𝛼,2
< 1 
𝐴𝐵(𝛼)
| 𝛿𝛼
(1 − 𝛼) 𝑛,𝑛−1
|
The same assumption is made again (Equation (6-190)), and under this assumption the condition is, 
𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 𝛼 2𝐷− 𝛿 𝐿
(1 − 𝛼) 𝑛,𝑛−1
+ 2𝑣𝑥𝜃  𝛿𝑚,𝑖 − 𝑣 𝛿 +(1 − 𝛼) 𝑥 (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2
𝐴𝐵(𝛼) 𝐷
+𝑣 𝜃 𝛿𝛼 + 𝐿𝑥 (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2
𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 𝐷 𝐴𝐵(𝛼)
−𝑣𝑥  𝛿
𝛼 + 𝑣 𝜃 𝛿𝛼 − 𝐿𝑚,𝑖 𝑥  𝑚,𝑖 2  + 𝑣𝑥𝜃  (2 − 2 cos𝜙)|𝛽 |(1 − 𝛼) (1 − 𝛼) (∆𝑥) (1 − 𝛼) 𝑚,𝐸𝛼,2 (6-210) 
𝐴𝐵(𝛼) 𝐴𝐵(𝛼)
+𝑣𝑥  (1 − 𝜃)(2 − 2 cos𝜙)𝛽(1 − 𝛼) 𝑚,𝐸
+ 𝛽
𝛼,2 (1 − 𝛼) 𝑛,𝐸𝛼,2
< 1 
𝐴𝐵(𝛼)
𝛿𝛼
(1 − 𝛼) 𝑛,𝑛−1
Simplifying, 
2𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 4𝐷
(𝛿𝛼 𝛼
𝐿
𝑛,𝑛−1 + 𝑣𝑥𝛿𝑚,𝑖 + 𝑣𝑥 cos𝜙 𝛽𝑚,𝐸 ) > (2𝑣𝑥𝛽𝑚,𝐸 + 4𝑣𝑥𝜃𝛿
𝛼
𝑚,𝑖 + 𝛽 ) +  (1 − 𝛼) 𝛼,2 (1 − 𝛼) 𝛼,2 𝑛,𝐸𝛼,2 (∆𝑥)2 (6-211) 
Thus, under this assumption, the second inductive stability condition for is upheld and conditionally 
stable under this condition.  
This concludes the stability analysis for the explicit upwind-downwind weighted scheme for the 
advection-dispersion equation with ABC fractional derivative, where the scheme is found to be 
180 
 
unstable under the assumption not presented and is conditionally stable under the assumption 
(Equation (6-190)), with the following stability conditions, 
𝐴𝐵(𝛼) 2𝐷𝐿 2𝐴𝐵(𝛼)
2𝑣𝑥(𝜃 − 1) 𝛿
𝛼 𝛼
(1 − 𝛼) 𝑚,𝑖
+ > 𝛿 , 
(∆𝑥)2 (1 − 𝛼) 𝑛,𝑛−1
2𝐴𝐵(𝛼) 2𝐴𝐵(𝛼) 4𝐷
(𝛿𝛼 𝛼
𝐿
𝑛,𝑛−1 + 𝑣𝑥𝛿𝑚,𝑖 + 𝑣𝑥 cos𝜙 𝛽𝑚,𝐸 ) > (2𝑣𝑥𝜃𝛿
𝛼
𝑚,𝑖 + 𝑣𝑥𝛽𝑚,𝐸 ) + , (1 − 𝛼) 𝛼,2 (1 − 𝛼) 𝛼,2 (∆𝑥)2
2𝐴𝐵(𝛼) 𝐴𝐵(𝛼)
(𝛿𝛼 + 𝑣 𝛿𝛼 + 𝑣 cos𝜙 𝛽 ) > (4𝑣 𝜃𝛿𝛼
4𝐷𝐿
+ 2𝑣 𝛽 + 𝛽 ) +  
(1 − 𝛼) 𝑛,𝑛−1 𝑥 𝑚,𝑖 𝑥 𝑚,𝐸𝛼,2 (1 − 𝛼) 𝑥 𝑚,𝑖 𝑥 𝑚,𝐸𝛼,2 𝑛,𝐸𝛼,2 (∆𝑥)2
6.3.5 Implicit upwind-downwind weighted scheme  
Substituting the induction method terms for the implicit upwind-downwind weighted numerical 
scheme discussed in Section 6.2.5, 
𝑢4?̂?
𝑗𝑘𝑖𝑚 𝑗𝑘𝑖𝑥(𝑚+∆𝑚)
𝑛𝑒 = 𝑣4?̂?𝑛𝑒 + 𝑟4?̂?𝑛𝑒
𝑗𝑘𝑖(𝑚−∆𝑚) + 𝑎 ?̂? 𝑒𝑗𝑘𝑖𝑚4 𝑛−1  
𝑛−2
−𝑔4∑(?̂? 𝑒
𝑗𝑘𝑖𝑚
𝑘+1 − ?̂?
𝑗𝑘𝑖𝑚 𝛼
𝑘𝑒 )𝛿𝑛,𝑘 (6-212) 
𝑘=0
𝑚
−𝑣 𝑔  ∑[𝜃(?̂? 𝑒𝑗𝑘𝑖𝑚 − ?̂? 𝑒𝑗𝑘𝑖(𝑚−∆𝑚)) + (1 − 𝜃)(?̂? 𝑒𝑗𝑘𝑖𝑥(𝑚+∆𝑚) − ?̂? 𝑒𝑗𝑘𝑖𝑚𝑥 4 𝑛 𝑛 𝑛 𝑛 )] 𝛿
𝛼
𝑚,𝑖 
𝑖=0
Multiple out and dividing by 𝑒𝑗𝑘𝑖𝑚, 
𝑛−2
𝑢 ?̂? = 𝑣 ?̂? 𝑒𝑗𝑘𝑖∆𝑚4 𝑛 4 𝑛 + 𝑟4?̂?𝑛𝑒
−𝑗𝑘𝑖∆𝑚 + 𝑎4?̂?𝑛−1 − 𝑔4∑(?̂?𝑘+1 − ?̂?𝑘)𝛿
𝛼
𝑛,𝑘 
𝑘=0
𝑚 (6-213) 
−𝑣 𝑔  ∑[𝜃(?̂? − ?̂? 𝑒−𝑗𝑘𝑖∆𝑚 𝑗𝑘𝑖∆𝑚 𝛼𝑥 4 𝑛 𝑛 ) + (1 − 𝜃)(?̂?𝑛𝑒 − ?̂?𝑛)] 𝛿𝑚,𝑖 
𝑖=0
As stated previously, the first procedure for the induction numerical stability analysis requires proving 
for a set ∀ 𝑛 > 1, that 
|?̂?𝑛| < |?̂?𝑜|  
If 𝑛 = 1, then 
𝑚
𝜃(?̂? − ?̂? 𝑒−𝑗𝑘𝑖∆𝑚) 
𝑢 ?̂? = 𝑣 ?̂? 𝑒𝑗𝑘𝑖∆𝑚 + 𝑟 ?̂? 𝑒−𝑗𝑘 ∆𝑚
1 1
𝑖 𝛼
4 1 4 1 4 1 + 𝑎4?̂?0 − 𝑣𝑥𝑔4  ∑[ ] 𝛿𝑚,𝑖  (6-214) 
+(1 − 𝜃)(?̂? 𝑒𝑗𝑘𝑖∆𝑚1 − ?̂?1)𝑖=0
A subset for 𝑚 is now considered, where 𝑚 = 0, 
𝑢 ?̂? = 𝑣 ?̂? 𝑒𝑗𝑘𝑖∆𝑚4 1 4 1 + 𝑟4?̂?1𝑒
−𝑗𝑘𝑖∆𝑚 + 𝑎4?̂?0 (6-215) 
Simplifying and rearranging, 
?̂?1 𝑎4
=  
?̂? 𝑢 − 𝑣 𝑒𝑗𝑘𝑖∆𝑚 − 𝑟 𝑒−𝑗𝑘 ∆𝑚
(6-216) 
0 4 4 4 𝑖
Applying a norm, the condition for the first induction requirement becomes, 
181 
 
|𝑎4|
< 1 (6-217) 
|𝑢4| + |𝑣4| + |𝑟4|
The term is expanded using the simplification terms associated with Equation (6-63), 
𝐴𝐵(𝛼)
| 𝛿𝛼
(1 − 𝛼) 𝑛,𝑛−1
|
< 1 
𝐴𝐵(𝛼)
| 𝛿𝛼
𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 2𝐷+ 2𝑣 𝜃 𝛿 𝛼 𝐿
(1 − 𝛼) 𝑛,𝑛−1 𝑥 (1 − 𝛼) 𝑚,𝑖
− 𝑣𝑥 (1 − 𝛼) 
𝛿𝑚,𝑖 + |(∆𝑥)2 (6-218) 
( )
𝐷𝐿 𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 𝛼 𝐷𝐿 𝐴𝐵(𝛼)+ | 2 − 𝑣𝑥  𝛿𝑚,𝑖 + 𝑣𝑥𝜃  𝛿𝑚,𝑖| + | 2 + 𝑣𝑥𝜃 𝛿
𝛼
(∆𝑥) (1 − 𝛼) (1 − 𝛼) (∆𝑥) (1 − 𝛼) 𝑚,𝑖
|
The assumption is made where, 
𝐷𝐿 𝐴𝐵(𝛼) 𝐴𝐵(𝛼)
+ 𝑣𝑥𝜃 𝛿
𝛼 𝛼
(∆𝑥)2 (1 − 𝛼) 𝑚,𝑖
> 𝑣𝑥 𝛿(1 − 𝛼) 𝑚,𝑖
 (6-219) 
Under the assumption (Equation (6-219)), the condition is 
𝐴𝐵(𝛼) 𝛼
(1 − 𝛼) 
𝛿𝑛,𝑛−1
< 1 
𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼)𝛿 𝛼
2𝐷𝐿 (6-220) 
(1 − 𝛼) 𝑛,𝑛−1
+ 2𝑣𝑥𝜃 𝛿 − 𝑣 𝛿 +(1 − 𝛼) 𝑚,𝑖 𝑥 (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2
( )
𝐷
+ 𝐿
𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 𝐷− 𝑣 𝛿 + 𝑣 𝜃 𝛿𝛼 + 𝐿
𝐴𝐵(𝛼) 𝛼
(∆𝑥)2 𝑥 (1 − 𝛼) 𝑚,𝑖 𝑥  𝑚,𝑖 2
+ 𝑣𝑥𝜃  𝛿(1 − 𝛼) (∆𝑥) (1 − 𝛼) 𝑚,𝑖
Simplifying, 
𝐴𝐵(𝛼) 2𝐷𝐿
(2𝜃 − 1)𝑣 𝛿𝛼𝑥 𝑚,𝑖 + > 0 (1 − 𝛼) (∆𝑥)2 (6-221) 
Remembering the weighting factor 𝜃, the condition for this subset of 𝑚 is unconditionally stable 
when 0.5 ≤ 𝜃 ≤ 1, but conditionally stable when 0 ≤ 𝜃 < 0.5. 
The opposed assumption is made, and the condition becomes, 
𝐴𝐵(𝛼)
𝛿𝛼
(1 − 𝛼) 𝑛,𝑛−1
< 1 
𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 2𝐷−  𝛿𝑛,𝑛−1 − 2𝑣𝑥𝜃  𝛿
𝛼
𝑚,𝑖 + 𝑣 𝛿
𝛼 − 𝐿
(1 − 𝛼) (1 − 𝛼) 𝑥 (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2 (6-222) 
( )
𝐷 𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 𝐷 𝐴𝐵(𝛼)
− 𝐿 2 + 𝑣
𝛼 𝛼 𝐿
𝑥  𝛿𝑚,𝑖 − 𝑣𝑥𝜃  𝛿𝑚,𝑖 + 2 + 𝑣𝑥𝜃 𝛿
𝛼
(∆𝑥) (1 − 𝛼) (1 − 𝛼) (∆𝑥) (1 − 𝛼) 𝑚,𝑖
Simplifying, 
𝐴𝐵(𝛼) 𝛼 𝛼 𝐴𝐵(𝛼) 𝐷𝐿𝑣𝑥 𝛿
𝛼
(1 − 𝛼) 𝑚,𝑖
> (𝛿𝑛,𝑛−1 + 𝑣𝑥𝜃𝛿𝑚,𝑖) +  (1 − 𝛼) (∆𝑥)2 (6-223) 
Hence, the first inductive stability condition for this subset is sustained and conditionally stable under 
the complementary assumption. 
A subset for 𝑚 is now considered for all 𝑚 ≥ 1, 
𝑚
𝜃(?̂? − ?̂? 𝑒−𝑗𝑘𝑖∆𝑚) 
𝑢 ?̂? = 𝑣 ?̂? 𝑒𝑗𝑘𝑖∆𝑚 −𝑗𝑘 ∆𝑚
1 1
+ 𝑟 ?̂? 𝑒 𝑖 + 𝑎 ?̂? − 𝑣 𝑔 ∑[ ]𝛿𝛼  (6-224) 4 1 4 1 4 1 4 0 𝑥 4
+(1 − 𝜃)(?̂? 𝑗𝑘𝑖∆𝑚
𝑚,𝑖
𝑖=0 1
𝑒 − ?̂?1)
Simplifying, 
182 
 
𝑚
𝑢 − 𝑣 𝑒𝑗𝑘𝑖∆𝑚 − 𝑟 𝑒−𝑗𝑘𝑖∆𝑚4 4 4 + 𝑣𝑥𝑔4𝜃(1 − 𝑒
−𝑗𝑘𝑖∆𝑚)∑𝛿𝛼
 𝑚,𝑖 
𝑖=0
 𝑚  ?̂?1 = 𝑎4?̂?    0 (6-225) 
+𝑣 𝑔 (1 − 𝜃)(𝑒𝑗𝑘𝑖∆𝑚𝑥 4 − 1)∑𝛿
𝛼
𝑚,𝑖
( 𝑖=0 )
Expanding the summation, and applying a norm on both sides as previously, 
(|𝑢| + |𝑣4| + |𝑟4| + 𝑣𝑥𝜃|𝑔 |(4 2 − 2 cos𝜙)|𝛽𝑚,𝐸 | + 𝑣𝑥|𝑔4|(1 − 𝜃)(2 − 2 cos𝜙)|𝛽𝑚,𝐸 |)|?̂?𝛼,2 𝛼,2 1| (6-226) 
= |𝑎4||?̂?0| 
Rearranging, 
|?̂?1| |𝑎4| (6-227) 
=  
|?̂?0| |𝑢4| + |𝑣4| + |𝑟4| + 𝑣𝑥𝜃|𝑔4|(2 − 2 cos𝜙)|𝛽𝑚,𝐸 | + 𝑣𝑥|𝑔4|(1 − 𝜃)(2 − 2 cos𝜙)|𝛽 |𝛼,2 𝑚,𝐸𝛼,2
Thus, the condition becomes, 
|𝑎4| (6-228) 
< 1 
|𝑢4| + |𝑣4| + |𝑟4| + 𝑣𝑥𝜃|𝑔4|(2 − 2 cos𝜙)|𝛽𝑚,𝐸 | + 𝑣𝑥|𝑔4|(1 − 𝜃)(2 − 2 cos𝜙)|𝛽𝛼,2 𝑚,𝐸 |𝛼,2
The term is expanded using the simplification terms associated with Equation (6-71), 
𝐴𝐵(𝛼)
| 𝛿𝛼
(1 − 𝛼) 𝑛,𝑛−1
|
< 1 
𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 2𝐷| 𝛿 + 2𝑣 𝜃 𝛿 − 𝑣 𝛿𝛼 + 𝐿 | 
(1 − 𝛼) 𝑛,𝑛−1 𝑥 (1 − 𝛼) 𝑚,𝑖 𝑥 (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2
  
𝐷 𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 𝐷 𝐴𝐵(𝛼) (6-229) 
 +| 𝐿 − 𝑣 𝛿𝛼 + 𝑣 𝜃 𝛿𝛼 | + | 𝐿 + 𝑣 𝜃 𝛿𝛼 |  
 (∆𝑥)2
𝑥 (1 − 𝛼) 𝑚,𝑖 𝑥 (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2 𝑥 (1 − 𝛼) 𝑚,𝑖  
𝐴𝐵(𝛼) 𝐴𝐵(𝛼)
+𝑣𝑥𝜃 |  | (2 − 2 cos𝜙)|𝛽 | + 𝑣 | | (1 − 𝜃)(2 − 2 cos𝜙)|𝛽 |( (1 − 𝛼) 𝑚,𝐸𝛼,2 𝑥 (1 − 𝛼) 𝑚,𝐸𝛼,2 )
The assumption in Equation (6-219) is made, and the resulting stability condition becomes, 
𝐴𝐵(𝛼) 𝛼 (6-230) 
 𝛿(1 − 𝛼) 𝑛,𝑛−1
< 1 
𝐴𝐵(𝛼)
𝛿𝛼
𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 2𝐷
 𝑛,𝑛−1 + 2𝑣𝑥𝜃
𝛼
 𝛿𝑚,𝑖 − 𝑣𝑥  𝛿
𝛼 + 𝐿  
(1 − 𝛼) (1 − 𝛼) (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2
  
𝐷𝐿 𝐴𝐵(𝛼) 𝛼 𝐴𝐵
(𝛼) 𝛼 𝐷𝐿 𝐴𝐵(𝛼)+ − 𝑣 𝛼
(∆𝑥)2 𝑥  
𝛿𝑚,𝑖 + 𝑣𝑥𝜃 𝛿 + + 𝑣 𝜃 𝛿   (1 − 𝛼) (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2
𝑥 (1 − 𝛼) 𝑚,𝑖  
𝐴𝐵(𝛼) 𝐴𝐵(𝛼)
+𝑣𝑥𝜃  (2 − 2 cos𝜙)𝛽𝑚,𝐸 + 𝑣( (1 − 𝛼) 𝛼,2 𝑥  
(1 − 𝜃)(2 − 2 cos𝜙)𝛽
(1 − 𝛼) 𝑚,𝐸𝛼,2)
Simplifying, 
𝐴𝐵(𝛼) 𝐴𝐵(𝛼)𝛼 2𝐷𝐿 (6-231) 𝑣𝑥(2𝜃 − 1) 𝛿𝑚,𝑖 + 𝑣𝑥(1 − cos𝜙) 𝛽𝑚,𝐸 + > 0 (1 − 𝛼) (1 − 𝛼) 𝛼,2 (∆𝑥)2
Therefore, under the assumption (Equation (6-219)), the first inductive stability condition for the 
second subset of m is defended and unconditionally stable when 0.5 ≤ 𝜃 ≤ 1; and conditionally 
stable when 0 ≤ 𝜃 < 0.5. 
The complementary assumption is made, and the condition becomes, 
183 
 
𝐴𝐵(𝛼)
𝛿𝛼
(1 − 𝛼) 𝑛,𝑛−1
< 1 
𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 2𝐷
− 𝛿𝛼 − 2𝑣 𝜃 𝛿𝛼 + 𝑣 𝛿𝛼 − 𝐿  
(1 − 𝛼) 𝑛,𝑛−1 𝑥 (1 − 𝛼) 𝑚,𝑖 𝑥 (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2
  (6-232) 
𝐷𝐿 𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 𝛼 𝐷𝐿 𝐴𝐵(𝛼) − + 𝑣 𝛿 − 𝑣 𝜃 𝛼
 (∆𝑥)2
𝑥 (1 − 𝛼) 𝑚,𝑖 𝑥 (1 − 𝛼) 
𝛿  𝑚,𝑖 + + 𝑣 𝜃 𝛿(∆𝑥)2 𝑥 (1 − 𝛼) 𝑚,𝑖  
𝐴𝐵(𝛼) 𝐴𝐵(𝛼)
+𝑣 𝜃 (2 − 2 cos𝜙)𝛽 + 𝑣 (1 − 𝜃)(2 − 2 cos𝜙)𝛽
( 𝑥 (1 − 𝛼) 𝑚,𝐸𝛼,2 𝑥 (1 − 𝛼) 𝑚,𝐸𝛼,2)
Simplifying, 
𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 𝐷
𝑣 (𝛽 𝛼 𝛼
𝐿
𝑥 𝑚,𝐸 + 𝛿𝑚,𝑖) > (𝛿𝑛,𝑛−1 + 𝑣𝑥𝜃𝛿
𝛼
𝑚,𝑖 + 𝑣𝑥 cos𝜙 𝛽 +  (1 − 𝛼) 𝛼,2 (1 − 𝛼) 𝑚,𝐸
)
𝛼,2 (∆𝑥)2 (6-233) 
Therefore, under this assumption the first inductive stability condition for the second subset of m is 
upheld and conditionally stable. 
The second procedure for the induction numerical stability analysis requires proving for a set ∀ 𝑛 ≥ 1, 
|?̂?𝑛| < |?̂?𝑜|  
Rearranging Equation (6-213) for ?̂?𝑛, 
𝑚
(𝑢4 − 𝑣 𝑒
𝑗𝑘𝑖∆𝑚
4 − 𝑟
−𝑗𝑘𝑖∆𝑚
4𝑒 + 𝑣𝑥𝑔4𝜃(1 − 𝑒
−𝑗𝑘𝑖∆𝑚)∑𝛿𝛼𝑚,𝑖
𝑖=0
𝑚 𝑛−2 (6-234) 
+ 𝑣𝑥𝑔4(1 − 𝜃)(𝑒
𝑗𝑘𝑖∆𝑚 − 1)∑𝛿𝛼𝑚,𝑖) ?̂?𝑛 = 𝑎4?̂?𝑛−1 − 𝑔4∑(?̂?𝑘+1 − ?̂?𝑘)𝛿
𝛼
𝑛,𝑘 
𝑖=0 𝑘=0
Following a similar simplification process as previously performed, 
𝑢 − 𝑣 𝑒𝑗𝑘𝑖∆𝑚 − 𝑟 −𝑗𝑘𝑖∆𝑚4 4 4𝑒 + 𝑣𝑥𝑔4𝜃(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)𝛽𝑚,𝐸𝛼,2
( ) ?̂?𝑛 
+𝑣𝑥𝑔4(1 − 𝜃)((cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙) − 1)𝛽𝑚,𝐸𝛼,2 (6-235) 
𝑛−2
= 𝑎4?̂?𝑛−1 − 𝑔
𝛼
4∑(?̂?𝑘+1 − ?̂?𝑘)𝛿𝑛,𝑘 
𝑘=0
Applying a norm, 
𝑢4 − 𝑣
𝑗𝑘𝑖∆𝑚 −𝑗𝑘𝑖∆𝑚
4𝑒 − 𝑟4𝑒 + 𝑣𝑥𝑔4𝜃(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)𝛽𝑚,𝐸𝛼,2
|( ) ?̂?𝑛| 
+𝑣𝑥𝑔4(1 − 𝜃)((cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙) − 1)𝛽𝑚,𝐸𝛼,2
𝑛−2 (6-236) 
= |𝑎4?̂?𝑛−1 − 𝑔4∑(?̂?𝑘+1 − ?̂?𝑘)𝛿
𝛼
𝑛,𝑘| 
𝑘=0
Therefore,  
𝑢4 − 𝑣4𝑒
𝑗𝑘𝑖∆𝑚 − 𝑟 𝑒−𝑗𝑘𝑖∆𝑚4 + 𝑣𝑥𝑔4𝜃(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)𝛽𝑚,𝐸𝛼,2
|( )| |?̂?𝑛| 
+𝑣𝑥𝑔4(1 − 𝜃)((cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙) − 1)𝛽𝑚,𝐸𝛼,2
𝑛−2 (6-237) 
< |𝑎4||?̂?𝑛−1| + |𝑔4| |∑(?̂?
𝛼
𝑘+1 − ?̂?𝑘)𝛿𝑛,𝑘| 
𝑘=0
Remembering that it has been proved that for a set ∀ 𝑛 > 1,   
184 
 
|?̂?𝑛−1| < |?̂?𝑜| 
Thus, 
𝑢 𝑗𝑘𝑖∆𝑚 −𝑗𝑘𝑖∆𝑚4 − 𝑣4𝑒 − 𝑟4𝑒 + 𝑣𝑥𝑔4𝜃(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)𝛽𝑚,𝐸𝛼,2
|( )| |?̂?𝑛| 
+𝑣𝑥𝑔4(1 − 𝜃)((cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙) − 1)𝛽𝑚,𝐸𝛼,2
𝑛−2 𝑛−2 (6-238) 
< |𝑎4||?̂?𝑛−1| + |𝑔4| |∑(?̂?𝑘+1 − ?̂?𝑘)𝛿
𝛼
𝑛,𝑘| < |𝑎4||?̂?𝑜| + |𝑔4| |∑(?̂?𝑘+1 − ?̂?𝑘)𝛿
𝛼
𝑛,𝑘| 
𝑘=0 𝑘=0
Therefore, it can be inferred that, 
𝑢 − 𝑣 𝑒𝑗𝑘𝑖∆𝑚 − 𝑟 𝑒−𝑗𝑘𝑖∆𝑚4 4 4 + 𝑣𝑥𝑔4𝜃(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)𝛽𝑚,𝐸𝛼,2
|( )| |?̂?𝑛| 
+𝑣𝑥𝑔4(1 − 𝜃)((cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙) − 1)𝛽𝑚,𝐸𝛼,2
𝑛−2 (6-239) 
< |𝑎4||?̂?𝑜| + |𝑔4| |∑(?̂?
𝛼
𝑘+1 − ?̂?𝑘)𝛿𝑛,𝑘| 
𝑘=0
The remaining summation is considered at the upper limit and expanded as previously performed, 
𝑢 − 𝑣 𝑒𝑗𝑘𝑖∆𝑚 − 𝑟 𝑒−𝑗𝑘𝑖∆𝑚4 4 4 + 𝑣𝑥𝑔4𝜃(1 − cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙)𝛽𝑚,𝐸𝛼,2
|( )| |?̂?𝑛| 
+𝑣𝑥𝑔4(1 − 𝜃)((cos𝜙 + 𝑖 𝑠𝑖𝑛 𝜙) − 1)𝛽𝑚,𝐸 (6-240) 𝛼,2
< (|𝑎4| + |𝑔4||𝛽𝑛,𝐸 |)|?̂?𝑜| 𝛼,2
Simplifying and rearranging, 
|?̂?𝑛| |𝑎4| + |𝑔4||𝛽𝑛,𝐸 |𝛼,2
<  (6-241) 
|?̂?𝑜| |𝑢4| + |𝑣4| + |𝑟4| + 𝑣𝑥𝜃|𝑔4|(2 − 2 cos𝜙)|𝛽𝑚,𝐸 | + 𝑣𝑥|𝑔4|(1 − 𝜃)(2 − 2 cos𝜙)|𝛽 |𝛼,2 𝑚,𝐸𝛼,2
Thus, the condition becomes, 
|𝑎4| + |𝑔4||𝛽𝑛,𝐸 |𝛼,2
< 1 (6-242) 
|𝑢4| + |𝑣4| + |𝑟4| + 𝑣𝑥𝜃|𝑔4|(2 − 2 cos𝜙)|𝛽𝑚,𝐸 | + 𝑣𝑥|𝑔4|(1 − 𝜃)(2 − 2 cos𝜙)|𝛽 |𝛼,2 𝑚,𝐸𝛼,2
The condition is expanded using the simplification terms associated with Equation (6-71), 
𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼)|  𝛿𝑛,𝑛−1| + |  | |𝛽 |(1 − 𝛼) (1 − 𝛼) 𝑛,𝐸𝛼,2
< 1 
𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 2𝐷| 𝛼 𝐿
(1 − 𝛼) 
𝛿𝑛,𝑛−1 + 2𝑣𝑥𝜃  𝛿𝑚,𝑖 − 𝑣𝑥 𝛿 + |(1 − 𝛼) (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2
(6-243) 
𝐷𝐿 𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 𝛼 𝐷𝐿 𝐴𝐵(𝛼)+ | 𝛼
(∆𝑥)2
− 𝑣𝑥 (1 − 𝛼) 
𝛿𝑚,𝑖 + 𝑣𝑥𝜃  𝛿(1 − 𝛼) 𝑚,𝑖
| + |
(∆𝑥)2
+ 𝑣𝑥𝜃 𝛿 |(1 − 𝛼) 𝑚,𝑖
𝐴𝐵(𝛼) 𝐴𝐵(𝛼)
+𝑣𝑥𝜃 |  | (2 − 2 cos𝜙)|𝛽𝑚,𝐸 | + 𝑣 | | (1 − 𝜃)(2 − 2 cos𝜙)|𝛽 |(1 − 𝛼) 𝛼,2 𝑥 (1 − 𝛼) 𝑚,𝐸𝛼,2
Similarly, the assumption in Equation (6-219) is made, and hence the conditions is, 
𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼)
(1 − 𝛼) 
𝛿𝑛,𝑛−1 + 𝛽(1 − 𝛼) 𝑛,𝐸𝛼,2
< 1 
𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 𝛼 2𝐷𝛿 𝐿
(1 − 𝛼) 𝑛,𝑛−1
+ 2𝑣𝑥𝜃 𝛿 − 𝑣 𝛿 +(1 − 𝛼) 𝑚,𝑖 𝑥 (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2
(6-244) 
𝐷𝐿 𝐴𝐵(𝛼) 𝐴𝐵(𝛼)+ − 𝑣 𝛿𝛼 + 𝑣 𝜃 𝛿𝛼
𝐷𝐿 𝐴𝐵(𝛼) 𝛼
(∆𝑥)2 𝑥 (1 − 𝛼) 𝑚,𝑖 𝑥
+ + 𝑣 𝜃 𝛿
(1 − 𝛼) 𝑚,𝑖 (∆𝑥)2 𝑥 (1 − 𝛼) 𝑚,𝑖
𝐴𝐵(𝛼) 𝐴𝐵(𝛼)
+𝑣𝑥𝜃  (2 − 2 cos𝜙)𝛽𝑚,𝐸 + 𝑣𝑥  (1 − 𝜃)(2 − 2 cos𝜙)𝛽(1 − 𝛼) 𝛼,2 (1 − 𝛼) 𝑚,𝐸𝛼,2
185 
 
Simplifying, 
𝐴𝐵(𝛼) 𝛼 4𝐷𝐿 𝐴𝐵(𝛼)2𝑣𝑥 (𝛿𝑚,𝑖(2𝜃 − 1) + (1 − cos𝜙)𝛽𝑚,𝐸 ) + > 𝛽  (1 − 𝛼) 𝛼,2 (∆𝑥)2 (1 − 𝛼) 𝑛,𝐸𝛼,2 (6-245) 
According to the assumption in Equation (6-219) and stability condition, the second inductive stability 
condition for is upheld and unconditionally stable when 0.5 ≤ 𝜃 ≤ 1, and conditionally stable 
when 0 ≤ 𝜃 < 0.5. The complementary assumption is made, and the condition becomes, 
𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼)𝛿
(1 − 𝛼) 𝑛,𝑛−1
+ 𝛽
(1 − 𝛼) 𝑛,𝐸𝛼,2
< 1 
𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 2𝐷−  𝛿𝑛,𝑛−1 − 2𝑣𝑥𝜃  𝛿 + 𝑣 𝛿
𝛼 − 𝐿
(1 − 𝛼) (1 − 𝛼) 𝑚,𝑖 𝑥 (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2 (6-246) 
𝐷𝐿 𝐴𝐵(𝛼) 𝛼 𝐴𝐵(𝛼) 𝛼 𝐷𝐿 𝐴𝐵(𝛼)− 2 + 𝑣𝑥  𝛿𝑚,𝑖 − 𝑣𝑥𝜃  𝛿𝑚,𝑖 + 2 + 𝑣𝑥𝜃 𝛿
𝛼
(∆𝑥) (1 − 𝛼) (1 − 𝛼) (∆𝑥) (1 − 𝛼) 𝑚,𝑖
𝐴𝐵(𝛼) 𝐴𝐵(𝛼)
+𝑣𝑥𝜃  (2 − 2 cos𝜙)𝛽𝑚,𝐸 + 𝑣𝑥  (1 − 𝜃)(2 − 2 cos𝜙)𝛽(1 − 𝛼) 𝛼,2 (1 − 𝛼) 𝑚,𝐸𝛼,2
Simplifying, 
𝐴𝐵(𝛼) 𝛼 2𝐷𝐿 𝐴𝐵(𝛼) 2𝑣𝑥 ((1 − 𝜃)𝛿𝑚,𝑖 + (1 − cos𝜙)𝛽
𝛼
(1 − 𝛼) 𝑚,𝐸
) − > (2𝛿 + 𝛽 ) 
𝛼,2 (∆𝑥)2 (1 − 𝛼) 𝑛,𝑛−1 𝑛,𝐸𝛼,2 (6-247) 
Under the complementary assumption and condition, the second inductive stability condition for is 
supported and the numerical scheme is conditionally stable.  
This completes the stability analysis for the implicit upwind-downwind weighted scheme for the 
advection-dispersion equation with ABC fractional derivative, where the scheme is stable under the 
following condition, 
𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 𝐷
𝑣 𝛿𝛼 > (𝛿𝛼 𝛼
𝐿
𝑥 (1 − 𝛼) 𝑚,𝑖 𝑛,𝑛−1
+ 𝑣𝑥𝜃𝛿𝑚,𝑖) + , (1 − 𝛼) (∆𝑥)2
𝐴𝐵(𝛼) ( )
𝑣 (𝛽 + 𝛿𝛼
𝐴𝐵 𝛼 𝐷𝐿
𝑥  𝑚,𝐸 𝑚,𝑖) > (𝛿
𝛼 𝛼
𝑛,𝑛−1 + 𝑣𝑥𝜃𝛿𝑚,𝑖 + 𝑣𝑥 cos𝜙 𝛽𝑚,𝐸 ) + , (1 − 𝛼) 𝛼,2 (1 − 𝛼) 𝛼,2 (∆𝑥)2
𝐴𝐵(𝛼) 4𝐷𝐿 𝐴𝐵(𝛼)
2𝑣𝑥  (𝛿
𝛼
𝑚,𝑖(2𝜃 − 1) + (1 − cos𝜙)𝛽𝑚,𝐸 ) + > 𝛽  (1 − 𝛼) 𝛼,2 (∆𝑥)2 (1 − 𝛼) 𝑛,𝐸𝛼,2
𝐴𝐵(𝛼) 2𝐷𝐿 𝐴𝐵(𝛼)
2𝑣𝑥 ((1 − 𝜃)𝛿
𝛼
𝑚,𝑖 + (1 − cos𝜙)𝛽𝑚,𝐸 ) − > (2𝛿
𝛼
𝑛,𝑛−1 + 𝛽𝑛,𝐸 ) (1 − 𝛼) 𝛼,2 (∆𝑥)2 (1 − 𝛼) 𝛼,2
Additional conditions are activated when the weighting factor is 0 ≤ 𝜃 < 0.5, 
𝐴𝐵(𝛼) 2𝐷𝐿
(2𝜃 − 1)𝑣 𝛼𝑥 𝛿(1 − 𝛼) 𝑚,𝑖
+ > 0, 
(∆𝑥)2
𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 2𝐷𝐿
𝑣𝑥(2𝜃 − 1) 𝛿
𝛼
𝑚,𝑖 + 𝑣𝑥(1 − cos𝜙) 𝛽𝑚,𝐸 + > 0 (1 − 𝛼) (1 − 𝛼) 𝛼,2 (∆𝑥)2
Only under these conditions, the error of the approximation made by the implicit upwind-downwind 
weighted numerical scheme is not propagated throughout the solution. 
186 
 
6.3.6 Evaluation of numerical stability results 
The stability conditions for the traditional upwind (implicit/explicit), upwind advection Crank-Nicolson 
and weighted upwind-downwind (implicit/explicit) numerical schemes are summarised in Table 6-1. 
The traditional implicit upwind scheme applied to the fractional advection-dispersion equation with 
ABC fractional derivative is conditionally stable under both assumptions made with a single condition 
for each assumption. There is only one practically applicable assumption for the customary explicit 
upwind scheme, which has three sub-conditions. The upwind advection Crank-Nicolson numerical 
scheme applied to the fractional advection-dispersion equation with ABC fractional derivative is 
unconditionally stable under the first assumption, and has a single condition under the second 
assumption made. The implicit upwind-downwind weighted numerical scheme is conditionally stable 
under both assumptions made, but unconditionally stable for the first assumption when the weighting 
factor is 0.5 ≤ 𝜃 ≤ 1. Similar to the explicit upwind scheme, the explicit upwind-downwind weighted 
numerical scheme has one practically applicable assumption, which has three conditions for stability.  
Of the numerical schemes analysed, the upwind advection Crank-Nicolson is the most stable 
numerical scheme, and would be suggested for use with the fractional advection-dispersion equation 
(ABC). The implicit upwind formulations are found to be more stable than the comparable explicit 
formulations. The proposed weighted implicit upwind-downwind scheme is more stable than the 
traditional upwind scheme when the weighting factor is 0.5 ≤ 𝜃 ≤ 1, which denotes at least half 
upwind weighted or more, with the downwind influence less than half. 
6.4  Chapter summary 
To improve the governing equation for groundwater transport modelling, the Atangana-Baleanu in 
Caputo sense (ABC) fractional derivative is applied to the advection-dispersion equation with a focus 
on the advection term to account for anomalous advection. Boundedness, existence and uniqueness 
for the developed advection-focused transport equation is presented. In addition, a semi-
discretisation analysis is performed to demonstrate the equation stability in time. The augmented 
upwind schemes are developed to facilitate the solution of the complex equation, and stability 
analysis conducted. The numerical stability analysis found the upwind Crank-Nicolson to be the most 
stable, and is thus recommended for use with the fractional advection-dispersion equation with ABC 
fractional derivative. 
 
187 
 
Table 6-1 Summary of the stability conditions for the upwind-based numerical schemes for the fractional advection-dispersion equation with ABC derivative 
Scheme Assumption Stability conditions 
𝐴𝐵(𝛼) 𝐷 2𝐷
𝑣 𝛿𝛼
𝐿 𝐿
𝑚,𝑖 >  2𝑣(𝛿
𝛼 + (1 − cos𝜙)𝛽 ) + > 𝛽  
Implicit (1 − 𝛼) (∆𝑥)2
𝑚,𝑖 𝑚,𝐸𝛼,2 (∆𝑥)2 𝑛,𝐸𝛼,2
upwind 𝐴𝐵(𝛼) 𝐷𝐿 4𝐷𝐿
𝑣 𝛿𝛼𝑚,𝑖 <  2𝑣(1 − cos𝜙)𝛽 + > 𝛽  ( 𝑚,𝐸1 − 𝛼) (∆𝑥)2 𝛼,2 (∆𝑥)2
𝑛,𝐸𝛼,2
Assumption not made Unstable 
Explicit 
𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 2𝐷
upwind 𝛼 𝐿
2𝐴𝐵(𝛼)
𝛿𝑛,𝑛−1 <  𝑣 𝛿
𝛼 +  𝑚𝑎𝑥(𝜆, 𝜇, 𝜌) < 𝛿𝛼  
(1 − 𝛼) (1 − 𝛼) 𝑚,𝑖 (∆𝑥)2 (1 − 𝛼) 𝑛,𝑛−1
( )
Upwind 𝐴𝐵 𝛼 𝐷0.5𝑣 𝛿𝛼
𝐿
𝑚,𝑖 >  Unconditionally stable  2
advection (1 − 𝛼) (∆𝑥)
Crank- 𝐴𝐵(𝛼) 𝐷 𝐴𝐵(𝛼) 4𝐷
𝛼 𝐿 𝛼 𝐿
Nicolson 0.5𝑣 𝛿 <  𝑣 𝛿(1 − 𝛼) 𝑚,𝑖 (∆𝑥)2 (1 − 𝛼) 𝑚,𝑖
<  
(∆𝑥)2
𝐴𝐵(𝛼) 2𝐷
𝛼 𝐿(2𝜃 − 1)𝑣 𝛿 + > 0 
(1 − 𝛼) 𝑚,𝑖 (∆𝑥)2
𝐴𝐵(𝛼)𝑣 𝐴𝐵(𝛼)𝑣 2𝐷
𝐷𝐿 𝐴𝐵(𝛼) 𝐴𝐵(𝛼)
𝐿
𝛼 𝛼 (2𝜃 − 1) 𝛿
𝛼
𝑚,𝑖 + (1 − cos𝜙) 𝛽 + > 0 + 𝑣𝜃 𝛿  
2  𝑚,𝑖
> 𝑣 𝛿  
 𝑚,𝑖 (1 − 𝛼) (1 − 𝛼)
 𝑚,𝐸𝛼,2 (∆𝑥)2
(∆𝑥) (1 − 𝛼) (1 − 𝛼) 𝐴𝐵(𝛼) 4𝐷 𝐴𝐵(𝛼)
Implicit 𝐿2𝑣 (𝛿𝛼𝑚,𝑖(2𝜃 − 1) + (1 − cos𝜙)𝛽 ) + > 𝛽  (1 − 𝛼) 𝑚,𝐸𝛼,2 (∆𝑥)2 (1 − 𝛼) 𝑛,𝐸𝛼,2upwind-
downwind *Unconditionally stable when 0.5 ≤ 𝜃 ≤ 1, or conditionally stable when 0 ≤ 𝜃 < 0.5 
weighted 𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 𝐷
𝛼 𝛼 𝛼 𝐿
scheme  𝑣 𝛿𝑚,𝑖 > (𝛿𝑛,𝑛−1 + 𝑣𝜃𝛿(1 − 𝛼) 𝑚,𝑖
) +  
(1 − 𝛼) (∆𝑥)2
𝐷𝐿 𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 𝐷
+ 𝑣𝜃 𝛿𝛼 < 𝑣 𝛿𝛼 𝛼 𝛼 𝛼
𝐿
(∆𝑥)2 (1 − 𝛼) 𝑚,𝑖 (1 − 𝛼) 𝑚,𝑖
 𝑣(𝛽𝑚,𝐸 + 𝛿𝑚,𝑖) > (𝛿𝑛,𝑛−1 + 𝑣𝜃𝛿𝑚,𝑖 + 𝑣 cos𝜙 𝛽𝛼,2 (1 − 𝛼) 𝑚,𝐸
) +  
𝛼,2 (1 − 𝛼) (∆𝑥)2
𝐴𝐵(𝛼) 2𝐷𝐿 𝐴𝐵(𝛼)
2𝑣 ((1 − 𝜃)𝛿𝛼𝑚,𝑖 + (1 − cos𝜙)𝛽 ) − > (2𝛿
𝛼
𝑚,𝐸 + 𝛽 )  𝛼,2 (1 − 𝛼) (∆𝑥)2 𝑛,𝑛−1 𝑛,𝐸𝛼,2 (1 − 𝛼) 
Assumption not made Unstable 
Explicit 𝐴𝐵(𝛼) 2𝐷𝐿 2𝐴𝐵(𝛼)
2𝑣(𝜃 − 1) 𝛿𝛼𝑚,𝑖 + > 𝛿
𝛼
upwind- (1 − 𝛼) (∆𝑥)2 (1 − 𝛼) 𝑛,𝑛−1
 
𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 2𝐷
downwind 𝛿𝛼
𝐿
𝑛,𝑛−1 + 𝑣 𝛿
𝛼
𝑚,𝑖 < 2𝑣𝜃 𝛿
𝛼
𝑚,𝑖 +  2𝐴𝐵(𝛼) 2𝐴𝐵(𝛼) 4𝐷   2 𝛼 𝛼 𝛼 𝐿
weighted (1 − 𝛼) (1 − 𝛼) (1 − 𝛼) (∆𝑥) (𝛿𝑛,𝑛−1 + 𝑣𝛿𝑚,𝑖 + 𝑣 cos𝜙 𝛽𝑚,𝐸 ) > (2𝑣𝜃𝛿 + 𝑣𝛽 ) +  𝛼,2 (1 − 𝛼) 𝑚,𝑖 𝑚,𝐸𝛼,2 (1 − 𝛼) (∆𝑥)2
scheme   2𝐴𝐵(𝛼) 𝐴𝐵(𝛼) 4𝐷𝐿
(𝛿𝛼 𝛼𝑛,𝑛−1 + 𝑣𝛿𝑚,𝑖 + 𝑣 cos𝜙 𝛽𝑚,𝐸 ) > (2𝑣𝛽
𝛼
𝛼,2 (1 − 𝛼) 𝑚,𝐸
+ 4𝑣𝜃𝛿 + 𝛽 ) +  
𝛼,2 𝑚,𝑖 𝑛,𝐸𝛼,2 (1 − 𝛼) (∆𝑥)2
188 
 
 
7  FRACTIONAL-FRACTAL  
 ADVECTION-DISPERSION EQUATION 
 
A fractal advection-dispersion equation and a fractional space-time advection-dispersion equation has 
been developed thus far to improve the simulation of groundwater transport specifically in fractured 
aquifers. However, the practicality of simulating a space-time fractional advection-dispersion 
equation is limited due to the computational power required and complexity of the required 
algorithms. On the other hand, the fractal advection-dispersion equation only considered fractal 
derivatives in space, and not time. Considering these two observations, it is theorised that combining 
these two approaches, fractional derivative in time and fractal derivative in space, could be the most 
efficient way of incorporating non-locality into both time and space components.  
Fractional and fractal derivatives have been considered together by many authors, but mostly 
separately for comparison and not fully combined in a single model (Nyikos and Pajkossy, 1985; Berry 
and Klein, 1996; Hilfer and Anton, 1995; El-Nabulsi, 2010; Chen et al., 2010a; Chen et al., 2010b; Sun 
et al., 2013; Chen and Liang, 2017; Sun et al., 2017b). Chen et al. (2010a) compared a fractal and 
fractional diffusion model, and found that both models can characterise the power law phenomena 
of anomalous diffusion, yet represent different processes from a statistical perspective. Namely, a 
stretch Gaussian process for the fractal model, and a Levy process for the fractional model. An 
important difference defined in terms of practical simulation, was that the fractal model is a local 
189 
 
operator and can be solved with local approximation techniques more efficiently than the fractional 
model.   
More recently, a few authors have combined the fractional and fractal derivative in a model or 
description (Feng, 2000; Jiang et al., 2010; Chen et al., 2012; Wang et al., 2015; Fan et al., 2016; 
Atangana, 2017; Yadav and Agarwal, 2019). Fan et al. (2016) develop a fractional-fractal diffusion 
model based on work done by Jiang et al. (2010), using Riemann-Liouville and Caputo fractional 
derivatives. The fractional-fractal model is applied to the diffusion process of methane gas in coal 
beds, and was found to capture anomalous diffusion in porous media. Additionally, a Bayesian method 
was applied to determine the fractional order and fractal dimension automatically, which did 
correspond with experimental data. Atangana (2017) developed new fractal-fractional differential and 
integral operators, and concludes with the new operators will provide tools for future investigations. 
Furthermore, superdiffusion and subdiffusion in heterogeneous aquifers were suggested as a specific 
application. A space-time fractional-fractal Boussinesq equation was developed Yadav and Agarwal 
(2019) to improve the simulation of groundwater flow in unconfined systems, where a Caputo 
fractional derivative was applied in time and the fractal derivative developed in this thesis applied to 
space.  
The application of a fractional-fractal advection-dispersion equation for transport in fractured aquifers 
is thus validated considering the application of the fractional-fractal diffusion models applied to 
anomalous gas diffusion and unconfined groundwater flow. 
7.1 Fractional-fractal advection-dispersion equation 
The fractional-fractal advection-dispersion equation is developed by applying the Caputo fractional 
derivative in time, and the fractal derivative as described in Section 3.2.1 in space for both the 
advection and dispersion terms. There are two non-local parameters, namely the fractional order (𝛼) 
and the fractal dimension, now denoted as 𝛽. The fractional-fractal equation can thus be defined as: 
2
𝐶 𝛼 𝛽
𝜕 𝜕
0𝐷𝑡  (𝑐(𝑥, 𝑡)) = 𝑉𝐹 (𝑥) 𝑐(
𝛽
𝑥, 𝑡) + 𝐷𝐹 (𝑥) 𝑐(𝑥, 𝑡) (7-1) 𝜕𝑥 𝜕𝑥2
Where, 
𝐶 𝛼
0𝐷𝑡  denotes the Caputo fractional derivative 
𝛽
𝑉𝐹  denotes the fractal velocity in the x-direction 
𝛽
𝐷𝐹  denotes the fractal dispersivity in the x-direction 
 
190 
 
7.2 Numerical approximation 
The typical numerical approximation is applied to the Caputo fractional derivative in time, as described 
in Section 5.1. A finite difference method is applied to approximate the advective term (central 
difference) and the dispersion term (standard second-order difference), resulting in: 
𝑛−1
(∆𝑡)−𝛼 𝑐𝑛−1 − 𝑐𝑛−1 𝑐𝑛−1 − 2𝑐𝑛−1 + 𝑐𝑛−1
𝑘+1 𝑘 𝛼 𝛽 𝛽 𝑚 [∑(𝑐𝑚 − 𝑐𝑚)𝛿𝑛,𝑘] = 𝑉
𝑚+1 𝑚−1
𝐹 ( ) + 𝐷
𝑚+1 𝑚−1
( ) (7-2) 
Γ(2 − 𝛼) 2∆𝑥 𝐹 (∆𝑥)2
𝑘=0
where, 
𝛿𝛼 1−𝛼 1−𝛼𝑛,𝑘 = (𝑛 − 𝑘) − (𝑛 − 𝑘 − 1)  
𝑥1−𝛼𝛽 (1 − 𝛼)
𝑉 = −𝑣 ( ) + 𝐷 ( )𝑥1−2𝛼𝐹 𝐿  𝛼 𝛼
𝑥2−2𝛼𝛽
𝐷𝐹 = 𝐷𝐿 ( 2 ) 𝛼
An algorithm is developed in Maple to solve the fractional-fractal equation using the numerical 
approximation scheme as described. The first step where 𝑛 = 1 is defined as: 
𝛽 𝑐[𝑚 + 1,0] − 𝑐[𝑚 − 1,0]
𝑉𝐹 ( )Γ(2 − 𝛼) 2∆𝑥  
𝑐[𝑚, 1] =   + 𝑐[𝑚, 0] (7-3) (∆𝑡)−𝛼 𝛽 𝑐[𝑚 + 1,0] − 2𝑐[𝑚, 0] + 𝑐[𝑚 − 1,0]
+ 𝐷𝐹 ( )
( (∆𝑥)
2
)
And, the recursive step where 𝑘 =  𝑛 − 1 is defined as: 
𝛽 𝑐[𝑚 + 1, 𝑛 − 1] − 𝑐[𝑚 − 1, 𝑛 − 1] 𝑉 ( ) 
1  Γ(2 − 𝛼) 
𝐹 2∆𝑥  
𝑐[𝑚, 𝑛] =   
(𝑛 − 1)1−𝛼  (∆𝑡)−𝛼  𝛽 𝑐[𝑚 + 1, 𝑛 − 1] − 2𝑐[𝑚, 0] + 𝑐[𝑚 − 1,0]
 
 + 𝐷𝐹 ( )
[ ( (∆𝑥)
2
) (7-4) 
𝑛−1  
 
−∑(𝑐[𝑚, 𝑘 + 1] − 𝑐[𝑚, 𝑘])(𝑛 − 𝑘)1−𝛼 − (𝑛 − 𝑘 − 1)1−𝛼 + 𝑐[𝑚, 𝑛 − 1] 
𝑘=0  
]
7.3 Relationship between fractional and fractal dimensions 
A fractional in time and fractal in space formulation of the advection-dispersion equation has two 
parameters to consider, namely the fractional order (𝛼) and the fractal dimension (𝛽). An investigation 
of the fractal velocity/dispersivity was conducted in Section 3.6, and found an exponential increase in 
the fractal velocity and dispersivity as the fractal dimension decreases. From this initial evaluation, it 
was suggested to use fractal dimensions below 0.5 with caution as the increase becomes ultra-
advection. A similar trend in the fractal dimension influence on the fractional-fractal 
191 
 
velocity/dispersivity is expected, but the additional influence of the fractional order requires 
exploration. To achieve this, variable fractal dimensions are plot for a series of fractional orders, i.e. 
𝛼, 𝛽: {0.9, 0.7, 0.5, 0.3, 0.1}. To facilitate this analysis, a fractional-fractal velocity/dispersivity 
(𝑉𝛼𝐹𝐹;  𝐷
𝛼
𝐹𝐹) is represented by applying the numerical approximation in Equation (7-4), 
Γ(2 − 𝛼) 𝑥(1−𝛽) 1 − 𝛽 𝑥(2−2𝛽)  
((−𝑣( )+𝐷 ( )𝑥(1−2𝛽))+𝐷 ( )) (7-5) 
(∆𝑡)−𝛼 𝛽 𝐿 𝛽 𝐿 𝛽2
where, 
∆𝑡 is the time step size 
𝛼 is the fractional order 
𝛽 is the fractal dimension 
𝑣 is the groundwater velocity 
𝐷𝐿 is the longitudinal hydrodynamic dispersivity 
For the analysis, a constant groundwater velocity (𝑣) is defined at 0.05 m/d, dispersivity (𝐷𝐿) of 0.3 
m2/d, and a fractional time step (∆𝑡) of 0.01 d. The first plot of 𝛼 = 1;  𝛽 = 1, forms the comparison 
base because this represents the traditional advection-dispersion model with no fractional order or 
fractal dimension (Figure 7-1). Initially the fractional-fractal velocity over space for a fractional order 
𝛼 = 1 (simplifying back to the local equation) and varying fractal dimensions (0.1 ≤ 𝛽 ≤ 1) is plot on 
a variable scale. These plots only assess the influence of the fractal derivative in space as the fractional 
order simplifies to the classical advection-dispersion equation in time, and as expected the influence 
of the fractal dimension follows the same trend as in Section 3.6 for the fractal advection-dispersion 
equation. However, the exponential increase is subdued due to the small fractional time step and 
discretisation (Figure 3-6 and Figure 7-1). In the subsequent plots (Figure 7-2 to Figure 7-6), the 
variable fractal dimension is evaluated at a series of fractional orders, 𝛼 = 0.9, 0.7, 0.5, 0.3, 0.1. Again, 
varying the fractal dimension produces the same trend at each fractional order, but a growth factor 
(on average 3) is found between each decrement of the fractional order. Thus, the fractional-fractal 
velocity/dispersivity increases not only exponentially with a decrease in the fractal order, but also with 
a decrease in the fractional order.  
The established relationship between the fractional order and fractal dimension with the fractional-
fractal velocity/dispersivity highlights the importance of selecting appropriate combinations. 
Considering the suggestion of using fractal dimensions of greater than 0.5 in Section 3.6, the range of 
appropriate fractional orders and fractal dimensions for the combined fractional-fractal model should 
be stricter due to the cumulative increase in velocity/dispersion. Thus, a recommendation of fractional 
order above 0.5, and a fractal dimension above 0.7 would be suggested for practical applications.  
192 
 
𝛼 = 1;  𝛽 = 1 𝛼 = 1;  𝛽 = 0.9 
𝛼 = 1;  𝛽 = 0.7 𝛼 = 1;  𝛽 = 0.5 
𝛼 = 1;  𝛽 = 0.3 𝛼 = 1;  𝛽 = 0.1 
 
Figure 7-1 Fractional-fractal velocity over space for a fractional order 𝜶 = 𝟏 (simplifying back to the local 
equation in time) and varying fractal dimensions (𝟎. 𝟏 ≤ 𝜷 ≤ 𝟏) on a variable plot scale. These plots solely 
evaluate the influence of the fractal derivative in space, and thus form a base of comparison, and the plot 
of 𝜶 = 𝟏 ; 𝜷 = 𝟏 represents the traditional-local model in both time and space. 
 
193 
 
Fractional-Fractal velocity/dispersivity 
Fractional-Fractal velocity/dispersivity Fractional-Fractal velocity/dispersivity 
𝛼 = 0.9;  𝛽 = 1 𝛼 = 0.9;  𝛽 = 0.9 
𝛼 = 0.9;  𝛽 = 0.7 𝛼 = 0.9;  𝛽 = 0.5 
𝛼 = 0.9;  𝛽 = 0.3 𝛼 = 0.9;  𝛽 = 0.1 
 
Figure 7-2 Fractional-fractal velocity over space for a fractional order 𝜶 = 𝟎. 𝟗 and varying fractal 
dimensions (𝟎. 𝟏 ≤ 𝜷 ≤ 𝟏) on a variable plot scale.  
 
194 
 
Fractional-Fractal velocity/dispersivity Fractional-Fractal velocity/dispersivity 
Fractional-Fractal velocity/dispersivity 
 
𝛼 = 0.7;  𝛽 = 1 𝛼 = 0.7;  𝛽 = 0.9 
𝛼 = 0.7;  𝛽 = 0.7 𝛼 = 0.7;  𝛽 = 0.5 
𝛼 = 0.7;  𝛽 = 0.3 𝛼 = 0.7;  𝛽 = 0.1 
 
Figure 7-3 Fractional-fractal velocity over space for a fractional order 𝜶 = 𝟎. 𝟕 and varying fractal 
dimensions (𝟎. 𝟏 ≤ 𝜷 ≤ 𝟏) on a variable plot scale. 
 
 
195 
 
Fractional-Fractal velocity/dispersivity Fractional-Fractal velocity/dispersivity Fractional-Fractal velocity/dispersivity 
 
𝛼 = 0.5;  𝛽 = 1 𝛼 = 0.5;  𝛽 = 0.9 
𝛼 = 0.5;  𝛽 = 0.5 
𝛼 = 0.5;  𝛽 = 0.7 
𝛼 = 0.5;  𝛽 = 0.1 
𝛼 = 0.5;  𝛽 = 0.3 
 
Figure 7-4 Fractional-fractal velocity over space for a fractional order 𝜶 = 𝟎. 𝟓 and varying fractal 
dimensions (𝟎. 𝟏 ≤ 𝜷 ≤ 𝟏) on a variable plot scale. 
 
 
196 
 
Fractional-Fractal velocity/dispersivity 
Fractional-Fractal velocity/dispersivity Fractional-Fractal velocity/dispersivity 
 
𝛼 = 0.3;  𝛽 = 1 𝛼 = 0.3;  𝛽 = 0.9 
𝛼 = 0.3;  𝛽 = 0.7 𝛼 = 0.3;  𝛽 = 0.5 
𝛼 = 0.3;  𝛽 = 0.3 𝛼 = 0.3;  𝛽 = 0.1 
 
Figure 7-5 Fractional-fractal velocity over space for a fractional order 𝜶 = 𝟎. 𝟑 and varying fractal 
dimensions (𝟎. 𝟏 ≤ 𝜷 ≤ 𝟏) on a variable plot scale. 
 
197 
 
Fractional-Fractal velocity/dispersivity 
Fractional-Fractal velocity/dispersivity Fractional-Fractal velocity/dispersivity 
 
𝛼 = 0.1;  𝛽 = 1 𝛼 = 0.1;  𝛽 = 0.9 
𝛼 = 0.1;  𝛽 = 0.5 
𝛼 = 0.1;  𝛽 = 0.7 
𝛼 = 0.1;  𝛽 = 0.1 
𝛼 = 0.1;  𝛽 = 0.3 
 
Figure 7-6 Fractional-fractal velocity over space for a fractional order 𝜶 = 𝟎. 𝟏 and varying fractal 
dimensions (𝟎. 𝟏 ≤ 𝜷 ≤ 𝟏) on a variable plot scale. 
198 
 
Fractional-Fractal velocity/dispersivity 
Fractional-Fractal velocity/dispersivity 
Fractional-Fractal velocity/dispersivity 
7.4 Simulation 
To test the recommendations made for the fractional order and fractal dimension, a fractional-fractal 
model is simulated for a simple transport problem in the software programme Maple (Appendix B). 
The simple one-dimensional transport problem used in Section 3.5 is applied again, and simulated in 
the software programme Maple (Figure 2-1). The parameters include a constant velocity within the 
aquifer at 0.5 m/d, longitudinal hydrodynamic dispersivity is 0.3 m2/d, and the initial contaminant 
concentration (C0) is 10 mg/l. The initial condition and boundary conditions for the defined problem 
are 
 
The presence of two fractional-fractal parameters (𝛼 and 𝛽) increases the number of combinations 
which are possible as seen in Section 7.3. The fractional-fractal velocity/dispersivity is highly sensitive 
to these parameters and a simulation of the potential combinations is simulated to investigate this 
relationship. The fractional-fractal model is applied to the simple transport model for fractional 
orders, 𝛼 = 0.9, 0.8, 0.7, 0.6, 0.5 and for each, the fractal dimension is varied from 0.9 to where the 
simulation becomes unstable or unrealistic. Graphical illustrations of the simulations are presented in 
Figure 7-7 to Figure 7-11, and the suitable 𝛼, 𝛽 combinations tabulated in Table 7-1. When the 
fractional order and fractal dimension are 1, the solution represents the classical advection-dispersion 
model, where the movement of contaminants form a symmetrical breakthrough curve along the x-
direction (Figure 7-7). When the fractional order is at 𝛼 = 1, and the derivative in time becomes an 
integer-order, the fractal dimension is appropriate from 0.5 ≤ 𝛽 ≤ 0.9, after which the solution 
becomes unstable. Varying the fractal dimension changes the shape of the breakthrough curve, 
changing from a Gaussian distribution, to a skewed distribution with a progressively heavier tail. In 
Figure 7-8, the fractional order is 𝛼 = 0.9, and the fractal dimension is appropriate from  0.5 ≤ 𝛽 ≤
0.9. The same trend is seen as the fractal dimension is decreased, yet the peak concentration in the 
breakthrough curve is more pronounced. It appears that the additional influence of a fractional order 
in time on a fractal dimension in space advection-dispersion model is a control on the breakthrough 
curve peak, while the fractal dimension controls the degree of tailing of the curve. These two controls 
could provide the tools to better represent measured anomalous breakthrough curves that cannot be 
fit to the classical model. The same trend is seen in Figure 7-9 for fractional order 𝛼 = 0.8, and in 
Figure 7-10 for fractional order 𝛼 = 0.7.  
199 
 
𝜶 = 𝟏;  𝜷 = 𝟏 𝜶 = 𝟏;  𝜷 = 𝟎. 𝟗 
10 mg/l 
0 mg/l 
𝑡 𝑡 
𝜶 = 𝟏;  𝜷 = 𝟎. 𝟖 𝜶 = 𝟏;  𝜷 = 𝟎. 𝟕 
𝑡 𝑡 
𝜶 = 𝟏;  𝜷 = 𝟎. 𝟔 𝜶 = 𝟏;  𝜷 = 𝟎. 𝟓 
𝑡 𝑡 
 
Figure 7-7 Simulation results illustrated for distance (m) in the x-direction along a line over time (d) for the 
fractional order 𝜶 = 𝟏 (simplifies to a local order), and varying fractal dimensions (𝟎. 𝟓 ≤ 𝜷 ≤ 𝟏)  
200 
 
𝑥 𝑥 
𝑥 
𝑥 𝑥 𝑥 
𝜶 = 𝟏;  𝜷 = 𝟏 𝜶 = 𝟎. 𝟗;  𝜷 = 𝟎. 𝟗 
10 mg/l 
0 mg/l 
𝑡 𝑡 
𝜶 = 𝟎. 𝟗;  𝜷 = 𝟎. 𝟖 𝜶 = 𝟎. 𝟗;  𝜷 = 𝟎. 𝟕 
𝑡 𝑡 
𝜶 = 𝟎. 𝟗;  𝜷 = 𝟎. 𝟔 𝜶 = 𝟎. 𝟗;  𝜷 = 𝟎. 𝟓 
𝑡 𝑡 
 
Figure 7-8 Simulation results illustrated for distance (m) in the x-direction along a line over time (d) for the 
fractional order 𝜶 = 𝟎. 𝟗, and varying fractal dimensions (𝟎. 𝟓 ≤ 𝜷 ≤ 𝟏) 
201 
 
𝑥 𝑥 𝑥 
𝑥 𝑥 𝑥 
𝜶 = 𝟏;  𝜷 = 𝟏 𝜶 = 𝟎. 𝟖;  𝜷 = 𝟎. 𝟗 
10 mg/l 
0 mg/l 
𝑡 𝑡 
𝜶 = 𝟎. 𝟖;  𝜷 = 𝟎. 𝟖 𝜶 = 𝟎. 𝟖;  𝜷 = 𝟎. 𝟕 
𝑡 𝑡 
𝜶 = 𝟎. 𝟖;  𝜷 = 𝟎. 𝟔 
𝑡 
 
Figure 7-9 Simulation results illustrated for distance (m) in the x-direction along a line over time (d) for the 
fractional order 𝜶 = 𝟎. 𝟖, and varying fractal dimensions (𝟎. 𝟔 ≤ 𝜷 ≤ 𝟏) 
202 
 
𝑥 𝑥 
𝑥 
𝑥 𝑥 
However, the range of appropriate fractal dimensions decreases systematically. For simulations 𝛼 =
0.9, 𝛽 = 0.5; 𝛼 = 0.8, 𝛽 = 0.6; and 𝛼 = 0.7, 𝛽 = 0.7 the influence of the exponential distribution of 
the fractional-fractal velocity/dispersivity becomes evident, where near the end of the x-directional 
line the velocity and dispersivity increase exponentially and the concentrations are preferentially 
peaked at this location at the beginning of the simulator time. This eventually causes instabilities as 
this rapid movement along the line becomes unrealistic. The result of this are clearly seen in 
simulations  𝛼 = 0.6, 𝛽 = 0.8 and 𝛼 = 0.5, 𝛽 = 0.9 where all the contaminants accumulate at the end 
of the x-direction line at the beginning of the simulation time (Figure 7-11).  
𝜶 = 𝟏;  𝜷 = 𝟏 𝜶 = 𝟎. 𝟕;  𝜷 = 𝟎. 𝟗 
10 mg/l 
0 mg/l 
𝑡 𝑡 
𝜶 = 𝟎. 𝟕;  𝜷 = 𝟎. 𝟖 𝜶 = 𝟎. 𝟕;  𝜷 = 𝟎. 𝟕 
𝑡 𝑡 
 
Figure 7-10 Simulation results illustrated for distance (m) in the x-direction along a line over time (d) for the 
fractional order 𝜶 = 𝟎. 𝟕, and varying fractal dimensions (𝟎. 𝟕 ≤ 𝜷 ≤ 𝟏) 
203 
 
𝑥 𝑥 
𝑥 𝑥 
𝜶 = 𝟏;  𝜷 = 𝟏 𝜶 = 𝟎. 𝟔;  𝜷 = 𝟎. 𝟗 
10 mg/l 
0 mg/l 
𝑡 𝑡 
𝜶 = 𝟎. 𝟔;  𝜷 = 𝟎. 𝟖 𝜶 = 𝟎. 𝟓;  𝜷 = 𝟎. 𝟗 
𝑡 𝑡  
Figure 7-11 Simulation results illustrated for distance (m) in the x-direction along a line over time (d) for the 
fractional order 𝜶 = 𝟎. 𝟔, 𝟎. 𝟓, and varying fractal dimensions (𝟎. 𝟖 ≤ 𝜷 ≤ 𝟏) 
Table 7-1 Tabulation of the fractional order (α) and fractal dimension (β) valid combination options for the 
fractional-fractal advection-dispersion equation. Valid combinations (x), Non-valid combinations (grey), 
recommended combinations (green). 
  β = 1 β = 0.9 β = 0.8 β = 0.7 β = 0.6 β = 0.5 β = 0.4 
α = 1 x x x x x x   
α = 0.9 x x x x x x   
α = 0.8 x x x x x     
α = 0.7 x x x x       
α = 0.6 x x x         
α = 0.5 x x           
α = 0.4               
204 
 
𝑥 𝑥 
𝑥 𝑥 
Thus, the recommendations made based on the relationship of the fractional-fractal parameters to 
the fractional-fractal velocity/dispersivity are validated, where fractional order above 0.5, and a fractal 
dimension above 0.7 would be suggested for practical applications. Yet, this recommendation is 
refined based on the simulations to a fractional order and fractal dimension of 0.7 ≤ 𝛼, 𝛽 ≤ 1. 
7.5 Chapter summary 
A fractional-fractal advection-dispersion model is developed from the fractional and fractal advection-
dispersion equations already considered, producing an efficient tool to model anomalous diffusion 
with the same non-local advantages. However, the fractional-fractal model has two parameters to 
consider, the fractional order (𝛼) and the fractal dimension (𝛽), and this increases the number of 
combinations available for simulation. An investigation on the influence of the fractional-fractal 
parameters on the fractional-fractal velocity/dispersivity found the fractional-fractal 
velocity/dispersivity increases not only exponentially with a decrease in the fractal order, but also with 
a decrease in the fractional order. A recommendation to use a fractional order above 0.5, and a fractal 
dimension above 0.7 was made based on the evaluation. The established relationship highlighted the 
importance of selecting appropriate combinations, and validated simulating the combinations for a 
simple transport problem. From the simulations, it is found that the fractional order controls the 
breakthrough curve peak, and the fractal dimension controls the position of the peak and tailing 
effect. These two controls potentially provide the tools to better represent measured anomalous 
breakthrough curves that cannot be fit to the classical model. The range of valid combinations 
decrease with decreasing fractional order and fractal dimension, and a final recommendation is made 
for a fractional order and fractal dimension of 0.7 ≤ 𝛼, 𝛽 ≤ 1. 
 
 
  
205 
 
 
8  DISCUSSION 
 
The endeavour to improve the representation of natural systems, to increase comprehension and 
management of those systems will be continued perpetually beyond this research because the 
complexity of natural phenomena is endless with many more facets to explore. Yet, each step made 
in progress improves the collective understanding and furthers the cause. Research conducted in this 
thesis is but a small step forward, but a step forward now the less. The numerical schemes and 
reformulated advection-dispersion equations will improve how groundwater transport, and 
potentially other systems, are simulated. Yet, it is important to understand the assumptions and 
limitations associated with these improvements.  
The numerical schemes developed to improve the classical advection-dispersion model are evaluated 
for numerical stability and application to a simple transport problem of limited extent and time. While 
this is appropriate from a research perspective, it should be noted that a full-scale application with 
measured observations has not been tested. For further applications, the augmented upwind 
numerical schemes should be extended to three-dimensions and tested on a large-scale testing facility 
for contaminant transport.  
The fractal advection-dispersion equation developed has the ability to model superdiffusion for fractal 
dimension less than 1, as well as subdiffusion for fractal dimensions greater than 1, without explicitly 
defining fractures or preferential pathways. The fractal advection-dispersion model is simulated on a 
206 
 
slightly larger scale of 200 m and 200 days due to local numerical methods being used. Yet, this is still 
limited and has not been applied with measured anomalous transport data. Similarly, for further 
applications, the fractal advection-dispersion model should be extended to three-dimensions and 
tested on a large-scale testing facility for contaminant transport.  
Fractional advection-dispersion equations are developed for fractured groundwater systems using the 
Caputo and Atangana-Baleanu fractional derivatives. The improvement from previous fractional 
advection-dispersion equations is the incorporation of the fractional derivative in time and space with 
the advection term in space as opposed to the typical application to the diffusion/dispersion term. 
The upwind schemes developed for the classical advection-dispersion equation are tested for the 
fractional advection-dispersion models in terms of numerical stability. Proving the numerical stability 
is important, but the simulation of these equations to a practical problem is restricted due to 
computational limitations, where a memory component in both space and time requires greater 
computational effort. For future analysis, the numerical schemes should be simulated for a transport 
problem to provide a complete evaluation. This particular limitation motivated the development of 
the fractional-fractal advection-dispersion model because the fractional in time and fractal in space 
provides similar advantages with less computational effort.  
The fractional-fractal advection-dispersion equation facilitates simulation on a simple transport 
problem to evaluate the combinations of the fractional order and fractal dimension. However, the 
computational effort is larger than that of the classical approach and the extent and time of the 
transport problem is limited, and only in one physical dimension (𝑥). The fractional-fractal is the most 
appropriate tool in terms of generally available computing power and in future research should be 
applied in three-dimensions to a field-scale transport modelling application.  
In general, a wide spread use of non-local approaches to solve practical problems remains hindered 
by the lack of a physical definition and the computational power required to simulate the intrinsic 
memory of fractional derivatives and integrals at large-scales. When our understand of fractional 
calculus becomes sufficiently tangible and computer processors advance, the use of fractional 
derivatives will change the way we model our world. Until then, small advancing steps, such as the 
research done here, will strive forward until this objective is reached.  
 
 
 
207 
 
 
9 CONCLUSIONS 
 
The complexity of real world systems inspire scientists to continually advance the methods used to 
represent these systems as understanding and technology advance. This fundamental principle is 
applied to groundwater transport, a real world problem where the current understanding often 
cannot describe what is observed. To improve the simulation of groundwater transport there are two 
approaches, 1) improve the physical characterisation of the heterogeneous system, or 2) improve the 
formulation of the governing equations used to simulate the system. The latter approach is followed 
by investigating numerical approximation schemes for the classical advection-dispersion equation, 
incorporating fractal and fractional derivatives into the formulation, and combining fractional and 
fractal derivatives.       
9.1 Local advection-dispersion equation 
The solution of the classical advection-dispersion equation for fractured groundwater systems is 
improved by developing augmented upwind finite difference numerical approximation schemes, 
which are better suited for advection-dominated systems. A numerical scheme combining the 
traditional upwind and Crank-Nicolson schemes is developed, along with a novel advection upwind 
Crank-Nicolson combination, and weighted upwind-downwind schemes. A stability analysis on the 
developed numerical schemes found the implicit formulations to be more stable and practically viable, 
and the new advection upwind Crank-Nicolson and weighted upwind-downwind schemes are 
208 
 
improvements from the traditional approaches. In conclusion, the augmented upwind schemes 
improve the simulation of the local advection-dispersion equation for fractured systems. 
9.2 Fractal advection-dispersion equation 
The simulation of anomalous transport in fractured aquifer systems is improved by providing a fractal 
advection-dispersion equation with numerical integration and approximation methods for solution. 
The fractal advection-dispersion model is applied to a simple transport problem; illustrating that the 
fractal formulation of the advection-dispersion equation can model superdiffusion for fractal 
dimension less than 1, and subdiffusion for fractal dimensions greater than 1, without explicitly 
defining fractures or preferential pathways. In this way, anomalous transport in fractured systems can 
be modelled without explicit locations and details of the fractures, especially where limited 
information is available on the preferential pathway causing the discrepancy. The relationship 
between the fractal velocity and dispersivity with the fractal dimension indicated that fractal 
dimensions 𝛼 ≥ 0.5 are the most appropriate for practical use, due to the exponential increase in 
velocity and dispersivity for fractal dimensions 0.1 ≤ 𝛼 ≤ 0.5. 
9.3 Fractional advection-dispersion equation 
To improve the governing equation for groundwater transport modelling, the Caputo and Atangana-
Baleanu in Caputo sense (ABC) fractional derivatives are applied to the advection-dispersion equation 
with a focus on the advection term to account for anomalous advection. The new advection-dispersion 
equation with ABC fractional derivative is proven to be bound, exist, unique and stable in time. The 
augmented upwind numerical schemes developed are applied to numerically approximate the 
solutions. A recursive method stability analysis, for the advection-dispersion model with Caputo 
fractional derivative, indicated the implicit weighted scheme to be an improvement on the traditional 
implicit upwind scheme, where the inclusion of the weighting factor (θ) provides a means to improve 
the likelihood of upholding the stability condition. Thus, the upwind advection Crank-Nicolson and 
implicit weighted upwind-downwind schemes are applicable for solution of the space-time fractional 
advection-dispersion equation (Caputo), when the stability criterion are upheld. The upwind Crank-
Nicolson scheme is recommended for use with the fractional advection-dispersion equation (ABC), as 
it was the most stable according to the recursive method stability analysis.   
9.4 Fractional-fractal advection-dispersion equation 
The fractional-fractal advection-dispersion equation is developed to provide an efficient non-local, in 
both space and time, modelling tool. The fractional-fractal model has two parameters, fractional order 
209 
 
(𝛼) and fractal dimension (𝛽), where simulations are valid for specific combinations. The range of valid 
combinations decrease with decreasing fractional order and fractal dimension, and a final 
recommendation is a fractional order and fractal dimension of 0.7 ≤ 𝛼, 𝛽 ≤ 1. The fractional-fractal 
model provides a flexible tool to model anomalous diffusion, where the fractional order controls the 
breakthrough curve peak, and the fractal dimension controls the position of the peak and tailing 
effect. These two controls potentially provide the tools to improve the representation of anomalous 
breakthrough curves that cannot be described by the classical model. 
In conclusion, the main aim of the research has been achieved by providing: 
 Improved numerical approximation schemes for the classical advection-dispersion model for 
fractured groundwater systems 
 
 Fractal advection-dispersion equation proven to simulate superdiffusion and subdiffusion by 
varying the fractal dimension, without explicit characterisation of fractures or preferential 
pathways 
 
 Fractional advection-dispersion equations for advection-dominated systems with the Caputo and 
Atangana-Baleanu fractional derivatives, and appropriate numerical approximation methods 
 
 Fractional-fractal advection-dispersion equation, which forms an efficient non-local modelling 
tool for practical applications, and provides a flexible, two-parameter system for representing 
anomalous diffusion in fractured groundwater systems without explicit characterisation of 
fractures or preferential pathways 
 
A modest step forward made, in the bigger scheme of progress, toward the collective mission of 
resolving the difference between modelled and observed to increase the comprehension and 
management of natural systems. 
  
210 
 
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219 
 
APPENDICES 
 
APPENDIX A – SCILAB CODES 
First-order upwind explicit scheme 
timer() 
dx=2*10^(-1);dt=4*10^(-2);T=10;Vx=0.5;DL=0.3;lambda=0.5; 
a=1-((dt*Vx)/dx)-((2*dt*DL)/(dx^2)); 
b=((dt*Vx)/dx)+(dt*DL)/(dx^2); 
c=(dt*DL)/(dx^2); 
L=30;N=L/dx;C0=10; 
 
B=a*diag(ones(N-1,1))+b*diag(ones(N-2,1),-1)+c*diag(ones(N-2,1),1); 
Vect=zeros(N-1,1); 
for i=1:N-1 
    U(i)=0; 
end 
k=T/dt; 
for m=1:k 
   Vect(1)=b*C0; 
    U=B*(U+Vect); 
end 
V(1)=C0; 
V(N+1)=0; 
for j=2:N 
    V(j)=U(j-1); 
end 
timer(); 
 
X=0:dx:L; 
plot2d(X,V,9) 
//tt=linspace(0,1,100); 
//uex=sin(%pi*tt)*exp(-1); 
//plot2d(tt,uex) 
//wex=sin(%pi*X)*exp(-1); 
//err=norm(wex'-V,'inf') 
 
First-order upwind implicit scheme 
timer() 
dx=2*10^(-1);dt=4*10^(-2);T=10;Vx=0.5;DL=0.3;lambda=0.5; 
a=1+((dt*Vx)/dx)+((2*dt*DL)/(dx^2)); 
b=-((dt*Vx)/dx)-((dt*DL)/(dx^2)); 
c=-((dt*DL)/(dx^2)); 
L=30;N=L/dx;C0=10; 
 
B=a*diag(ones(N-1,1))+b*diag(ones(N-2,1),-1)+c*diag(ones(N-2,1),1); 
Vect=zeros(N-1,1); 
for i=1:N-1 
    U(i)=0; 
end 
k=T/dt; 
for m=1:k 
I 
 
   Vect(1)=-b*C0; 
    U=inv(B)*(U+Vect); 
end 
V(1)=C0; 
V(N+1)=0; 
for j=2:N 
    V(j)=U(j-1); 
end 
timer(); 
 
X=0:dx:L; 
plot2d(X,V,3) 
//tt=linspace(0,1,100); 
//uex=sin(%pi*tt)*exp(-1); 
//plot2d(tt,uex) 
//wex=sin(%pi*X)*exp(-1); 
//err=norm(wex'-V,'inf') 
 
First-order upwind Crank-Nicolson scheme 
timer(); 
dx=2*10^(-1);dt=4*10^(-2);T=10;Vx=0.5;DL=0.3;lambda=0.5; 
L=30;N=L/dx;C0=10; 
a=1+(0.5*Vx*dt)/dx+(dt*DL)/(dx^2); 
b=-((0.5*Vx*dt)/dx)-((0.5*dt*DL)/(dx^2)); 
c=-(0.5*dt*DL)/(dx^2); 
d=1-(0.5*Vx*dt)/dx-((dt*DL)/(dx^2)); 
e=((0.5*Vx*dt)/dx)+((0.5*dt*DL)/(dx^2)); 
f=(0.5*dt*DL)/(dx^2); 
 
B=a*diag(ones(N-1,1))+b*diag(ones(N-2,1),-1)+c*diag(ones(N-2,1),1); 
A=d*diag(ones(N-1,1))+e*diag(ones(N-2,1),-1)+f*diag(ones(N-2,1),1); 
Vect=zeros(N-1,1); 
for i=1:N-1 
    U(i)=0; 
end 
k=T/dt; 
for m=1:k 
   Vect(1)=(e-b)*C0; 
   U=inv(B)*(A*(U+Vect)); 
end 
V(1)=C0; 
V(N+1)=0; 
for j=2:N 
    V(j)=U(j-1); 
end 
timer() 
 
X=0:dx:L; 
plot2d(X,V,2) 
//tt=linspace(0,1,100); 
//uex=sin(%pi*tt)*exp(-1); 
//plot2d(tt,uex) 
//wex=sin(%pi*X)*exp(-1); 
//err=norm(wex'-V,'inf') 
II 
 
 
First-order upwind advection Crank-Nicolson scheme (explicit) 
timer() 
dx=2*10^(-1);dt=4*10^(-2);T=10;Vx=0.5;DL=0.3;lambda=0.5; 
L=30;N=L/dx;C0=10; 
a=1+(0.5*Vx*dt)/dx; 
b=-(0.5*Vx*dt)/dx; 
c=1-((0.5*Vx*dt)/dx)-((2*dt*DL)/(dx^2)); 
d=(dt*DL)/(dx^2); 
f=(dt*DL)/(dx^2)+(0.5*Vx*dt)/dx; 
 
B=a*diag(ones(N-1,1))+b*diag(ones(N-2,1),-1); 
A=c*diag(ones(N-1,1))+f*diag(ones(N-2,1),-1)+d*diag(ones(N-2,1),1) 
Vect=zeros(N-1,1); 
for i=1:N-1 
    U(i)=0; 
end 
k=T/dt; 
for m=1:k 
   Vect(1)=(f-b)*C0; 
   U=inv(B)*(A*(U+Vect)); 
end 
V(1)=C0; 
V(N+1)=0; 
for j=2:N 
    V(j)=U(j-1); 
end 
timer(); 
 
X=0:dx:L; 
plot2d(X,V,1) 
//tt=linspace(0,1,100); 
//uex=sin(%pi*tt)*exp(-1); 
//plot2d(tt,uex) 
//wex=sin(%pi*X)*exp(-1); 
//err=norm(wex'-V,'inf') 
 
First-order upwind advection Crank-Nicolson scheme (implicit) 
timer(); 
dx=2*10^(-1);dt=4*10^(-2);T=10;Vx=0.5;DL=0.3;lambda=0.5; 
L=30;N=L/dx;C0=10; 
 
a=1+(0.5*Vx*dt)/dx+(2*dt*DL)/(dx^2); 
b=-((0.5*Vx*dt)/dx)-((dt*DL)/(dx^2)) 
c=-(dt*DL)/(dx^2) 
d=1-(0.5*Vx*dt)/dx 
e=(0.5*Vx*dt)/dx 
 
B=a*diag(ones(N-1,1))+b*diag(ones(N-2,1),-1)+c*diag(ones(N-2,1),1); 
A=d*diag(ones(N-1,1))+e*diag(ones(N-2,1),-1) 
Vect=zeros(N-1,1); 
for i=1:N-1 
    U(i)=0; 
end 
III 
 
k=T/dt; 
for m=1:k 
   Vect(1)=(e-b)*C0; 
   U=inv(B)*(A*(U+Vect)); 
end 
V(1)=C0; 
V(N+1)=0; 
for j=2:N 
    V(j)=U(j-1); 
end 
timer() 
 
X=0:dx:L; 
plot2d(X,V,1) 
//tt=linspace(0,1,100); 
//uex=sin(%pi*tt)*exp(-1); 
//plot2d(tt,uex) 
//wex=sin(%pi*X)*exp(-1); 
//err=norm(wex'-V,'inf') 
 
First-order weighted upwind-downwind scheme (explicit) 
timer() 
dx=2*10^(-1);dt=4*10^(-2);T=10;Vx=0.5;DL=0.3;theta=0.9;lambda=0.5; 
a=1+(2*dt*DL)/(dx^2)+(theta*dt*Vx)/dx-((1-theta)*dt*Vx)/dx; 
b=-((theta*dt*Vx)/dx)-((dt*DL)/(dx^2)); 
c=(((1-theta)*dt*Vx)/dx)-((dt*DL)/(dx^2)); 
L=30;N=L/dx;C0=10; 
B=a*diag(ones(N-1,1))+b*diag(ones(N-2,1),-1)+c*diag(ones(N-2,1),1); 
Vect=zeros(N-1,1); 
for i=1:N-1 
    U(i)=0; 
end 
k=T/dt; 
for m=1:k 
   Vect(1)=-b*C0; 
    U=inv(B)*(U+Vect); 
end 
V(1)=C0; 
V(N+1)=0; 
for j=2:N 
    V(j)=U(j-1); 
end 
timer(); 
 
X=0:dx:L; 
plot2d(X,V,13) 
//tt=linspace(0,1,100); 
//uex=sin(%pi*tt)*exp(-1); 
//plot2d(tt,uex) 
//wex=sin(%pi*X)*exp(-1); 
//err=norm(wex'-V,'inf') 
 
 
 
IV 
 
First-order weighted upwind-downwind scheme (implicit) 
timer(); 
dx=2*10^(-1);dt=4*10^(-2);T=10;Vx=0.5;DL=0.3;theta=0.5;lambda=0.5; 
a=1-(theta*((dt*Vx)/dx))-(1-theta)*((dt*Vx)/dx)-((2*dt*DL)/(dx^2)); 
b=theta*((dt*Vx)/dx)+(dt*DL)/(dx^2); 
c=-(1-theta)*((dt*Vx)/dx)+(dt*DL)/(dx^2); 
L=30;N=L/dx;C0=10; 
 
B=a*diag(ones(N-1,1))+b*diag(ones(N-2,1),-1)+c*diag(ones(N-2,1),1); 
Vect=zeros(N-1,1); 
for i=1:N-1 
    U(i)=0; 
end 
k=T/dt; 
for m=1:k 
   Vect(1)=b*C0; 
    U=B*(U+Vect); 
end 
V(1)=C0; 
V(N+1)=0; 
for j=2:N 
    V(j)=U(j-1); 
end 
timer() 
 
X=0:dx:L; 
plot2d(X,V,2) 
//tt=linspace(0,1,100); 
//uex=sin(%pi*tt)*exp(-1); 
//plot2d(tt,uex) 
//wex=sin(%pi*X)*exp(-1); 
//err=norm(wex'-V,'inf') 
 
 
  
V 
 
APPENDIX B – MAPLE CODES 
Fractal advection-dispersion model 
 
Fractional-fractal advection-dispersion model 
 
 
VI 
 
 
 
 
 
 
VII