Modelling of groundwater flow within a leaky aquifer with fractal-fractional differential operators

Abstract

The major leading problem in groundwater modelling is to produce a suitable model for leaky aquifers. An attempt to answer this, researchers devoted their focus to propose new models for complex real-world physical problems within a leaky aquifer. Their proposed models include the nonlocal fractional differentiation operator with kernel power law, fractional differentiation and integral with exponential decay, and fractional differentiation and integral with the generalized Mittag-Leffler function. However, for these operators, none of these models were suitable to model anomalous realworld physical problems. In this study, the main purpose of this project is to develop an operator that is suitable to capture groundwater flow within a leaky aquifer, and this operator will also aim to attract non-local problems that display at the same time. The solution to this problem, this project will introduce a new powerful operator called fractal-fractional differentiation and integral which have been awarded by many researchers with both self-similar and memory effects. For solution, we firstly derive solution using Hantush extended equation with respect to time and space. We also extend by employing the Predictor-Corrector method and Atangana Baleanu derivative to obtain numerical solution for non-linear differentiation and integral, and the exact numerical solution for groundwater flow within a leaky aquifer. We further presented the special solution, uniqueness and also the stability analysis. While conducting the research, our finding also indicated that fractal-fractional operators depict real-world physical problems to capture groundwater flow within a leaky aquifer than classical equation.