Analysis of general groundwater flow equation with non-local operators
Mathobo, Mashudu C.
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The main aim of this research was to analyze the general groundwater flow equation with nonlocal operators. Non-local operators are able to introduce the mathematical formulation of memory and random walk. Recently, Mashudu and Mathobo (2017) introduced a more general groundwater flow model which considers the forgotten terms by Theis; however, their model uses the concept of classical differential operators and therefore could not help capture complexities of the media through which the sub-surface water flows. It is in this regard that their model was extended to the concept of non-local operators. The general groundwater flow equation for a confined aquifer was analyzed using the Caputo fractional derivative, Caputo-Fabrizio fractional derivative, AtanganaBaleanu fractional derivative and fractal-fractional differential operators. New numerical schemes were developed using the classical Adam Bash fort method and the newly developed Newton Polynomial method which was developed by Atangana and Araz in 2019. Stability analysis of the numerical schemes were presented. Numerical simulations were developed and showed evidence of three different types of flow according to fractional orders. Slow flow was observed in fractional order from 0.99 to 0.51, depicting that the geological media is not well connected and thus not helping water to pass easily. When the fractional order is below 0.5, fast flow was observed and when the fractional order is 1, normal flow was recovered and the geology was assumed to be homogenous.