Application of the New Numerical Method with Atangana–Baleanu Fractal-Fractional Derivative on the Self-Similar Leaky Aquifer Equations

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It has been proven that fractional Laplacian well describes nonlocal diffusion phenomena. The practical question one would ask is the following: Why do we need such differential operator to model the flow within a leaky aquifer? Well, one needs to understand that each differential operator depicts mean square displacement that helps to capture how the water maneuvers within a geological genesis. While this point has not been the main focus in this field and was not considered to be very critical, it is important to account for every particle movement within a media, where the geometry of such media leads the movement of the particle. As the water moves within a geological formation, it encounters some challenges or resistance resulting from different properties of the media. Because of heterogeneities, the water can flow in one direction/portion and follow a specific natural law, for example, the exponential decay law, which is more associated with the flow within the clay. The clay hinders the water from passing very quickly compared to its behavior with sand or silt, thus the velocity of the water decreases until eventually it becomes stagnant; such a process can be depicted very well using the differential operator, whose kernels are exponential decay law. Nevertheless, if instead the water goes through such geological formation and is able to maintain a constant velocity for a longer time, one would be oblige to use the differential operator whose kernel is power law decay. However, there are more complex situations, where, for example, the aquifer has dual or multi-layers horizontally, this could pose serious problem to modeling such a situation – in this situation the flow can follow first the power law decay process and later the exponential law. This scenario can be practically related to the situation where water is flowing slowly within a portion of a geological formation, but later becomes stagnant because of poor porosity and permeability. This situation cannot be depicted using the classical differentiation or fractional differentiation with power and exponential decay law kernels, as they do not accommodate crossover behavior for mean square displacement. Therefore, a differential operator whose kernel has crossover behavior can be used, which is the generalized Mittag-Leffler kernel.