Stability and Convergence of a Time-Fractional Variable Order Hantush Equation for a Deformable Aquifer

Abstract

The medium through which the groundwater moves varies in time and space. The Hantush equation describes the movement of groundwater through a leaky aquifer. To include explicitly the deformation of the leaky aquifer into the mathematical formulation, we modify the equation by replacing the partial derivative with respect to time by the time-fractional variable order derivative. The modified equation is solved numerically via the Crank-Nicolson scheme. The stability and the convergence in this case are presented in details.

On Time-Fractional Diffusion Equations with Space-Dependent Variable Order

“…[21]. As a matter of fact, several VO diffusion models have been successfully applied to numerous areas of applied sciences and engineering, such as chemistry [4], rheology [18], biology [6], hydrogeology [2] and physics [20,22]. We point out that the VO time-fractional kinetic equation is frequently considered as a macroscopic model to continuous time random walk (CRTW) associated with stochastic diffusion processes with space-dependent diffusion coefficient, and we refer to [13,15] for the derivation of VO time-fractional diffusion models from a CTRW scheme.…”

Applications of variable-order fractional operators: a review

“…The VO diffusion equation formed the basis of several interesting investigations involving anomalous advection-diffusion systems, especially those seen in nature. For example, the flow of groundwater through underground aquifers has been modelled in [134][135][136] through VO timefractional operators. Groundwater diffuses through porous, fractured, layered and heterogeneous aquifers, whose structure changes with space as well as time, leading to anomalous diffusion and a VO scaling of the MSD with time.…”

“…Hence, this flow process cannot be modelled accurately through CO models. Thus, in [134,135] this anomalous flow is modelled through a VO version of the groundwater flow equation (called the Theis equation) given as…”

On the Generalized Mass Transport Equation to the Concept of Variable Fractional Derivative

“…Inside the discredited problem domain, the variable internal properties, boundaries, and stresses of the system are approximated. Deterministic, distributedparameter, and numerical models can relax the rigid idealized conditions of analytical models or lumped-parameter models, and they can therefore be more realistic and flexible for simulating fields' conditions [19][20][21][22][23][24][25][26][27][28][29]. The finite difference schemes for constant-order time or space fractional diffusion equations have been widely studied [19][20][21][22][23][24][25][26][27][28][29].…”