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

and Applied Analysis 3 3. Numerical Solution Environmental phenomena, such as groundwater flow described by variational order derivative, are highly complex phenomena, which do not lend themselves readily to analysis of analytical models. The discussion presented in this section will therefore be devoted to the derivation of numerical solution to the modified Hantush equation (6). Solving difficult equations with numerical scheme has been passionate exercise for many scholars [9–20]. However, there exists numerous of this scheme in the literature [14– 20]. Some of these numerical techniques are very accurate while approximating solutions of difficult equations. These numerical methods yield approximate solutions to the governing equation through the discretization of space and time [7]. Within the discredited problem domain, the variable internal properties, boundaries, and stresses of the system are approximated. Deterministic, distributed-parameter, numerical models can relax the rigid idealised conditions of analytical models or lumped-parameter models, and they can therefore be more realistic and flexible for simulating fields conditions [7]. Recently Atangana and Botha [7] have extended the groundwater flow model to the concept of time-fractional variable order derivative; they solved the resulting equation via Crank-Nicolson numerical scheme. The finite difference schemes for constant-order timeor space-fractional diffusion equations have beenwidely studied in [9–14]. Recently, Sun et al. [21] studied the solution of the advection dispersion equation with time-fractional variable order derivative. The study of the implicit difference approximation scheme for constant-order time-fractional diffusion equations was presented in [15]. Recently, the weighted average finite difference method was introduced [16]. The matrix approach for fractional diffusion equations was proposed [17], and Hanert proposed a flexible numerical scheme for the discretization of the space-time fractional diffusion equation (see [18]). Recently, the numerical scheme for VO space-fractional advection-dispersion equation was considered [19]. The investigation of the explicit scheme for VO nonlinear space-fractional diffusion equation was done (see [20]). 3.1. Crank-Nicolson Scheme [22]. Before performing the numerical methods, we assume that (3) has a unique and sufficiently smooth solution. To establish the numerical schemes for the above equation, we let x l = lh, 0 ≤ l ≤ M, Mh = L, t k = kτ, 0 ≤ k ≤ N, Nτ = T, h is the step and τ is the time size, andM andN are grid points. We introduce the Crank-Nicolson scheme as follows. Firstly, the discretization of firstand second-order space derivative is stated as ∂s ∂r = 1 2 (( s (r l+1 , t k+1 ) − Φ (r l−1 , t k+1 ) 2 (h) ) +( s (r l+1 , t k ) − s (r l−1 , t k ) 2 (h) )) + O (h) , (7) ∂2s ∂r = 1 2 (( s (r l+1 , t k+1 ) − 2s (r l , t k+1 ) + s (r l−1 , t k+1 ) (h)2 ) +( s (r l+1 , t k ) − 2s (r l , t k ) + s (r l−1 , t k ) (h)2 ))