On an accurate discretization of a variable-order fractional reaction-diffusion equation

The aim of this paper is to develop an accurate discretization technique to solve a class of variable-order fractional (VOF) reaction-diffusion problems. In the spatial direction, the problem is first discretized by using a compact finite difference operator. Then, a weighted-shifted Grünwald formula is applied for the temporal discretization of fractional derivatives. To solve the derived nonlinear discrete system, an accurate iterative algorithm is also formulated. The solvability, stability and L2-convergence of the proposed scheme are derived for all variable-order α(t) ∈ (0, 1). The proposed method is of accuracy-order O(τ3+h4), where τ and h are temporal and spatial step sizes, respectively. Through some numerical simulations, the theoretical analysis and high-accuracy of the proposed method are verified. Comparative results also indicate that the accuracy of the new discretization technique is superior to the other methods available in the literature. Finally, the feasibility of the proposed VOF model is demonstrated by using the reported experimental data.