Numerical Solution for a Nonlinear Time-Space Fractional Convection-Diffusion Equation

Abstract

In this article, we take a time–space fractional convection-diffusion problem with a nonlinear reaction term on a finite domain. We use the L1 operator to discretize the Caputo fractional derivative and the weighted shifted Grünwald difference (WSGD) method to approximate the Riesz fractional derivative. Furthermore, we apply the Crank Nicolson difference scheme with weighted shifted Grünwald–Letnikov and obtain that the numerical method is unconditionally stable and convergent with the accuracy of O(τ2−α+h2), where α∈(0,1]. For finding the numerical solution of the nonlinear system of equation, we apply the fixed iteration method. In the end, numerical simulations are treated to verify the effectiveness and consistency of the proposed method.