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

In this article, we take a time-space fractional convection-diffusion problem with a nonlinear reaction term on a finite domain. We use the L-1 operator to discretize the Caputo fractional derivative and the weighted shifted Grunwald difference (WSGD) method to approximate the Riesz fractional derivative. Furthermore, we apply the Crank Nicolson difference scheme with weighted shifted Grunwald-Letnikov and obtain that the numerical method is unconditionally stable and convergent with the accuracy of O(tau(2-alpha) + h(2)), where alpha is an element of (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.