General Padé approximation method for time-space fractional diffusion equation

In recent years, fractional differential equations have attracted much attention due to their wide application. In this paper, we present a novel numerical method for the space-time Riesz-Caputo fractional diffusion equation, which discrete the Riesz derivative by a fourth-order fractional-compact difference scheme, then the above space-time Riesz-Caputo fractional diffusion equation change into a fractional ordinary differential equation (FODE) system. Again using Laplace and inverse Laplace transforms, one can get the analytical solution of the FODE system, furthermore, we approximate the Mittag-Leffler function by the global Padé approximation and obtain the numerical method for the space-time Riesz-Caputo fractional diffusion equation. Finally, two numerical examples are presented to show that the numerical results are in good line with our theoretical analysis.