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In this paper, applying a novel second-order numerical approximation formula for the Riesz derivative and Crank–Nicolson technique for the temporal derivative, a numerical algorithm is constructed for the two-dimensional spatial fractional diffusion equation with convergence order, whereτ,andare the temporal and spatial step sizes, respectively. It is proved that the proposed algorithm is unconditionally stable and convergent by using the energy method. Meanwhile, by adding the high-order perturbation items for the above numerical scheme, an alternating direction implicit difference scheme is also constructed. Finally, some numerical results are presented to demonstrate the validity of theoretical analysis and show the accuracy and effectiveness of the method described herein. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]