An approximate solution for the time-fractional diffusion equation

In this paper, a numerical method based on a finite difference scheme is proposed for solving the time-fractional diffusion equation (TFDE). The TFDE is obtained from the standard diffusion equation by replacing the first-order time derivative with Caputo fractional derivative. At first, we introduce a time discrete scheme. Then, we prove the proposed method is unconditionally stable and the approximate solution converges to the exact solution with order O(Δt2−α)O(Δt2−α), where ΔtΔt is the time step size and αα is the order of Caputo derivative. Finally, some examples are presented to verify the order of convergence and show the application of the present method.