A high order finite difference/spectral approximations to the time fractional diffusion equations

In this paper, we consider the numerical solution of a time-fractional diffusion equation, which is obtained from the standard diffusion equation by replacing the first order time derivative with a fractional derivative of order alpha, with 0 <= alpha <= 1. The main purpose of this work is to construct high order scheme to efficiently solve the time-fractional diffusion equation. The proposed method is based on a finite difference scheme in time and Legendre spectral methods in space. The numerical examples show the convergence rate is O(Delta t(3-alpha)+N-m), where Delta t, N and m are the time step size, the polynomial degree and the regularity of the exact solution, respectively.