A new high-order method for the time-fractional diffusion equation with a source

In this paper, we propose a new high-order finite difference method to solve the time-fractional diffusion equation with a source. We first construct a finite difference approximation of the Caputo fractional derivative of order α (0 < α < 1), and show that the convergence rate of our approximation is (4 − α). We then investigate the properties of the fractional differentiation matrix for our new approximations, and introduce an implicit finite difference method which employs such approximations for the time discretization of the fractional diffusion equation, coupled with a Fourier-type expansion in space. By taking advantage of the special structure of our fractional differentiation matrix, each of the linear systems resulted from our new high-order approximations for each mode of time-fractional diffusion equation can be solved in order O(N²). Numerical experiments about the performance of our method in evaluating fractional derivatives, and solving fractional ordinary differential equations and time-fractional diffusion equation are also presented, to demonstrate the efficiency of our method.