Numerical solution of fractional diffusion equation over a long time domain

In this paper, we propose a method to compute approximate solutions to one dimensional fractional diffusion equation which requires solution for a long time domain. For this, we use a set of shifted Legendre polynomials for the space domain and a set of Legendre rational functions for the time domain. The unknown solution is approximated by using these sets of orthogonal functions with unknown coefficients and the fractional derivative of the approximate solution is represented by an operational matrix, resulting in a linear system with the unknown coefficients. Numerical examples are given to demonstrate the effectiveness of the method.