A pseudo-spectral scheme for the approximate solution of a time-fractional diffusion equation

We consider an initial-boundary-value problem for a time-fractional diffusion equation with initial condition u(0)(x) and homogeneous Dirichlet boundary conditions in a bounded interval [0, L]. We study a semidiscrete approximation scheme based on the pseudo-spectral method on Chebyshev-Gauss-Lobatto nodes. In order to preserve the high accuracy of the spectral approximation we use an approach based on the evaluation of the Mittag-Leffler function on matrix arguments for the integration along the time variable. Some examples are presented and numerical experiments illustrate the effectiveness of the proposed approach.