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

Abstract

We consider an initial-boundary-value problem for a time-fractional diffusion equation with initial condition u₀(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.

Keywords:

• fractional derivatives
• time-fractional diffusion equations
• spectral methods
• differentiation matrix
• matrix functions

2010 AMS Subject Classification::

• 65M70
• 34A08
• 26A33
• 33C47
• 33E12

Acknowledgements

The work of R.Garrappa has been supported under the GNCS - INdAM Project 2014.