A Numerical Procedure for Fractional-Time-Space Differential Equations with the Spectral Fractional Laplacian

The aim of this chapter is to device a computationally effective procedure for numerically solving fractional-time-space differential equations with the spectral fractional Laplacian. A truncated spectral representation of the solution in terms of the eigenfunctions of the usual integer-order Laplacian is considered. Time-dependent coefficients in this representation, which are solutions to some linear fractional differential equations, are evaluated by means of a generalized exponential time-differencing method, with some advantages in terms of accuracy and computational effectiveness. Rigorous a priori error estimates are derived, and they are verified by means of some numerical experiments.