A pseudo-spectral based method for time-fractional advection-diffusion equation

In this paper, a pseudo-spectral method with the Lagrange polynomial basis is proposed to solve the time-fractional advection-diffusion equation. A semi-discrete approximation scheme is used for conversion of this equation to a system of ordinary fractional differential equations. Also, to protect the high accuracy of the spectral approximation, the Mittag-Leffler function is used for the integration along the time variable. Some examples are performed to illustrate the accuracy and efficiency of the proposed method.