Computationally efficient numerical methods for time- and space-fractional Fokker-Planck equations

Fractional Fokker-Planck equations have been used to model several physical situations that present anomalous diffusion. In this paper, a class of time-and space-fractional Fokker-Planck equations (TSFFPE), which involve the Riemann-Liouville time-fractional derivative of order 1 - alpha (alpha is an element of (0, 1)) and the Riesz space-fractional derivative (RSFD) of order mu is an element of (1, 2), are considered. The solution of TSFFPE is important for describing the competition between subdiffusion and Levy flights. However, effective numerical methods for solving TSFFPE are still in their infancy. We present three computationally efficient numerical methods to deal with the RSFD, and approximate the Riemann-Liouville time-fractional derivative using the Grunwald method. The TSFFPE is then transformed into a system of ordinary differential equations (ODE), which is solved by the fractional implicit trapezoidal method (FITM). Finally, numerical results are given to demonstrate the effectiveness of these methods. These techniques can also be applied to solve other types of fractional partial differential equations.