Numerical methods for time-space fractional partial differential equations

In this chapter, numerical methods for time-space fractional partial differential equations are presented. Firstly, some preliminary knowledge for fractional operators is introduced. Secondly, finite difference methods for the one-dimensional time-space fractional advection-dispersion equation and time-space Caputo-Reisz fractional diffusion equation in two dimensions are considered, respectively. Thirdly, an unstructured mesh finite element method for the two-dimensional time-space Riesz fractional diffusion equation on an irregular convex domain is presented. Finally, a two-dimensional time-space fractional diffusion equation based on the fractional Laplacian operator is investigated.