Numerical solution of the space-time fractional diffusion equation: Alternatives to finite differences

One of the ongoing issues with fractional-order diffusion models is the design of efficient numerical schemes for the space and time discretizations. Until now, most models have relied on a low-order finite difference method to discretize both the fractional-order space and time derivatives. While the finite difference method is simple and straightforward to solve integer-order differential equations, its appeal is reduced for fractional-order differential equations as it leads to systems of linear equation defined by large full matrices. Alternatives to the finite difference method exist but a unified presentation and comparison of these methods is still missing. In this paper, we compare 4 different numerical discretizations of the space-time fractional diffusion model. These consist of the finite difference, finite element, pseudo-spectral and radial basis functions methods. We suggest that non-local methods, like the pseudo-spectral and radial basis functions method, are well-suited to discretize the non-local operators like fractional-order derivatives. These methods naturally take the global behavior of the solution into account and thus do not result in an extra computational cost when moving from an integer-order to a fractional-order diffusion model.