Finite difference approximations for two-sided space-fractional partial differential equations

Fractional order partial differential equations are generalizations of classical partial differential equations. Increasingly, these models are used in applications such as fluid flow, finance and others. In this paper we examine some practical numerical methods to solve a class of initial-boundary value fractional partial differential equations with variable coefficients on a finite domain. We examine the case when a left-handed or a right-handed fractional spatial derivative may be present in the partial differential equation. Stability, consistency, and (therefore) convergence of the methods are discussed. The stability (and convergence) results in the fractional PDE unify the corresponding results for the classical parabolic and hyperbolic cases into a single condition. A numerical example using a finite difference method for a two-sided fractional PDE is also presented and compared with the exact analytical solution.