Finite difference approximations for one-dimensional Riesz fractional diffusion equation with fractional boundary condition

Abstract:

In this paper, we examine a practical numerical method to solve a one-dimensional Riesz fractional diffusion equation with fractional boundary conditions. In order to propose an implicit finite difference method, we use the fractional centered derivative approach to approximate the Riesz fractional derivative and use the standard Grünwald-Letnikov fractional order operator to discrete the Riemann-Liouville fractional derivative in fractional boundary conditions. Then we discuss the existence and uniqueness of solution for the method. The stability, consistency and convergence of the method are also established. Finally, a numerical experiment is proposed to show the effectiveness of the method.