Efficient numerical methods for Riesz space-fractional diffusion equations with fractional Neumann boundary conditions

Abstract

This paper is concerned with the study of second order scheme for finding the approximate solutions of space-fractional diffusion equations with fractional Neumann boundary conditions, which are based on the Crank-Nicholson method in time and the shifted Grünwald-Letnikov operator and Taylor expansion method in space. The fractional differential operators are taken in sense of the Riesz fractional derivative. The convergence, solvability and stability of the proposed scheme are proved. In addition, a new numerical scheme is proposed by using different approaches to deal with the boundary conditions. The accuracy and efficiency of the presented method is shown by conducting two numerical examples.