Numerical investigation of a Riesz fractional diffusion equation with Dirichlet boundary conditions via the Crank-Nicolson method

Abstract

This study examines a finite difference method based on the Crank-Nicolson scheme, specifically developed for solving fractional diffusion equations involving the Riesz spatial derivative and the Caputo-Fabrizio time derivative, subject to Dirichlet boundary conditions. The proposed numerical scheme employs a Taylor series expansion for temporal discretization in conjunction with shifted Grünwald-Letnikov operators for spatial approximation. A rigorous technical proof of the scheme’s unconditional stability is provided, and its convergence properties are thoroughly analyzed. In addition, the practical effectiveness of the method is demonstrated through a representative numerical example, with corresponding graphical results obtained via MATLAB simulations.