Local discontinuous Galerkin approximations to variable-order time-fractional diffusion model based on the Caputo–Fabrizio fractional derivative

Abstract

In this paper, we construct and investigate an accurate numerical scheme for solving a class of variable-order (VO) fractional diffusion equation based on the Caputo–Fabrizio fractional derivative. The scheme is presented by using a finite difference method in temporal variable and a local discontinuous Galerkin method (LDG) in space. For all variable-order α(t)∈(0,1), we derive the stability and L2 convergence of proposed scheme and prove that the method is of accuracy-order O(τ+hk+1), where τ, h and k are temporal step sizes, spatial step sizes and the degree of piecewise Pk polynomials, respectively. Several numerical tests are given to validate the theoretical analysis and efficiency of the proposed algorithm.