Analysis of Numerical Method for Diffusion Equation with Time-Fractional Caputo–Fabrizio Derivative

Abstract

In this paper, we propose a high-precision discrete scheme for the time-fractional diffusion equation (TFDE) with Caputo-Fabrizio type. First, a special discrete scheme of C-F derivative is used in time direction and a compact difference operator is used in space direction. Second, we discuss the convergence of the proposed method in discrete $L^1$ -norm and $L^2$ -norm. The convergence order of our discrete scheme is $O\left({\tau }^2+h^4\right)$ , where $\tau$ and $h$ are the temporal and spatial step sizes, respectively. The aim of this paper is to show that fractional operator without singular term is very useful for improving the accuracy of discrete scheme.