A Legendre spectral-finite difference method for Caputo-Fabrizio time-fractional distributed-order diffusion equation

In this paper, we introduce a hybrid method based on a finite difference method and a spectral method for solving the multi-term time-fractional diffusion equations (TFDEs) based on Caputo-Fabrizio fractional operator. We apply a finite difference scheme for discretizing the time derivatives and consider a Legendre-spectral approximation in space discretization to semi-discrete problem. It is known that the spectral method has been an efficient tool for computing numerical solutions of differential equations because of its high-order accuracy. We discuss the convergence of the proposed method in discrete L-2-norm. Furthermore, we extend the multi-term TFDE to the distributed order and analyze the method for the considered equation. In the end, we confirm the proven theoretical results with the help of some numerical examples.