Efficient numerical method for multi-term time-fractional diffusion equations with Caputo-Fabrizio derivatives

TL;DR

This paper introduces an efficient, stable, and convergent numerical scheme combining fast finite difference and spectral collocation methods for solving multi-term Caputo-Fabrizio time-fractional diffusion equations, with proven theoretical guarantees.

Contribution

The paper presents a novel fast finite difference scheme with optimal complexity for multi-term Caputo-Fabrizio derivatives, coupled with spectral collocation for spatial discretization, and provides rigorous stability and convergence analysis.

Findings

Scheme is unconditionally stable and convergent.

Achieves $O(igl(rac{ au}{N}igr)^2 + N^{-m})$ accuracy.

Numerical results confirm theoretical predictions.

Abstract

In this paper, we consider a numerical method for the multi-term Caputo-Fabrizio time-fractional diffusion equations (with orders $\alpha_i\in(0,1)$, $i=1,2,\cdots,n$). The proposed method employs a fast finite difference scheme to approximate multi-term fractional derivatives in time, requiring only $O(1)$ storage and $O(N_T)$ computational complexity, where $N_T$ denotes the total number of time steps. Then we use a Legendre spectral collocation method for spatial discretization. The stability and convergence of the scheme have been thoroughly discussed and rigorously established. We demonstrate that the proposed scheme is unconditionally stable and convergent with an order of $O(\left(\Delta t\right)^{2}+N^{-m})$, where $\Delta t$, $N$, and $m$ represent the timestep size, polynomial degree, and regularity in the spatial variable of the exact solution, respectively. Numerical results are presented to validate the theoretical predictions.