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

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 $O\left({\left(\mathrm{\Delta }t\right)}^2+N^{-m}\right)$, where $\mathrm{\Delta }t$, $N$, 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.