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.

翻译：本文研究多时间项Caputo-Fabrizio分数阶扩散方程（阶数$\alpha_i\in(0,1)$，$i=1,2,\cdots,n$）的数值方法。所提方法采用快速有限差分格式逼近时间方向上的多时间项分数阶导数，仅需$O(1)$存储量和$O(N_T)$计算复杂度，其中$N_T$表示总时间步数。随后，我们使用Legendre谱配置法进行空间离散。本文对该格式的稳定性与收敛性进行了深入讨论与严格论证。我们证明，所提格式无条件稳定且收敛，其收敛阶为$O(\left(\Delta t\right)^{2}+N^{-m})$，其中$\Delta t$、$N$和$m$分别代表时间步长、多项式阶数以及精确解在空间变量中的正则性。数值结果验证了理论预测的准确性。