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：本文研究了多时间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。所提出的方法采用快速有限差分格式逼近时间方向上的多时间分数阶导数，仅需$O(1)$存储空间和$O(N_T)$计算复杂度，其中$N_T$表示总时间步数。随后，我们利用Legendre谱配置法进行空间离散。本文对该格式的稳定性和收敛性进行了全面讨论并严格证明。我们证明该格式无条件稳定且收敛阶为$O(\left(\Delta t\right)^{2}+N^{-m})$，其中$\Delta t$、$N$和$m$分别表示时间步长、多项式次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有效性。