| Summary: | In this study, we propose a novel second-order numerical formula that approximates the Caputo-Fabrizio (CF) fractional derivative at node $t_{k+\frac{1}{2}}$. The nonlocal property of the CF fractional operator requires $O(M^2)$ operations and $O(M)$ memory storage, where $M$ denotes the numbers of divided intervals. To improve the efficiency, we further develop a fast algorithm based on the novel approximation technique that reduces the computing complexity from $O(M^2)$ to $O(M)$, and the memory storage from $O(M)$ to $O(1)$. Rigorous arguments for convergence analyses of the direct method and fast method are provided, and two numerical examples are implemented to further confirm the theoretical results and efficiency of the fast algorithm.
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