Research article

Fast algorithm based on the novel approximation formula for the Caputo-Fabrizio fractional derivative

  • Received: 08 November 2019 Accepted: 04 February 2020 Published: 14 February 2020
  • MSC : 65M06, 65M12

  • 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.

    Citation: Yang Liu, Enyu Fan, Baoli Yin, Hong Li. Fast algorithm based on the novel approximation formula for the Caputo-Fabrizio fractional derivative[J]. AIMS Mathematics, 2020, 5(3): 1729-1744. doi: 10.3934/math.2020117

    Related Papers:

  • 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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