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

Yang Liu; Enyu Fan; Baoli Yin; Hong Li; Yang Liu; Enyu Fan; Baoli Yin; Hong Li

doi:10.3934/math.2020117

AIMS Mathematics

2020, Volume 5, Issue 3: 1729-1744. doi: 10.3934/math.2020117

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Research article

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

School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China

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.
- Caputo-Fabrizio fractional derivative,
- novel approximation formula,
- fast algorithm,
- second-order convergence rate,
- computing complexity
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

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Abstract

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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