Volume 17, Issue 2
A Finite Difference Scheme for Caputo-Fabrizio Fractional Differential Equations

Int. J. Numer. Anal. Mod., 17 (2020), pp. 195-211.

Published online: 2020-02

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• Abstract

In this work, we consider a new fractional derivative with nonsingular kernel introduced by Caputo–Fabrizio (CF) and propose a finite difference method for computing the CF fractional derivatives. Based on an iterative technique, we can reduce the computational complexity from $O$($J$2$N$) to $O$($JN$), and the corresponding storage will be cut down from $O$($JN$) to $O$($N$), which makes the computation much more efficient. Besides, by adopting piece-wise Lagrange polynomials of degrees 1, 2, and 3, we derive the second, third, and fourth order discretization formulas respectively. The error analysis and numerical experiments are carefully provided for the validation of the accuracy and efficiency of the presented method.

• Keywords

Caputo–Fabrizio derivative, fractional differential equations, higher order scheme.

26A33, 35R11, 65N06

guoxu5861@gmail.com (Xu Guo)

liyutian@cuhk.edu.cn (Yutian Li)

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@Article{IJNAM-17-195, author = {Xu and Guo and guoxu5861@gmail.com and 5256 and Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong. and Xu Guo and Yutian and Li and liyutian@cuhk.edu.cn and 10605 and and Yutian Li and Tieyong and Zeng and and 10960 and and Tieyong Zeng}, title = {A Finite Difference Scheme for Caputo-Fabrizio Fractional Differential Equations}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2020}, volume = {17}, number = {2}, pages = {195--211}, abstract = {

In this work, we consider a new fractional derivative with nonsingular kernel introduced by Caputo–Fabrizio (CF) and propose a finite difference method for computing the CF fractional derivatives. Based on an iterative technique, we can reduce the computational complexity from $O$($J$2$N$) to $O$($JN$), and the corresponding storage will be cut down from $O$($JN$) to $O$($N$), which makes the computation much more efficient. Besides, by adopting piece-wise Lagrange polynomials of degrees 1, 2, and 3, we derive the second, third, and fourth order discretization formulas respectively. The error analysis and numerical experiments are carefully provided for the validation of the accuracy and efficiency of the presented method.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/13647.html} }
TY - JOUR T1 - A Finite Difference Scheme for Caputo-Fabrizio Fractional Differential Equations AU - Guo , Xu AU - Li , Yutian AU - Zeng , Tieyong JO - International Journal of Numerical Analysis and Modeling VL - 2 SP - 195 EP - 211 PY - 2020 DA - 2020/02 SN - 17 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/13647.html KW - Caputo–Fabrizio derivative, fractional differential equations, higher order scheme. AB -

In this work, we consider a new fractional derivative with nonsingular kernel introduced by Caputo–Fabrizio (CF) and propose a finite difference method for computing the CF fractional derivatives. Based on an iterative technique, we can reduce the computational complexity from $O$($J$2$N$) to $O$($JN$), and the corresponding storage will be cut down from $O$($JN$) to $O$($N$), which makes the computation much more efficient. Besides, by adopting piece-wise Lagrange polynomials of degrees 1, 2, and 3, we derive the second, third, and fourth order discretization formulas respectively. The error analysis and numerical experiments are carefully provided for the validation of the accuracy and efficiency of the presented method.

Xu Guo, Yutian Li & Tieyong Zeng. (2020). A Finite Difference Scheme for Caputo-Fabrizio Fractional Differential Equations. International Journal of Numerical Analysis and Modeling. 17 (2). 195-211. doi:
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