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Volume 43, Issue 3
A High Order Scheme for Fractional Differential Equations with the Caputo-Hadamard Derivative

Xingyang Ye, Junying Cao & Chuanju Xu

DOI: 10.4208/jcm.2312-m2023-0098

J. Comp. Math., 43 (2025), pp. 615-640.

Published online: 2024-11

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  • Abstract

In this paper, we consider numerical solutions of the fractional diffusion equation with the $α$ order time fractional derivative defined in the Caputo-Hadamard sense. A high order time-stepping scheme is constructed, analyzed, and numerically validated. The contribution of the paper is twofold: 1) regularity of the solution to the underlying equation is investigated, 2) a rigorous stability and convergence analysis for the proposed scheme is performed, which shows that the proposed scheme is $3 + α$ order accurate. Several numerical examples are provided to verify the theoretical statement.

  • Keywords

Caputo-Hadamard derivative, Fractional differential equations, High order scheme, Stability and convergence analysis.

  • AMS Subject Headings

34A08, 65L05, 65L20

  • Copyright

COPYRIGHT: © Global Science Press

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  • TXT
@Article{JCM-43-615, author = {Ye , XingyangCao , Junying and Xu , Chuanju}, title = {A High Order Scheme for Fractional Differential Equations with the Caputo-Hadamard Derivative}, journal = {Journal of Computational Mathematics}, year = {2024}, volume = {43}, number = {3}, pages = {615--640}, abstract = {

In this paper, we consider numerical solutions of the fractional diffusion equation with the $α$ order time fractional derivative defined in the Caputo-Hadamard sense. A high order time-stepping scheme is constructed, analyzed, and numerically validated. The contribution of the paper is twofold: 1) regularity of the solution to the underlying equation is investigated, 2) a rigorous stability and convergence analysis for the proposed scheme is performed, which shows that the proposed scheme is $3 + α$ order accurate. Several numerical examples are provided to verify the theoretical statement.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2312-m2023-0098}, url = {http://global-sci.org/intro/article_detail/jcm/23552.html} }
TY - JOUR T1 - A High Order Scheme for Fractional Differential Equations with the Caputo-Hadamard Derivative AU - Ye , Xingyang AU - Cao , Junying AU - Xu , Chuanju JO - Journal of Computational Mathematics VL - 3 SP - 615 EP - 640 PY - 2024 DA - 2024/11 SN - 43 DO - http://doi.org/10.4208/jcm.2312-m2023-0098 UR - https://global-sci.org/intro/article_detail/jcm/23552.html KW - Caputo-Hadamard derivative, Fractional differential equations, High order scheme, Stability and convergence analysis. AB -

In this paper, we consider numerical solutions of the fractional diffusion equation with the $α$ order time fractional derivative defined in the Caputo-Hadamard sense. A high order time-stepping scheme is constructed, analyzed, and numerically validated. The contribution of the paper is twofold: 1) regularity of the solution to the underlying equation is investigated, 2) a rigorous stability and convergence analysis for the proposed scheme is performed, which shows that the proposed scheme is $3 + α$ order accurate. Several numerical examples are provided to verify the theoretical statement.

Ye , XingyangCao , Junying and Xu , Chuanju. (2024). A High Order Scheme for Fractional Differential Equations with the Caputo-Hadamard Derivative. Journal of Computational Mathematics. 43 (3). 615-640. doi:10.4208/jcm.2312-m2023-0098
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