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A High Order Scheme for Fractional Differential Equations with the Caputo-Hadamard Derivative
Xingyang Ye, Junying Cao and Chuanju Xu

J. Comp. Math. DOI: 10.4208/jcm.2312-m2023-0098

Publication Date : 2024-04-01

  • 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.

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