Numerical Analysis of a High-Order Scheme with Nonuniform Time Grids for Caputo-Hadamard Fractional Reaction Sub-diffusion Equations

doi:10.1007/s42967-024-00441-7

Communications on Applied Mathematics and Computation ›› 2026, Vol. 8 ›› Issue (1): 338-365.doi: 10.1007/s42967-024-00441-7

Numerical Analysis of a High-Order Scheme with Nonuniform Time Grids for Caputo-Hadamard Fractional Reaction Sub-diffusion Equations

Chunxiu Liu¹, Junying Cao², Tong Lyu¹, Xingyang Ye¹

1. School of Science, Jimei University, Xiamen, 361021, Fujian, China;
2. School of Data Science and Information Engineering, Guizhou Minzu University, Guiyang, 550025, Guizhou, China

收稿日期:2024-02-19 修回日期:2024-05-25 出版日期:2026-02-20 发布日期:2026-02-11
通讯作者: Xingyang Ye,E-mail:yexingyang@jmu.edu.cn E-mail:yexingyang@jmu.edu.cn
基金资助:
This work is supported by the Natural Science Foundation of Fujian Province of China (2022J01338), Fujian Alliance of Mathematics (2024SXLMMS03), the NSFC (12361083 and 11961009), and the Natural Science Research Project of Department of Education of Guizhou Province, China (QJJ2023012).

Numerical Analysis of a High-Order Scheme with Nonuniform Time Grids for Caputo-Hadamard Fractional Reaction Sub-diffusion Equations

Chunxiu Liu¹, Junying Cao², Tong Lyu¹, Xingyang Ye¹

1. School of Science, Jimei University, Xiamen, 361021, Fujian, China;
2. School of Data Science and Information Engineering, Guizhou Minzu University, Guiyang, 550025, Guizhou, China

Received:2024-02-19 Revised:2024-05-25 Online:2026-02-20 Published:2026-02-11
Contact: Xingyang Ye,E-mail:yexingyang@jmu.edu.cn E-mail:yexingyang@jmu.edu.cn
Supported by:
This work is supported by the Natural Science Foundation of Fujian Province of China (2022J01338), Fujian Alliance of Mathematics (2024SXLMMS03), the NSFC (12361083 and 11961009), and the Natural Science Research Project of Department of Education of Guizhou Province, China (QJJ2023012).

摘要/Abstract

摘要： In this paper, we propose a numerical approach for the fractional reaction sub-diffusion equation with a Caputo-Hadamard derivative of fractional order α∈(0,1), whose solutions typically exhibit singular behavior at the initial time. The approach involves the use of an L2-type discrete operator of order 3-α to approximate the fractional derivative on nonuniform temporal meshes, while a standard second-order difference method is employed for uniform spatial meshes. Detailed discussions are presented on the truncation errors and coefficients of the proposed discrete fractional derivative operator. The stability as well as the accuracy of the resulting numerical scheme is rigorously analyzed on a unique nonuniform mesh. Several numerical examples are provided to validate the theoretical results and to demonstrate the efficiency of the proposed method.

关键词: Caputo-Hadamard derivative, Fractional reaction sub-diffusion equations, Nonuniform meshes, Stability and convergence

Abstract: In this paper, we propose a numerical approach for the fractional reaction sub-diffusion equation with a Caputo-Hadamard derivative of fractional order α∈(0,1), whose solutions typically exhibit singular behavior at the initial time. The approach involves the use of an L2-type discrete operator of order 3-α to approximate the fractional derivative on nonuniform temporal meshes, while a standard second-order difference method is employed for uniform spatial meshes. Detailed discussions are presented on the truncation errors and coefficients of the proposed discrete fractional derivative operator. The stability as well as the accuracy of the resulting numerical scheme is rigorously analyzed on a unique nonuniform mesh. Several numerical examples are provided to validate the theoretical results and to demonstrate the efficiency of the proposed method.

Key words: Caputo-Hadamard derivative, Fractional reaction sub-diffusion equations, Nonuniform meshes, Stability and convergence

中图分类号:

Chunxiu Liu, Junying Cao, Tong Lyu, Xingyang Ye. Numerical Analysis of a High-Order Scheme with Nonuniform Time Grids for Caputo-Hadamard Fractional Reaction Sub-diffusion Equations[J]. Communications on Applied Mathematics and Computation, 2026, 8(1): 338-365.

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Numerical Analysis of a High-Order Scheme with Nonuniform Time Grids for Caputo-Hadamard Fractional Reaction Sub-diffusion Equations

Numerical Analysis of a High-Order Scheme with Nonuniform Time Grids for Caputo-Hadamard Fractional Reaction Sub-diffusion Equations

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