球对称域上Caputo-Hadamard分数阶扩散方程源项反演问题

兰州理工大学学报 ›› 2025, Vol. 51 ›› Issue (5): 154-162.

球对称域上Caputo-Hadamard分数阶扩散方程源项反演问题

张晨雨, 杨帆^*

兰州理工大学理学院, 甘肃兰州 730050

收稿日期:2023-08-09 出版日期:2025-10-28 发布日期:2025-10-25
通讯作者: 杨帆(1976-),男,甘肃灵台人,教授. Email:yfggd114@163.com
基金资助:
国家自然科学基金(12461083),甘肃省自然科学基金(21JR7RA214)

Source term inversion of Caputo-Hadamard fractional diffusion equation on spherically symmetric domain

ZHANG Chen-yu, YANG Fan

School of Science, Lanzhou University of Technology, Lanzhou 730050, China

Received:2023-08-09 Online:2025-10-28 Published:2025-10-25

摘要/Abstract

摘要： 研究了球对称域上Caputo-Hadamard分数阶扩散方程源项反演问题.应用Laplace变换和 Laplace 逆变换得到了问题的精确解. 对精确解进行分析, 发现问题是不适定的.在此基础上,采用拟边界正则化方法解决解的稳定性问题,并分别给出了在先验正则化参数选择规则和后验正则化参数选择规则下的两个收敛误差估计.采用有限差分离散得到迭代格式,通过数值算例说明了该正则化方法的有效性和稳定性.

关键词: 反问题, Caputo-Hadamard分数阶扩散方程, 球对称域, 识别未知源, 拟边界正则化方法

Abstract: The problem of identifying the source term for the Caputo-Hadamard fractional diffusion equation on the spherically symmetric domain is studied. The exact solution to the problem is obtained by using Laplace transform and inverse Laplace transform. Analytical examination of the exact solution reveals that the problem is ill-posed. On this basis, the Quasi-boundary regularization method is used to solve the stability problem of the solution. Furthermore, two convergence error estimates under both a priori regularization parameter selection rule and a posteriori regularization parameter selection rule are provided respectively. The finite difference discretization method is used to obtain the iterative scheme. The effectiveness and stability of the regularization method are demonstrated through numerical examples.

Key words: inverse problem, Caputo-Hadamard fractional diffusion equation, spherically symmetric domain, identifying unknown source, Quasi-boundary regularization method

中图分类号:

O241.1

张晨雨, 杨帆. 球对称域上Caputo-Hadamard分数阶扩散方程源项反演问题[J]. 兰州理工大学学报, 2025, 51(5): 154-162.

ZHANG Chen-yu, YANG Fan. Source term inversion of Caputo-Hadamard fractional diffusion equation on spherically symmetric domain[J]. Journal of Lanzhou University of Technology, 2025, 51(5): 154-162.

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