Simultaneous Recovery of Two Time-Dependent Coefficients in a Multi-Term Time-Fractional Diffusion Equation

IF 1 4区数学 Q3 MATHEMATICS, APPLIED

Computational Methods in Applied Mathematics Pub Date : 2023-07-12 DOI:10.1515/cmam-2022-0210

Wenjun Ma, Liangliang Sun

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Abstract

Abstract This paper deals with an inverse problem on simultaneously determining a time-dependent potential term and a time source function from two-point measured data in a multi-term time-fractional diffusion equation. First we study the existence, uniqueness and some regularities of the solution for the direct problem by using the fixed point theorem. Then a nice conditional stability estimate of inversion coefficients problem is obtained based on the regularity of the solution to the direct problem and a fine property of the Caputo fractional derivative. In addition, the ill-posedness of the inverse problem is illustrated and we transfer the inverse problem into a variational problem. Moreover, the existence and convergence of the minimizer for the variational problem are given. Finally, we use a modified Levenberg–Marquardt method to reconstruct numerically the approximate functions of two unknown time-dependent coefficients effectively. Numerical experiments for three examples in one- and two-dimensional cases are provided to show the validity and robustness of the proposed method.

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多项时间分数阶扩散方程中两个时间相关系数的同时恢复

摘要研究了多项时间分数扩散方程中两点测量数据同时确定时变势项和时变源函数的反问题。首先利用不动点定理研究了直接问题解的存在唯一性和一些规律。然后利用直接问题解的正则性和分数阶导数的优良性质，得到了反演系数问题的条件稳定性估计。此外，还说明了反问题的病态性，并将反问题转化为变分问题。此外，还给出了变分问题的最小值的存在性和收敛性。最后，利用改进的Levenberg-Marquardt方法对两个未知时相关系数的近似函数进行了数值重构。通过三个一维和二维实例的数值实验，验证了该方法的有效性和鲁棒性。

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来源期刊

Computational Methods in Applied Mathematics MATHEMATICS, APPLIED-

CiteScore

2.40

自引率

7.70%

发文量

期刊介绍： The highly selective international mathematical journal Computational Methods in Applied Mathematics　(CMAM) considers original mathematical contributions to computational methods and numerical analysis with applications mainly related to PDEs. CMAM seeks to be interdisciplinary while retaining the common thread of numerical analysis, it is intended to be readily readable and meant for a wide circle of researchers in applied mathematics. The journal is published by De Gruyter on behalf of the Institute of Mathematics of the National Academy of Science of Belarus.