一个分数阶生长-抑制系统的参数反问题

doi:10.12286/jssx.j2022-0928

计算数学 ›› 2023, Vol. 45 ›› Issue (2): 215-229.DOI: 10.12286/jssx.j2022-0928

一个分数阶生长-抑制系统的参数反问题

杨冰, 李功胜

山东理工大学数学与统计学院, 淄博 255049

收稿日期:2022-03-21 出版日期:2023-05-14 发布日期:2023-05-13
通讯作者: 李功胜, Email:ligs@sdut.edu.cn
基金资助:
国家自然科学基金(11871313)和山东省自然科学基金(ZR2019MA021)资助.

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杨冰, 李功胜. 一个分数阶生长-抑制系统的参数反问题[J]. 计算数学, 2023, 45(2): 215-229.

Yang Bing, Li Gongsheng. AN INVERSE COEFFICIENT PROBLEM FOR A FRACTIONAL-ORDER ACTIVATOR-INHIBITOR SYSTEM[J]. Mathematica Numerica Sinica, 2023, 45(2): 215-229.

AN INVERSE COEFFICIENT PROBLEM FOR A FRACTIONAL-ORDER ACTIVATOR-INHIBITOR SYSTEM

Yang Bing, Li Gongsheng

School of Math and Statist, Shandong University of Technology, Zibo 255049, China

Received:2022-03-21 Online:2023-05-14 Published:2023-05-13

本文研究一个分数阶生长-抑制线性系统模型及其参数反问题.首先利用Laplace逆变换得到正问题解的唯一存在性.其次, 考虑一个利用内点观测数据确定微分阶数与衰减率的反问题, 应用极值原理在Laplace像空间中证明反演的唯一性.最后, 基于正问题的有限差分解, 应用同伦正则化算法进行数值反演.计算结果表明算法的收敛性及反问题的数值稳定性.

关键词： 分数阶生长-抑制系统, Laplace变换, 有限差分格式, 参数反问题, 唯一性, 数值反演

This article deals with a linear fractional-order activator-inhibitor model and a related inverse coefficient problem. The unique existence of the solution to the forward problem is obtained based on the Laplace inverse transform. An inverse problem of determining the fractional order and the attenuation rate is investigated using the measurements at one interior point, and the uniqueness of the inverse problem is proved in the mapping space of Laplace transform by the maximum principle. An implicit finite difference scheme is established to give the numerical solution of the forward problem, and numerical inversions with noisy data are performed by the homotopy regularization algorithm. The inversion solutions approximate to the exact solution demonstrate the algorithm’s convergence and the numerical stability of the inverse problem.

Key words: Fractional-order activator-inhibitor system, Laplace transform, finite difference scheme, inverse coefficient problem, uniqueness, numerical inversion

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一个分数阶生长-抑制系统的参数反问题

AN INVERSE COEFFICIENT PROBLEM FOR A FRACTIONAL-ORDER ACTIVATOR-INHIBITOR SYSTEM

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