修正的Helmholtz方程未知源识别反问题

数学物理学报 ›› 2012, Vol. 32 ›› Issue (3): 557-565.

修正的Helmholtz方程未知源识别反问题

杨帆^1**|傅初黎²|李晓晓¹

1.兰州理工大学理学院兰州 730050|2.兰州大学数学与统计学院兰州 730000

收稿日期:2010-10-08 修回日期:2012-03-23 出版日期:2012-06-25 发布日期:2012-06-25
基金资助:
国家自然科学基金(10671085)和兰州理工大学优秀青年基金(Q2010015)资助

The Inverse Problem of Identifying the Unknown Source for the Modified Helmholtz Equation

YANG Fan^1**, FU Chu-Li², LI Xiao-Xiao¹

1.School of Science, Lanzhou University of Technology, |Lanzhou |730050|2.School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000

Received:2010-10-08 Revised:2012-03-23 Online:2012-06-25 Published:2012-06-25
Supported by:
国家自然科学基金(10671085)和兰州理工大学优秀青年基金(Q2010015)资助

摘要/Abstract

摘要：

探讨半带状区域上二维修正的Helmholtz方程只含有一个空间变量的未知源识别反问题. 这类问题是不适定的, 即问题的解(如果存在的话) 不连续依赖于测量数据. 利用拟可逆正则化方法, 得到问题的一个正则近似解, 并且给出正则解和精确解之间收敛的误差估计. 数值实验表明拟可逆正则化方法对于这种未知源识别非常有效.

关键词: 修正的Helmholtz方程, 未知源, 正则化, 拟可逆

Abstract:

This paper discusses the inverse problem of determining a spacewise unknonwn source for the two dimensional modified Helmholtz equation in a half strip domain. This problem is ill-posed, i.e., the solution does not depend continuously on the data. A regularization solution of the inverse problem is obtained by the quasi-reversibility regularization method. For the regularization solution, convergence
estimate is obtained between the regularization solution and the exact solution. The numerical example shows that the quasi-reversibility regularization method works effectively for identification of the unknown source.

Key words: Modified Helmholtz equation, Unknown source, Regularization, Quasi-reversibility

中图分类号:

35J05

杨帆, 傅初黎, 李晓晓. 修正的Helmholtz方程未知源识别反问题[J]. 数学物理学报, 2012, 32(3): 557-565.

YANG Fan, FU Chu-Li, LI Xiao-Xiao. The Inverse Problem of Identifying the Unknown Source for the Modified Helmholtz Equation[J]. Acta mathematica scientia,Series A, 2012, 32(3): 557-565.

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