修正的Helmholtz方程未知源识别的Fourier截断正则化方法

数学物理学报 ›› 2014, Vol. 34 ›› Issue (4): 1040-1047.

• 论文 • 上一篇

修正的Helmholtz方程未知源识别的Fourier截断正则化方法

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:2013-01-13 修回日期:2014-03-05 出版日期:2014-08-25 发布日期:2014-08-25
基金资助:
国家自然科学基金(11171136, 11261032)、兰州理工大学优秀青年基金学校基金(Q201015)、甘肃省高校基本科研业务费和甘肃省自然科学基金(1310RJYA021)资助.

The Fourier Regularization Method for 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:2013-01-13 Revised:2014-03-05 Online:2014-08-25 Published:2014-08-25
Supported by:
国家自然科学基金(11171136, 11261032)、兰州理工大学优秀青年基金学校基金(Q201015)、甘肃省高校基本科研业务费和甘肃省自然科学基金(1310RJYA021)资助.

摘要/Abstract

摘要：

探讨半无界区域上二维修正的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Fourier截断正则化方法, 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. 数值例子表明Fourier截断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, Fourier截断

Abstract:

This paper discusses the problem of determining an unknown source which depends only on one variable for the two dimensional modified Helmholtz equation in a half infinite domain. This problem is ill-posed in the sense that the
solution (if it exists) does not depend continuously on the data. The regularization solution is obtained by the Fourier
regularization method. Convergence estimate is presented between the exact solution and the regularization solution. Numerical examples show that our proposed method is effective and stable.

Key words: Modified Helmholtz equation, Unknown source, Regularization, Fourier truncation

中图分类号:

35J05

杨帆, 傅初黎, 李晓晓. 修正的Helmholtz方程未知源识别的Fourier截断正则化方法[J]. 数学物理学报, 2014, 34(4): 1040-1047.

YANG Fan, FU Chu-Li, LI Xiao-Xiao. The Fourier Regularization Method for Identifying the Unknown Source for the Modified Helmholtz Equation[J]. Acta mathematica scientia,Series A, 2014, 34(4): 1040-1047.

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修正的Helmholtz方程未知源识别的Fourier截断正则化方法

The Fourier Regularization Method for Identifying the Unknown Source for the Modified Helmholtz Equation

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