[1] Wang Zewen,. Determination of pollution point source in parabolic system model [J]. Journal of Southeast University (English Edition), 2009, 25 (2): 278-285. [doi:10.3969/j.issn.1003-7985.2009.02.029]

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Determination of pollution point source in parabolic system model()

确定抛物型方程组模型中的污染点源

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Journal of Southeast University (English Edition)[ISSN:1003-7985/CN:32-1325/N]

Volumn:

25

Issue:

2009 2

Page:

278-285

Research Field:

Mathematics, Physics, Mechanics

Publishing date:

2009-06-30

Info

Title:

Determination of pollution point source in parabolic system model

确定抛物型方程组模型中的污染点源

Author(s):

Wang Zewen

Department of Mathematics, Southeast University, Nanjing 210096, China
School of Mathematics and Informational Science, East China Institute of Technology, Fuzhou 344000, China

王泽文

东南大学数学系, 南京 210096; 东华理工大学数学与信息科学学院, 抚州 344000

Keywords:

inverse source problem;  parabolic system;  uniqueness;  local Lipschitz stability;  pollution source

源项反演;  抛物型方程组;  惟一性;  局部Lipschitz稳定;  污染源

PACS:

O175.24

DOI:

10.3969/j.issn.1003-7985.2009.02.029

Abstract:

This paper considers an inverse problem for a partial differential equation to identify a pollution point source in a watershed.The mathematical model of the problem is a weakly coupled system of two linear parabolic equations for the concentrations u(x, t)and v(x, t)with an unknown point source F(x, t)=λ(t)δ(x-s)related to the concentration u(x, t), where s is the point source location and λ(t)is the amplitude of the pollution point source.Assuming that source F becomes inactive after time T^*, it is proved that it can be uniquely determined by the indirect measurements {v(0, t), v(a, t), v(b, t), v(l, t), 0<t≤T, T^*<T}, and, thus, the local Lipschitz stability for this inverse source problem is obtained.Based on the proof of its uniqueness, an inversion scheme is presented to determine the point source.Finally, two numerical examples are given to show the feasibility of the inversion scheme.

考虑了在流域中确定单个污染点源的偏微分方程反问题.该反问题的数学模型是关于浓度u(x, t)和v(x, t)的弱耦合线性抛物型方程组, 其中关于浓度u(x, t)的点源F(x, t)=λ(t)δ(x-s)是未知的, 这里s表示点源位置, λ(t)表示污染点源的排放强度.在已知污染源于时刻T^*停止排放的条件下, 证明了F(x, t)=λ(t)δ(x-s)可由间接测量数据{v(0, t), v(a, t), v(b, t), v(l, t), 0<t≤T, T^*<T}惟一决定, 且该反问题是局部Lipschitz稳定的.基于惟一性的证明方法, 提出了决定点源的反演算法.最后, 给出的2个数值例子表明了反演算法是可行的.

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Memo

Memo:

Biography: Wang Zewen(1974—), male, graduate, associate professor, zwwang@ecit.edu.cn.
Foundation items: The National Natural Science Foundation of China(No.10861001), the Natural Science Foundation of Jiangxi Province.
Citation: Wang Zewen.Determination of pollution point source in parabolic system model[J].Journal of Southeast University(English Edition), 2009, 25(2):278-285.

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