Open Access

ARTICLE

Cauchy Problem for the Laplace Equation in 2D and 3D Doubly Connected Domains

by Ji-Chuan Liu¹, Quan-Guo Zhang²

1 Corresponding author. Department of Mathematics, China University of Mining and Technology, Xuzhou, Jiangsu 221116, PR China. Email: liujichuan2003@126.com
2 School of mathematical sciences, Luoyang Normal University, Luoyang, Henan 471000, PR China. Email: zhangqg07@163.com

Computer Modeling in Engineering & Sciences 2013, 93(3), 203-220. https://doi.org/10.3970/cmes.2013.093.203

Abstract

In this paper, we propose an algorithm to solve a Cauchy problem of the Laplace equation in doubly connected domains for 2D and 3D cases in which the Cauchy data are given on the outer boundary. We want to seek a solution in the form of the single-layer potential and discrete it by parametrization to yield an ill-conditioned system of algebraic equations. Then we apply the Tikhonov regularization method to solve this ill-posed problem and obtain a stable numerical solution. Based on the regularization parameter chosen suitably by GCV criterion, the proposed method can get the approximate temperature and heat flux on the inner boundary. Numerical examples illustrate that the proposed method is reasonable and feasible.

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Keywords

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