A regularization method for a Cauchy problem of Laplace's equation in an annular domain
type: Item
instance of: scholarly article
title: A regularization method for a Cauchy problem of Laplace's equation in an annular domain
zbMATH Open document ID: 1259.65138
author: Xianqiang Yang
published in: Mathematics and Computers in Simulation
publication date: 2012-11-15T00:00:00Z
review text: The Cauchy problem \[ \Delta u=0\quad \text{in} \quad S, \] \[ u= f \quad on \quad \Gamma,\quad \frac{du}{d\nu}= g \quad \text{on} \quad\Gamma, \] is defined in an annular domain \( S\subset \mathbb R^{2}\) with the boundary \( dS=\Gamma\;U \gamma \), where \( \Gamma\) and \(\gamma\) are two circles of radius \( r_{1}>r_{2}\). The functions \(f\) and \(g\) cannot be known exactly and it is assumed that \[ \|f^{\delta} - f\|\leq \delta, \quad \|g^{\delta} -g \|\leq \delta. \] To obtain a stable approximate solution of this problem the authors use the Tikhonov regularization parameter \(\alpha\) in the minimization of the functional \[ I(v)= \|v - f^{\delta}\|^{2} + \omega\|\frac{dv}{d\nu}-g^{\delta}\|^{2}+\alpha\|v\|^{2} \] in the space of fundamental solutions corresponding to uniformly located points \(Q_{1i}\) and \(Q_{2i}\) on the circles \(R_{1}\) and \(R_{2}\).
Mathematics Subject Classification ID: 65M30
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 35R25
Mathematics Subject Classification ID: 65M12
Mathematics Subject Classification ID: 65M80
zbMATH DE Number: 6107081
zbMATH Keywords: Tikhonov regularization
zbMATH Keywords: convergence
zbMATH Keywords: method of fundamental solutions
zbMATH Keywords: Cauchy problem
zbMATH Keywords: Laplace equation
reviewed by: Ivan Secrieru
MaRDI profile type: Publication
full work available at URL: j.matcom.2012.05.009
OpenAlex ID: W2000427788
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DOI: 10.1016/J.MATCOM.2012.05.009
description: scientific article; zbMATH DE number 6107081