An iterative method for the Cauchy problem for second-order elliptic equations


The problem of reconstructing the solution to a second-order elliptic equation in a doubly-connected domain from knowledge of the solution and its normal derivative on the outer part of the boundary of the solution domain, that is from Cauchy data, is considered. An iterative method is given to generate a stable numerical approximation to this inverse ill-posed problem. The procedure is physically feasible in that boundary data is updated with data of the same type in the iterations, meaning that Dirichlet values is updated with Dirichlet values from the previous step and Neumann values by Neumann data. Proof of convergence and stability are given by showing that the proposed method is an extension of the Landweber method for an operator equation reformulation of the Cauchy problem. Connection with the alternating method is discussed. Numerical examples are included confirming the feasibility of the suggested approach.

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Divisions: College of Engineering & Physical Sciences > School of Informatics and Digital Engineering > Mathematics
College of Engineering & Physical Sciences > Systems analytics research institute (SARI)
Additional Information: © 2018, Elsevier. Licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International
Publication ISSN: 1879-2162
Full Text Link: http://linkingh ... 020740318302376
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PURE Output Type: Article
Published Date: 2018-07
Published Online Date: 2018-04-24
Accepted Date: 2018-04-21
Authors: Baravdish, George
Borachok, Ihor
Chapko, Roman
Tomas Johansson, B. (ORCID Profile 0000-0001-9066-7922)
Slodička, Marián

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