On the numerical solution of a Cauchy problem for the Laplace equation via a direct integral equation approach

Chapko, Roman and Johansson, B. Tomas (2012). On the numerical solution of a Cauchy problem for the Laplace equation via a direct integral equation approach. Inverse Problems and Imaging, 6 (1), 25–38.

Abstract

We investigate the problem of determining the stationary temperature field on an inclusion from given Cauchy data on an accessible exterior boundary. On this accessible part the temperature (or the heat flux) is known, and, additionally, on a portion of this exterior boundary the heat flux (or temperature) is also given. We propose a direct boundary integral approach in combination with Tikhonov regularization for the stable determination of the temperature and flux on the inclusion. To determine these quantities on the inclusion, boundary integral equations are derived using Green’s functions, and properties of these equations are shown in an L2-setting. An effective way of discretizing these boundary integral equations based on the Nystr¨om method and trigonometric approximations, is outlined. Numerical examples are included, both with exact and noisy data, showing that accurate approximations can be obtained with small computational effort, and the accuracy is increasing with the length of the portion of the boundary where the additionally data is given.

Publication DOI: https://doi.org/10.3934/ipi.2012.6.25
Divisions: Engineering & Applied Sciences > Mathematics
Uncontrolled Keywords: cauchy problem,integral equation of the first kind,logarithmic- and hypersingularities,single- and double layer potentials,Green’s function,Laplace equation,Tikhonov regularization
Published Date: 2012-02

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