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

Roman Chapko; B. Tomas Johansson

doi:10.3934/ipi.2012.6.25

Article Contents

2012, Volume 6, Issue 1: 25-38. Doi: 10.3934/ipi.2012.6.25

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

Roman Chapko^1, and
B. Tomas Johansson^2,

1.
Faculty of Applied Mathematics and Computer Science, Ivan Franko National University of Lviv, 79000 Lviv, Ukraine
2.
School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT

Received: December 31, 2010

Revised: November 30, 2011

Published: January 2012

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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 $L^2$-setting. An effective way of discretizing these boundary integral equations based on the Nyström 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.

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Mathematics Subject Classification: Primary: 35R25; Secondary: 35J05, 65R20.

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