An alternating boundary integral based method for a Cauchy problem for the Laplace equation in semi-infinite regions

Roman Chapko, B. Tomas Johansson

Research output: Contribution to journalArticle

Abstract

We consider a Cauchy problem for the Laplace equation in a two-dimensional semi-infinite region with a bounded inclusion, i.e. the region is the intersection between a half-plane and the exterior of a bounded closed curve contained in the half-plane. The Cauchy data are given on the unbounded part of the boundary of the region and the aim is to construct the solution on the boundary of the inclusion. In 1989, Kozlov and Maz'ya [10] proposed an alternating iterative method for solving Cauchy problems for general strongly elliptic and formally self-adjoint systems in bounded domains. We extend their approach to our setting and in each iteration step mixed boundary value problems for the Laplace equation in the semi-infinite region are solved. Well-posedness of these mixed problems are investigated and convergence of the alternating procedure is examined. For the numerical implementation an efficient boundary integral equation method is proposed, based on the indirect variant of the boundary integral equation approach. The mixed problems are reduced to integral equations over the (bounded) boundary of the inclusion. Numerical examples are included showing the feasibility of the proposed method.
Original languageEnglish
Pages (from-to)317-333
Number of pages17
JournalInverse Problems and Imaging
Volume2
Issue number3
DOIs
Publication statusPublished - Aug 2008

Fingerprint

Laplace equation
Boundary Integral
Laplace's equation
Cauchy Problem
Boundary integral equations
Inclusion
Mixed Problem
Half-plane
Iterative methods
Iteration
Adjoint System
Boundary Integral Equation Method
Boundary value problems
Integral equations
Mixed Boundary Value Problem
Closed curve
Boundary Integral Equations
Well-posedness
Cauchy
Bounded Domain

Keywords

  • Laplace equation
  • Cauchy problem
  • semi-infinite region
  • alternating method
  • Green's functions
  • integral equation of the first kind
  • quadrature method
  • trigonometric- and sinc- quadrature rules

Cite this

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abstract = "We consider a Cauchy problem for the Laplace equation in a two-dimensional semi-infinite region with a bounded inclusion, i.e. the region is the intersection between a half-plane and the exterior of a bounded closed curve contained in the half-plane. The Cauchy data are given on the unbounded part of the boundary of the region and the aim is to construct the solution on the boundary of the inclusion. In 1989, Kozlov and Maz'ya [10] proposed an alternating iterative method for solving Cauchy problems for general strongly elliptic and formally self-adjoint systems in bounded domains. We extend their approach to our setting and in each iteration step mixed boundary value problems for the Laplace equation in the semi-infinite region are solved. Well-posedness of these mixed problems are investigated and convergence of the alternating procedure is examined. For the numerical implementation an efficient boundary integral equation method is proposed, based on the indirect variant of the boundary integral equation approach. The mixed problems are reduced to integral equations over the (bounded) boundary of the inclusion. Numerical examples are included showing the feasibility of the proposed method.",
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An alternating boundary integral based method for a Cauchy problem for the Laplace equation in semi-infinite regions. / Chapko, Roman; Johansson, B. Tomas.

In: Inverse Problems and Imaging, Vol. 2, No. 3, 08.2008, p. 317-333.

Research output: Contribution to journalArticle

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