American Institute of Mathematical Sciences

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August  2008, 2(3): 317-333. doi: 10.3934/ipi.2008.2.317

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

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, United Kingdom

Received  December 2007 Revised  June 2008 Published  July 2008

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.
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..
Mathematics Subject Classification: Primary: 35R25; Secondary: 35J05, 65R2.
Citation: Roman Chapko, B. Tomas Johansson. An alternating boundary integral based method for a Cauchy problem for the Laplace equation in semi-infinite regions. Inverse Problems & Imaging, 2008, 2 (3) : 317-333. doi: 10.3934/ipi.2008.2.317
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show all references

References:
[1]

ZAMM, 86 (2006), 268-280. doi: 10.1002/zamm.200410238.  Google Scholar

[2]

J. Inv. Ill-Posed Probl., 9 (2001), 13-29.  Google Scholar

[3]

Amer. J. Math., 80 (1958), 16-36. doi: 10.2307/2372819.  Google Scholar

[4]

Ark. Mat., Astr. Fys., 26 (1939), 1-9.  Google Scholar

[5]

Vol. 2'' (ed. Agarwal), World Scientific, Singapore, (1993), 127-140.  Google Scholar

[6]

Numer. Funct. Anal. Optim., 22 (2001), 861-884. doi: 10.1081/NFA-100108313.  Google Scholar

[7]

J. Inverse Ill-Posed Probl., 14 (2006), 251-266. doi: 10.1515/156939406777340928.  Google Scholar

[8]

Academic Press, London, 1977.  Google Scholar

[9]

Numer. Algorithms, 21 (1999), 247-260. doi: 10.1023/A:1019134102565.  Google Scholar

[10]

Algebra i Analiz, 1 (1989), 144-170. English transl.: Leningrad Math. J., 1 (1990), 1207-1228. Google Scholar

[11]

Zh. Vychisl. Mat. i Mat. Fiz., 31 (1991), 64-74. English transl.: U.S.S.R. Comput. Math. and Math. Phys., 31 (1991), 45-52.  Google Scholar

[12]

2nd edition, Springer-Verlag, Heidelberg 1999.  Google Scholar

[13]

Eng. Anal. Bound. Elem., 20 (1997), 123-133. doi: 10.1016/S0955-7997(97)00056-8.  Google Scholar

[14]

Cambridge University Press, 2000.  Google Scholar

[15]

D. Maxwell, M. Truffer, S. Avdonin and M. Stuefer, Determining glacier velocities and stresses with inverse methods: an iterative scheme,, to appear in Journal of Glaciology., ().   Google Scholar

[16]

Springer-Verlag, New-York, 1970.  Google Scholar

[17]

Chapman & Hall/CRC Press, 2002.  Google Scholar

[18]

Springer-Verlag, Heidelberg, 1993.  Google Scholar

[19]

Nauka Publ., Moscow, 1986 (in Russian).  Google Scholar

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