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
References:
[1]

G. Bastay, T. Johansson, V. A. Kozlov and D. Lesnic, An alternating method for the stationary Stokes system, ZAMM, 86 (2006), 268-280. doi: 10.1002/zamm.200410238.  Google Scholar

[2]

J. Baumeister and A. Leitāo, On iterative methods for solving ill-posed problems modeled by partial differential equations, J. Inv. Ill-Posed Probl., 9 (2001), 13-29.  Google Scholar

[3]

A.-P. Calderón, Uniqueness in the Cauchy problem for partial differential equations, Amer. J. Math., 80 (1958), 16-36. doi: 10.2307/2372819.  Google Scholar

[4]

T. Carleman, Sur un probléme d'unicité pur les systémes d'équations aux dérivées partielles á deux variables indépendantes, (French) Ark. Mat., Astr. Fys., 26 (1939), 1-9.  Google Scholar

[5]

R. Chapko and R. Kress, On a quadrature method for a logarithmic integral equation of the first kind, in "World Scientific Series in Applicable Analysis, Contributions in Numerical Mathematics, Vol. 2'' (ed. Agarwal), World Scientific, Singapore, (1993), 127-140.  Google Scholar

[6]

H. W. Engl and A. Leitāo, A Mann iterative regularization method for elliptic Cauchy problems, Numer. Funct. Anal. Optim., 22 (2001), 861-884. doi: 10.1081/NFA-100108313.  Google Scholar

[7]

U. Hämarik and T. Raus, On the choice of the regularization parameter in ill-posed problems with approximately given noise level of data, J. Inverse Ill-Posed Probl., 14 (2006), 251-266. doi: 10.1515/156939406777340928.  Google Scholar

[8]

M. A. Jawson and G. Symm, "Integral Equations Methods in Potential Theory and Elastostatics,'' Academic Press, London, 1977.  Google Scholar

[9]

M. Jourhmane and A. Nachaoui, An alternating method for an inverse Cauchy problem, Numer. Algorithms, 21 (1999), 247-260. doi: 10.1023/A:1019134102565.  Google Scholar

[10]

V. A. Kozlov and V. G. Maz'ya, On iterative procedures for solving ill-posed boundary value problems that preserve differential equations, Algebra i Analiz, 1 (1989), 144-170. English transl.: Leningrad Math. J., 1 (1990), 1207-1228. Google Scholar

[11]

V. A. Kozlov, V. G. Maz'ya and A. V. Fomin, An iterative method for solving the Cauchy problem for elliptic equations, 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]

R. Kress, "Linear Integral Equations," 2nd edition, Springer-Verlag, Heidelberg 1999.  Google Scholar

[13]

D. Lesnic, L. Elliot and D. B. Ingham, An iterative boundary element method for solving numerically the Cauchy problem for the Laplace equation, Eng. Anal. Bound. Elem., 20 (1997), 123-133. doi: 10.1016/S0955-7997(97)00056-8.  Google Scholar

[14]

W. McLean, "Strongly Elliptic Systems and Boundary Integral Equations,'' 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]

C. Miranda, "Partial Differential Equations of Elliptic Type,'' Springer-Verlag, New-York, 1970.  Google Scholar

[17]

A. Polyanin, "Handbook of Linear Partial Differential Equations for Engineers and Scientists,'' Chapman & Hall/CRC Press, 2002.  Google Scholar

[18]

F. Stenger, "Numerical Methods Based on Sinc and Analytic Functions,'' Springer-Verlag, Heidelberg, 1993.  Google Scholar

[19]

G. M. Vainikko and A. Y. Veretennikov, "Iteration Procedures in Ill-Posed Problems,'' Nauka Publ., Moscow, 1986 (in Russian).  Google Scholar

show all references

References:
[1]

G. Bastay, T. Johansson, V. A. Kozlov and D. Lesnic, An alternating method for the stationary Stokes system, ZAMM, 86 (2006), 268-280. doi: 10.1002/zamm.200410238.  Google Scholar

[2]

J. Baumeister and A. Leitāo, On iterative methods for solving ill-posed problems modeled by partial differential equations, J. Inv. Ill-Posed Probl., 9 (2001), 13-29.  Google Scholar

[3]

A.-P. Calderón, Uniqueness in the Cauchy problem for partial differential equations, Amer. J. Math., 80 (1958), 16-36. doi: 10.2307/2372819.  Google Scholar

[4]

T. Carleman, Sur un probléme d'unicité pur les systémes d'équations aux dérivées partielles á deux variables indépendantes, (French) Ark. Mat., Astr. Fys., 26 (1939), 1-9.  Google Scholar

[5]

R. Chapko and R. Kress, On a quadrature method for a logarithmic integral equation of the first kind, in "World Scientific Series in Applicable Analysis, Contributions in Numerical Mathematics, Vol. 2'' (ed. Agarwal), World Scientific, Singapore, (1993), 127-140.  Google Scholar

[6]

H. W. Engl and A. Leitāo, A Mann iterative regularization method for elliptic Cauchy problems, Numer. Funct. Anal. Optim., 22 (2001), 861-884. doi: 10.1081/NFA-100108313.  Google Scholar

[7]

U. Hämarik and T. Raus, On the choice of the regularization parameter in ill-posed problems with approximately given noise level of data, J. Inverse Ill-Posed Probl., 14 (2006), 251-266. doi: 10.1515/156939406777340928.  Google Scholar

[8]

M. A. Jawson and G. Symm, "Integral Equations Methods in Potential Theory and Elastostatics,'' Academic Press, London, 1977.  Google Scholar

[9]

M. Jourhmane and A. Nachaoui, An alternating method for an inverse Cauchy problem, Numer. Algorithms, 21 (1999), 247-260. doi: 10.1023/A:1019134102565.  Google Scholar

[10]

V. A. Kozlov and V. G. Maz'ya, On iterative procedures for solving ill-posed boundary value problems that preserve differential equations, Algebra i Analiz, 1 (1989), 144-170. English transl.: Leningrad Math. J., 1 (1990), 1207-1228. Google Scholar

[11]

V. A. Kozlov, V. G. Maz'ya and A. V. Fomin, An iterative method for solving the Cauchy problem for elliptic equations, 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]

R. Kress, "Linear Integral Equations," 2nd edition, Springer-Verlag, Heidelberg 1999.  Google Scholar

[13]

D. Lesnic, L. Elliot and D. B. Ingham, An iterative boundary element method for solving numerically the Cauchy problem for the Laplace equation, Eng. Anal. Bound. Elem., 20 (1997), 123-133. doi: 10.1016/S0955-7997(97)00056-8.  Google Scholar

[14]

W. McLean, "Strongly Elliptic Systems and Boundary Integral Equations,'' 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]

C. Miranda, "Partial Differential Equations of Elliptic Type,'' Springer-Verlag, New-York, 1970.  Google Scholar

[17]

A. Polyanin, "Handbook of Linear Partial Differential Equations for Engineers and Scientists,'' Chapman & Hall/CRC Press, 2002.  Google Scholar

[18]

F. Stenger, "Numerical Methods Based on Sinc and Analytic Functions,'' Springer-Verlag, Heidelberg, 1993.  Google Scholar

[19]

G. M. Vainikko and A. Y. Veretennikov, "Iteration Procedures in Ill-Posed Problems,'' Nauka Publ., Moscow, 1986 (in Russian).  Google Scholar

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