# American Institute of Mathematical Sciences

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

 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.
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 and 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. [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. [3] A.-P. Calderón, Uniqueness in the Cauchy problem for partial differential equations, Amer. J. Math., 80 (1958), 16-36. doi: 10.2307/2372819. [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. [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. [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. [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. [8] M. A. Jawson and G. Symm, "Integral Equations Methods in Potential Theory and Elastostatics,'' Academic Press, London, 1977. [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. [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. [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. [12] R. Kress, "Linear Integral Equations," 2nd edition, Springer-Verlag, Heidelberg 1999. [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. [14] W. McLean, "Strongly Elliptic Systems and Boundary Integral Equations,'' Cambridge University Press, 2000. [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. [16] C. Miranda, "Partial Differential Equations of Elliptic Type,'' Springer-Verlag, New-York, 1970. [17] A. Polyanin, "Handbook of Linear Partial Differential Equations for Engineers and Scientists,'' Chapman & Hall/CRC Press, 2002. [18] F. Stenger, "Numerical Methods Based on Sinc and Analytic Functions,'' Springer-Verlag, Heidelberg, 1993. [19] G. M. Vainikko and A. Y. Veretennikov, "Iteration Procedures in Ill-Posed Problems,'' Nauka Publ., Moscow, 1986 (in Russian).

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. [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. [3] A.-P. Calderón, Uniqueness in the Cauchy problem for partial differential equations, Amer. J. Math., 80 (1958), 16-36. doi: 10.2307/2372819. [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. [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. [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. [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. [8] M. A. Jawson and G. Symm, "Integral Equations Methods in Potential Theory and Elastostatics,'' Academic Press, London, 1977. [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. [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. [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. [12] R. Kress, "Linear Integral Equations," 2nd edition, Springer-Verlag, Heidelberg 1999. [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. [14] W. McLean, "Strongly Elliptic Systems and Boundary Integral Equations,'' Cambridge University Press, 2000. [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. [16] C. Miranda, "Partial Differential Equations of Elliptic Type,'' Springer-Verlag, New-York, 1970. [17] A. Polyanin, "Handbook of Linear Partial Differential Equations for Engineers and Scientists,'' Chapman & Hall/CRC Press, 2002. [18] F. Stenger, "Numerical Methods Based on Sinc and Analytic Functions,'' Springer-Verlag, Heidelberg, 1993. [19] G. M. Vainikko and A. Y. Veretennikov, "Iteration Procedures in Ill-Posed Problems,'' Nauka Publ., Moscow, 1986 (in Russian).
 [1] Noui Djaidja, Mostefa Nadir. Comparison between Taylor and perturbed method for Volterra integral equation of the first kind. Numerical Algebra, Control and Optimization, 2021, 11 (4) : 487-493. doi: 10.3934/naco.2020039 [2] Yanqun Liu, Ming-Fang Ding. A ladder method for linear semi-infinite programming. Journal of Industrial and Management Optimization, 2014, 10 (2) : 397-412. doi: 10.3934/jimo.2014.10.397 [3] Cheng Ma, Xun Li, Ka-Fai Cedric Yiu, Yongjian Yang, Liansheng Zhang. On an exact penalty function method for semi-infinite programming problems. Journal of Industrial and Management Optimization, 2012, 8 (3) : 705-726. doi: 10.3934/jimo.2012.8.705 [4] Burcu Özçam, Hao Cheng. A discretization based smoothing method for solving semi-infinite variational inequalities. Journal of Industrial and Management Optimization, 2005, 1 (2) : 219-233. doi: 10.3934/jimo.2005.1.219 [5] Ke Su, Yumeng Lin, Chun Xu. A new adaptive method to nonlinear semi-infinite programming. Journal of Industrial and Management Optimization, 2022, 18 (2) : 1133-1144. doi: 10.3934/jimo.2021012 [6] Roman Chapko, B. Tomas Johansson. On the numerical solution of a Cauchy problem for the Laplace equation via a direct integral equation approach. Inverse Problems and Imaging, 2012, 6 (1) : 25-38. doi: 10.3934/ipi.2012.6.25 [7] Imtiaz Ahmad, Siraj-ul-Islam, Mehnaz, Sakhi Zaman. Local meshless differential quadrature collocation method for time-fractional PDEs. Discrete and Continuous Dynamical Systems - S, 2020, 13 (10) : 2641-2654. doi: 10.3934/dcdss.2020223 [8] Liqun Qi, Zheng yan, Hongxia Yin. Semismooth reformulation and Newton's method for the security region problem of power systems. Journal of Industrial and Management Optimization, 2008, 4 (1) : 143-153. doi: 10.3934/jimo.2008.4.143 [9] Marco Berardi, Fabio V. Difonzo. A quadrature-based scheme for numerical solutions to Kirchhoff transformed Richards' equation. Journal of Computational Dynamics, 2022, 9 (2) : 69-84. doi: 10.3934/jcd.2022001 [10] Jie-Wen He, Chi-Chon Lei, Chen-Yang Shi, Seak-Weng Vong. An inexact alternating direction method of multipliers for a kind of nonlinear complementarity problems. Numerical Algebra, Control and Optimization, 2021, 11 (3) : 353-362. doi: 10.3934/naco.2020030 [11] Graham W. Alldredge, Ruo Li, Weiming Li. Approximating the $M_2$ method by the extended quadrature method of moments for radiative transfer in slab geometry. Kinetic and Related Models, 2016, 9 (2) : 237-249. doi: 10.3934/krm.2016.9.237 [12] Jinchuan Zhou, Changyu Wang, Naihua Xiu, Soonyi Wu. First-order optimality conditions for convex semi-infinite min-max programming with noncompact sets. Journal of Industrial and Management Optimization, 2009, 5 (4) : 851-866. doi: 10.3934/jimo.2009.5.851 [13] Van Duong Dinh. On the Cauchy problem for the nonlinear semi-relativistic equation in Sobolev spaces. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1127-1143. doi: 10.3934/dcds.2018047 [14] Sanjit Kumar Mohanty, Rajani Ballav Dash. A quadrature rule of Lobatto-Gaussian for numerical integration of analytic functions. Numerical Algebra, Control and Optimization, 2021  doi: 10.3934/naco.2021031 [15] Darya V. Verveyko, Andrey Yu. Verisokin. Application of He's method to the modified Rayleigh equation. Conference Publications, 2011, 2011 (Special) : 1423-1431. doi: 10.3934/proc.2011.2011.1423 [16] Yones Esmaeelzade Aghdam, Hamid Safdari, Yaqub Azari, Hossein Jafari, Dumitru Baleanu. Numerical investigation of space fractional order diffusion equation by the Chebyshev collocation method of the fourth kind and compact finite difference scheme. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2025-2039. doi: 10.3934/dcdss.2020402 [17] V. Varlamov, Yue Liu. Cauchy problem for the Ostrovsky equation. Discrete and Continuous Dynamical Systems, 2004, 10 (3) : 731-753. doi: 10.3934/dcds.2004.10.731 [18] Adrien Dekkers, Anna Rozanova-Pierrat. Cauchy problem for the Kuznetsov equation. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 277-307. doi: 10.3934/dcds.2019012 [19] Andreas Kirsch. An integral equation approach and the interior transmission problem for Maxwell's equations. Inverse Problems and Imaging, 2007, 1 (1) : 159-179. doi: 10.3934/ipi.2007.1.159 [20] Beatrice Bugert, Gunther Schmidt. Analytical investigation of an integral equation method for electromagnetic scattering by biperiodic structures. Discrete and Continuous Dynamical Systems - S, 2015, 8 (3) : 435-473. doi: 10.3934/dcdss.2015.8.435

2020 Impact Factor: 1.639