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An alternating boundary integral based method for a Cauchy problem for the Laplace equation in semiinfinite 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 
References:
[1] 
G. Bastay, T. Johansson, V. A. Kozlov and D. Lesnic, An alternating method for the stationary Stokes system, ZAMM, 86 (2006), 268280. doi: 10.1002/zamm.200410238. 
[2] 
J. Baumeister and A. Leitāo, On iterative methods for solving illposed problems modeled by partial differential equations, J. Inv. IllPosed Probl., 9 (2001), 1329. 
[3] 
A.P. Calderón, Uniqueness in the Cauchy problem for partial differential equations, Amer. J. Math., 80 (1958), 1636. 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), 19. 
[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), 127140. 
[6] 
H. W. Engl and A. Leitāo, A Mann iterative regularization method for elliptic Cauchy problems, Numer. Funct. Anal. Optim., 22 (2001), 861884. doi: 10.1081/NFA100108313. 
[7] 
U. Hämarik and T. Raus, On the choice of the regularization parameter in illposed problems with approximately given noise level of data, J. Inverse IllPosed Probl., 14 (2006), 251266. 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), 247260. doi: 10.1023/A:1019134102565. 
[10] 
V. A. Kozlov and V. G. Maz'ya, On iterative procedures for solving illposed boundary value problems that preserve differential equations, Algebra i Analiz, 1 (1989), 144170. English transl.: Leningrad Math. J., 1 (1990), 12071228. 
[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), 6474. English transl.: U.S.S.R. Comput. Math. and Math. Phys., 31 (1991), 4552. 
[12] 
R. Kress, "Linear Integral Equations," 2nd edition, SpringerVerlag, 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), 123133. doi: 10.1016/S09557997(97)000568. 
[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,'' SpringerVerlag, NewYork, 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,'' SpringerVerlag, Heidelberg, 1993. 
[19] 
G. M. Vainikko and A. Y. Veretennikov, "Iteration Procedures in IllPosed 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), 268280. doi: 10.1002/zamm.200410238. 
[2] 
J. Baumeister and A. Leitāo, On iterative methods for solving illposed problems modeled by partial differential equations, J. Inv. IllPosed Probl., 9 (2001), 1329. 
[3] 
A.P. Calderón, Uniqueness in the Cauchy problem for partial differential equations, Amer. J. Math., 80 (1958), 1636. 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), 19. 
[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), 127140. 
[6] 
H. W. Engl and A. Leitāo, A Mann iterative regularization method for elliptic Cauchy problems, Numer. Funct. Anal. Optim., 22 (2001), 861884. doi: 10.1081/NFA100108313. 
[7] 
U. Hämarik and T. Raus, On the choice of the regularization parameter in illposed problems with approximately given noise level of data, J. Inverse IllPosed Probl., 14 (2006), 251266. 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), 247260. doi: 10.1023/A:1019134102565. 
[10] 
V. A. Kozlov and V. G. Maz'ya, On iterative procedures for solving illposed boundary value problems that preserve differential equations, Algebra i Analiz, 1 (1989), 144170. English transl.: Leningrad Math. J., 1 (1990), 12071228. 
[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), 6474. English transl.: U.S.S.R. Comput. Math. and Math. Phys., 31 (1991), 4552. 
[12] 
R. Kress, "Linear Integral Equations," 2nd edition, SpringerVerlag, 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), 123133. doi: 10.1016/S09557997(97)000568. 
[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,'' SpringerVerlag, NewYork, 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,'' SpringerVerlag, Heidelberg, 1993. 
[19] 
G. M. Vainikko and A. Y. Veretennikov, "Iteration Procedures in IllPosed Problems,'' Nauka Publ., Moscow, 1986 (in Russian). 
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