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The Riemann-Hilbert problem and the generalized Neumann kernel on multiply connected regions. (English) Zbl 1157.45303
Summary: This paper presents and studies Fredholm integral equations associated with the linear Riemann-Hilbert problems on multiply connected regions with smooth boundary curves. The kernel of these integral equations is the generalized Neumann kernel. The approach is similar to that for simply connected regions [see R. Wegmann, A. H. M. Murid and M. M. S. Nasser, J. Comput. Appl. Math. 182, No. 2, 388–415 (2005; Zbl 1070.30017)]. There are, however, several characteristic differences, which are mainly due to the fact that the complement of a multiply connected region has a quite different topological structure. This implies that there is no longer perfect duality between the interior and exterior problems. We investigate the existence and uniqueness of solutions of the integral equations. In particular, we determine the exact number of linearly independent solutions of the integral equations and their adjoints. The latter determine the conditions for solvability. An analytic example on a circular annulus and several numerically calculated examples illustrate the results.

MSC:
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
30E25 Boundary value problems in the complex plane
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[1] Atkinson, K.E., The numerical solution of integral equations of the second kind, (1997), Cambridge University Press Cambridge · Zbl 0155.47404
[2] Baker, C.T.H., The numerical treatment of integral equations, (1977), Clarendon Press Oxford · Zbl 0217.53103
[3] Murid, A.H.M.; Nasser, M.M.S., Eigenproblem of the generalized Neumann kernel, Bull. Malaysian math. sci. soc., 26, 2, 13-33, (2003) · Zbl 1185.45003
[4] Murid, A.H.M.; Razali, M.R.M.; Nasser, M.M.S., Solving Riemann problem using Fredholm integral equation of the second kind, (), 171-178
[5] Muskhelishvili, N.I., Singular integral equations, (1953), Noordhoff Groningen · Zbl 0051.33203
[6] Polyanin, A.D.; Manzhirov, A.V., Handbook of integral equations, (1998), CRC Press Boca Raton · Zbl 1021.45001
[7] Vekua, I.N., Generalized analytic functions, (1992), Pergamon London · Zbl 0698.47036
[8] Wegert, E., An iterative method for solving nonlinear riemann – hilbert problems, J. comput. appl. math., 29, 311-327, (1990) · Zbl 0705.65020
[9] Wegmann, R., Convergence proofs and error estimates for an iterative method for conformal mapping, Numer. math., 44, 435-461, (1984) · Zbl 0526.30008
[10] Wegmann, R., Fast conformal mapping of multiply connected regions, J. comput. appl. math., 130, 119-138, (2001) · Zbl 1058.30032
[11] Wegmann, R.; Murid, A.H.M.; Nasser, M.M.S., The riemann – hilbert problem and the generalized Neumann kernel, J. comput. appl. math., 182, 388-415, (2005) · Zbl 1070.30017
[12] Wendland, W., Elliptic systems in the plane, (1979), Pitman London · Zbl 0396.35001
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