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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:
 4.5e+11 Integral equations of the convolution type 3e+26 Boundary value problems, complex analysis
##### References:
 [1] Atkinson, K. E.: The numerical solution of integral equations of the second kind, (1997) [2] Baker, C. T. H.: The numerical treatment of integral equations, (1977) · Zbl 0373.65060 [3] Murid, A. H. M.; Nasser, M. M. S.: Eigenproblem of the generalized Neumann kernel, Bull. malaysian math. Sci. soc. 26, No. 2, 13-33 (2003) · Zbl 1185.45003 · doi:emis:journals/BMMSS/vol26_1_2.htm [4] Murid, A. H. M.; Razali, M. R. M.; Nasser, M. M. S.: Solving Riemann problem using Fredholm integral equation of the second kind, Proceedings of simposium kebangsaan sains matematik ke-10, 171-178 (2002) [5] Muskhelishvili, N. I.: Singular integral equations, (1953) · Zbl 0051.33203 [6] Polyanin, A. D.; Manzhirov, A. V.: Handbook of integral equations, (1998) [7] Vekua, I. N.: Generalized analytic functions, (1992) [8] Wegert, E.: An iterative method for solving nonlinear Riemann – Hilbert problems, J. comput. Appl. math. 29, 311-327 (1990) · Zbl 0705.65020 · doi:10.1016/0377-0427(90)90014-Q [9] Wegmann, R.: Convergence proofs and error estimates for an iterative method for conformal mapping, Numer. math. 44, 435-461 (1984) · Zbl 0526.30008 · doi:10.1007/BF01405574 [10] Wegmann, R.: Fast conformal mapping of multiply connected regions, J. comput. Appl. math. 130, 119-138 (2001) · Zbl 1058.30032 · doi:10.1016/S0377-0427(99)00387-8 [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 · doi:10.1016/j.cam.2004.12.019 [12] Wendland, W.: Elliptic systems in the plane, (1979)