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Scalar Riemann–Hilbert Problem for Multiply Connected Domains

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Book cover Functional Equations in Mathematical Analysis

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 52))

Abstract

We solve the scalar Riemann–Hilbert problem for circular multiply connected domains. The method is based on the reduction of the boundary value problem to a system of functional equations (without integral terms). In the previous works, the Riemann–Hilbert problem and its partial cases such as the Dirichlet problem were solved under geometrical restrictions to the domains. In the present work, the solution is constructed for any circular multiply connected domain in the form of modified Poincaré series.

Mathematics Subject Classification (2000): Primary 30E25

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Notes

  1. 1.

    Despite the method of truncation can be effective in numeric computations, one can hardly accept that this method yields a closed form solution. Any way it depends on the definition of the term “closed form solution”. By my private definition, reduction of a boundary value problem to an integral equation does not yields a closed form solution. A regular infinite system [18] can be considered as an equation with compact operator, i.e., it is no more than a discreet form of an Fredholm integral equation.

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Authors and Affiliations

  1. Department of Computer Science and Computer Methods, Pedagogical University, Podchora̧żych 2, 30-084, Kraków, Poland

    Vladimir V. Mityushev

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  1. Vladimir V. Mityushev
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Correspondence to Vladimir V. Mityushev .

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  1. , Department of Mathematics, National Technical University of Athens, Iroon Polytechneiou 9, Athens, 15780, Greece

    Themistocles M. Rassias

  2. Institute of Mathematics, Pedagogical University, ul. Podchorazych 2, Krakow, 30-084, Poland

    Janusz Brzdek

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Mityushev, V.V. (2011). Scalar Riemann–Hilbert Problem for Multiply Connected Domains. In: Rassias, T., Brzdek, J. (eds) Functional Equations in Mathematical Analysis. Springer Optimization and Its Applications(), vol 52. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0055-4_38

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