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Modern Problems in Applied Analysis pp 135–152Cite as

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Mixed Problem for Laplace’s Equation in an Arbitrary Circular Multiply Connected Domain

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Abstract

Mixed boundary value problems for the two-dimensional Laplace’s equation in a domain D are reduced to the Riemann-Hilbert problem Re \(\overline {\lambda (t)}\psi (t) = 0\), t ∈ ∂D, with a given Hölder continuous function λ(t) on ∂D except at a finite number of points where a one-sided discontinuity is admitted. The celebrated Keldysh-Sedov formulae were used to solve such a problem for a simply connected domain. In this paper, a method of functional equations is developed to mixed problems for multiply connected domains. For definiteness, we discuss a problem having applications in composites with a discontinuous coefficient λ(t) on one of the boundary components. It is assumed that the domain D is a canonical domain, the lower half-plane with circular holes. A constructive iterative algorithm to obtain an approximate solution in analytical form is developed in the form of an expansion in the radius of the holes.

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Notes

  1. 1.

    The standard form for the \(\mathbb R\)-linear problem is additive [9, 12], e.g. \(\phi (t)= \phi _k(t) + \overline {\phi _k(t)}+ c(t)\). In this case \(X(t)=\exp \phi (t)\), \(X_k(t)=\exp \phi _k(t)\).

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Author information

Authors and Affiliations

  1. Institute of Computer Sciences, Pedagogical University, ul. Podchorazych 2, Krakow, 30-084, Poland

    Vladimir Mityushev

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  1. Vladimir Mityushev
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Correspondence to Vladimir Mityushev .

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Editors and Affiliations

  1. Department of Mathematics & Natural Sciences, Rzeszow University, Rzeszów, Poland

    Piotr Drygaś

  2. Department of Economics, Belarusian State University, Minsk, Belarus

    Sergei Rogosin

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Mityushev, V. (2018). Mixed Problem for Laplace’s Equation in an Arbitrary Circular Multiply Connected Domain. In: Drygaś, P., Rogosin, S. (eds) Modern Problems in Applied Analysis. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-72640-3_10

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