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

  • Vladimir Mityushev
Conference paper
Part of the Trends in Mathematics book series (TM)

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.

Keywords

Mixed boundary value problem Keldysh-Sedov formulae Riemann-Hilbert problem Multiply connected domain Iterative functional equation 

Mathematics Subject Classification (2010)

30E25 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Computer SciencesPedagogical UniversityKrakowPoland

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