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Abstract <p>This paper, which continues the authors' 2024 article, is devoted to the development of an analytical-numerical multipole method as applied to the Zaremba problem, i.e., a mixed boundary value problem with Dirichlet–Neumann boundary conditions for the Laplace equation in planar, simply connected domains <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(g\)</EquationSource> <!--ComMat2570147Bagapsh-m1--> </InlineEquation> of complex shape, whose boundaries may contain singularities. The method allows one to obtain not only the solution, but also its derivatives on certain smooth parts of the boundary near the singularities. The effectiveness of the method was demonstrated by examples of constructing a conformal mapping, and in previous works (with other coauthors), by examples of constructing a harmonic mapping of domains with complex curved boundaries.</p>

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Multipole Method for Solving the Zaremba Problem in Complex Domains and Its Application to Constructing a Conformal Mapping

  • 期刊论文

  • 发表日期：

    2025年11月1日

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  • A. O. Bagapsh,
  • V. I. Vlasov

摘要

Abstract

This paper, which continues the authors' 2024 article, is devoted to the development of an analytical-numerical multipole method as applied to the Zaremba problem, i.e., a mixed boundary value problem with Dirichlet–Neumann boundary conditions for the Laplace equation in planar, simply connected domains \(g\) of complex shape, whose boundaries may contain singularities. The method allows one to obtain not only the solution, but also its derivatives on certain smooth parts of the boundary near the singularities. The effectiveness of the method was demonstrated by examples of constructing a conformal mapping, and in previous works (with other coauthors), by examples of constructing a harmonic mapping of domains with complex curved boundaries.