On the solution of the Poincaré boundary value problem for generalised harmonic functions in simply connected domains

Authors

  • Tatyana R. Nagornaya Smolensk State University, 4 Przhevalskogo Street, Smolensk 214000, Russia
  • Karim M. Rasulov Smolensk State University, 4 Przhevalskogo Street, Smolensk 214000, Russia

Keywords:

differential equation, generalised harmonic function, Poincaré boundary value problem, generalised Hilbert boundary value problem, integral equation, simply connected domain

Abstract

In this paper, a boundary value problem of the Poincaré type is considered for one second-order elliptic differential equation, generating a class of generalised harmonic functions, in simply connected domains with smooth boundaries. It is established that for sufficiently general assumptions about the coefficients of the boundary value condition of the considered problem, its solution reduces to the sequential solution of the well-studied integro-differential Hilbert boundary value problem and the differential Hilbert boundary value problem in classes of analytic functions of a complex variable. In addition, necessary and sufficient solvability conditions of the considered problem are obtained and its Noetherian property is proved.

Author Biographies

  • Tatyana R. Nagornaya, Smolensk State University, 4 Przhevalskogo Street, Smolensk 214000, Russia

    senior lecturer at the department of mathematical analysis, faculty of physics and mathematics

     

  • Karim M. Rasulov, Smolensk State University, 4 Przhevalskogo Street, Smolensk 214000, Russia

    doctor of science (physics and mathematics), full professor; head of the department of mathematical analysis, faculty of physics and mathematics

     

References

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  8. Muskhelishvili NI. Singulyarnye integral’nye uravneniya: granichnye zadachi teorii funktsii i nekotorye ikh prilozheniya k matematicheskoi fizike [Singular integral equations: boundary value problems of function theory and some of their applications to mathematical physics]. 3rd edition. Moscow: Nauka; 1968. 511 p. Russian.

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Published

2024-04-01

Issue

No. 1 (2024): Journal of the Belarusian State University. Mathematics and Informatics

Section

Real, Complex and Functional Analysis

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How to Cite

[1]
Nagornaya, T.R. and Rasulov, K.M. 2024. On the solution of the Poincaré boundary value problem for generalised harmonic functions in simply connected domains. Journal of the Belarusian State University. Mathematics and Informatics. 1 (Apr. 2024), 6–15.