Vestnik Samarskogo Universiteta. Estestvenno-Nauchnaya Seriya
 RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Vestnik SamU. Estestvenno-Nauchnaya Ser.: Year: Volume: Issue: Page: Find

 Vestnik SamU. Estestvenno-Nauchnaya Ser., 2019, Volume 25, Issue 1, Pages 7–20 (Mi vsgu584)

Mathematics

The correctness of a Dirichlet type problem for the degenerate multidimensional hyperbolic-elliptic equations

S. A. Aldashev

Institute of Mathematics and Mathematical Modeling, 125, Pushkin street, Almaty, 050010, Republic of Kazakhstan

Abstract: Multidimensional hyperbolic-elliptic equations describe important physical, astronomical, and geometric processes. It is known that the oscillations of elastic membranes in space according to Hamilton’s prism can be modeled by multidimensional degenerate hyperbolic equations. Assuming that the membrane is in equilibrium in half the bend, from Hamilton’s principle we also obtain degenerate elliptic equations. Consequently, vibrations of elastic membranes in space can be modeled as multidimensional degenerate hyperbolic-elliptic equations. When studying these applications, it is necessary to obtain an explicit representation of the investigated boundary value problems. The author has previously studied the Dirichlet problem for multidimensional hyperbolic-elliptic equations, where the unique solvability of this problem is shown, which essentially depends on the height of the cylindrical domain under consideration. However, the Dirichlet problem in a cylindrical domain for multidimensional degenerate hyperbolic-elliptic equations has not been studied previously. In this paper, the Dirichlet problem is studied for a class of degenerate multidimensional hyperbolic-elliptic equations. Moreover, the existence and uniqueness of the solution depends on the height of the considered cylindrical domain and on the degeneration of the equation. A uniqueness criterion for a regular solution is also obtained.

Keywords: correctness, Dirichlet problem, cylindrical domain, degeneration of Bessel function, criteria.

DOI: https://doi.org/10.18287/2541-7525-2019-25-1-7-20

Full text: PDF file (245 kB)
Full text: https:/.../6922 (published under the terms of the Creative Commons Attribution 4.0 International License)
References: PDF file   HTML file

UDC: 517.956
Accepted:20.02.2019

Citation: S. A. Aldashev, “The correctness of a Dirichlet type problem for the degenerate multidimensional hyperbolic-elliptic equations”, Vestnik SamU. Estestvenno-Nauchnaya Ser., 25:1 (2019), 7–20

Citation in format AMSBIB
\Bibitem{Ald19} \by S.~A.~Aldashev \paper The correctness of a Dirichlet type problem for the degenerate multidimensional hyperbolic-elliptic equations \jour Vestnik SamU. Estestvenno-Nauchnaya Ser. \yr 2019 \vol 25 \issue 1 \pages 7--20 \mathnet{http://mi.mathnet.ru/vsgu584} \crossref{https://doi.org/10.18287/2541-7525-2019-25-1-7-20} \elib{https://elibrary.ru/item.asp?id=39454580}