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Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2023, Volume 33, Issue 2, Pages 312–328
DOI: https://doi.org/10.35634/vm230209
(Mi vuu852)
 

MATHEMATICS

On one problem for a fourth-order mixed-type equation that degenerates inside and on the boundary of a domain

A. K. Urinovab, D. A. Usmonova

a Fergana State University, ul. Murabbiylar, 19, Fergana, 150100, Uzbekistan
b Institute of Mathematics named after V. I. Romanovsky of the Academy of Sciences of the Republic of Uzbekistan, ul. Universiteteskaya, 46, Tashkent, 100174, Uzbekistan
References:
Abstract: In the article, a nonlocal boundary value problem has been investigated for a fourth-order mixed-type equation degenerating inside and on the boundary of a domain. Applying the method of separation of variables to the problem under study, the spectral problem for an ordinary differential equation is obtained. The Green function of the last problem is constructed, with the help of which it is equivalently reduced to the Fredholm integral equation of the second kind with a symmetric kernel, which implies the existence of eigenvalues and the system of eigenfunctions for the spectral problem. The theorem of expansion of a given function into a uniformly convergent series with respect to the system of eigenfunctions is proved. Using the found integral equation and Mercer's theorem, a uniform convergence of some bilinear series depending on the found eigenfunctions is proved. The order of the Fourier coefficients is established. The solution of the problem under study is written as the sum of the Fourier series with respect to the system of eigenfunctions of the spectral problem. An estimate for the problem's solution is obtained, from which its continuous dependence on the given functions follows.
Keywords: degenerate mixed-type equations, spectral problem, Green's function, integral equation, Fourier series, method of separation of variables.
Received: 29.12.2022
Accepted: 22.03.2023
Bibliographic databases:
Document Type: Article
UDC: 517.956
MSC: 35G15
Language: Russian
Citation: A. K. Urinov, D. A. Usmonov, “On one problem for a fourth-order mixed-type equation that degenerates inside and on the boundary of a domain”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 33:2 (2023), 312–328
Citation in format AMSBIB
\Bibitem{UriUsm23}
\by A.~K.~Urinov, D.~A.~Usmonov
\paper On one problem for a fourth-order mixed-type equation that degenerates inside and on the boundary of a domain
\jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki
\yr 2023
\vol 33
\issue 2
\pages 312--328
\mathnet{http://mi.mathnet.ru/vuu852}
\crossref{https://doi.org/10.35634/vm230209}
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    Вестник Удмуртского университета. Математика. Механика. Компьютерные науки
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    References:32
     
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