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Chelyabinskiy Fiziko-Matematicheskiy Zhurnal, 2024, Volume 9, Issue 1, Pages 111–123
DOI: https://doi.org/10.47475/2500-0101-2024-9-1-111-123 (Mi chfmj362) 		 
Mathematics

Mixed problem for a nonlinear impulsive differential equation of parabolic type

T. K. Yuldasheva, A. K. Fayziyeva, F. D. Rakhmonovb

a Tashkent State University of Economics, Tashkent, Uzbekistan
b National University of Uzbekistan, Tashkent, Uzbekistan
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DOI: https://doi.org/10.47475/2500-0101-2024-9-1-111-123
Abstract: In this paper, we consider a nonlinear impulsive parabolic type partial differential equation with nonlinear impulsive conditions. Dirichlet type boundary value conditions with respect to spatial variable is used, and eigenvalues and eigenfunctions of the spectral problem are founded. The Fourier method of the separation of variables is applied. A countable system of nonlinear functional equations is obtained with respect to the Fourier coefficients of the unknown function. A theorem on a unique solvability of the countable system of nonlinear functional equations is proved by the method of successive approximations. A criteria of uniqueness and existence of a solution for the nonlinear impulsive mixed problem is obtained. A solution of the mixed problem is derived in the form of the Fourier series. The absolute and uniform convergence of the Fourier series is proved.
Keywords: mixed problem, nonlinear parabolic equation, nonlinear impulsive conditions, involution, unique solvability.
Funding agency Grant number
Ministry of Innovative Development of the Republic of Uzbekistan F-FA-2021-424
The research of first author is funded by the Ministry of Innovative Development of the Republic of Uzbekistan (grant F-FA-2021-424).
Received: 24.12.2023
Revised: 20.02.2024
Document Type: Article
UDC: 517.956.42
Language: English
Citation: T. K. Yuldashev, A. K. Fayziyev, F. D. Rakhmonov, “Mixed problem for a nonlinear impulsive differential equation of parabolic type”, Chelyab. Fiz.-Mat. Zh., 9:1 (2024), 111–123
Citation in format AMSBIB
\Bibitem{YulFayRak24}
\by T.~K.~Yuldashev, A.~K.~Fayziyev, F.~D.~Rakhmonov
\paper Mixed problem for a nonlinear impulsive differential equation of parabolic type
\jour Chelyab. Fiz.-Mat. Zh.
\yr 2024
\vol 9
\issue 1
\pages 111--123
\mathnet{http://mi.mathnet.ru/chfmj362}
\crossref{https://doi.org/10.47475/2500-0101-2024-9-1-111-123}
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