# Vestnik КRAUNC. Fiz.-Mat. Nauki. 2023. vol. 42. no. 1. P. 123-139. ISSN 2079-6641

**MATHEMATICS **

https://doi.org/10.26117/2079-6641-2023-42-1-123-139

Research Article

Full text in Russian

MSC 35R11

**Non-Local Initial-Boundary Value Problem for a Degenerate Fourth-Order Equation with a Fractional Gerasimov-Caputo Derivative**

**A. K. Urinov^*, D. A. Usmonov^***

Fergana state university, Uzbekistan, 150100, Fergana, 19, Murabbiylar st.

Abstract. Recently, initial-boundary problems in a rectangular domain for differential equations in partial derivatives of both even and odd order have been intensively studied. In this case, non-degenerate equations or equations that degenerate on one side of the quadrilateral are taken as the object of study. But initialboundary problems (both local and non-local) for equations with two or three lines of degeneracy remain unexplored. In this paper, in a rectangular domain, a fourth-order equation degene-rating on three sides of the rectangular and contains the Gerasimov-Caputo fractional diffe-rentiation operator has been considered. For this equation, an initial-boundary problem is formulated and investigated, with non-local conditions connecting the values of the desired function and its derivatives up to the third order (inclusive), taken on the sides of the rectangle. From the beginning, the uniqueness of the solution of the formulated problem was proved by the method of energy integrals. Then, the spectral problem that arises when applying the Fourier method based on the separation of variables to the considered initial-boundary problem has been investigated. The Green’s function of the spectral problem was constructed, with the help of which it is equivalently reduced to an integral Fredholm equation of the second kind with a symmetric kernel, which implies the existence of a countable number of eigenvalues and eigenfunctions of the spectral problem. A theorem is proved for expanding a given function into a uniformly convergent series in terms of a system of eigenfunctions. Using the found integral equation and Mercer’s theorem, we prove the uniform convergence of some bilinear series depending on the found eigenfunctions. The order of the Fourier coeffi-cients have been established. The solution of the considered is written as the sum of a Fourier series with respect to the system of eigenfunctions of the spectral problem. The uniform convergence of this series and the series obtained from it by term-by-term differentiation is studied. An estimate for solution to problem is obtained, from which follows its continuous dependence on the given functions.

*Key words: degenerate fourth order equation, initial boundary value problem, method of separation of variables, spectral problem, Green’s function, integral equation, existence, uniqueness and stability of the solution.*

Received: 07.11.2023; Revised: 20.02.2023; Accepted: 24.03.2023; First online: 16.04.2023

**For citation.** Urinov A. K.∗, Usmonov D. A. Non-local initial-boundary value problem for a degenerate fourth-order equation with a fractional Gerasimov-Caputo derivative. Vestnik KRAUNC. Fiz.-mat. nauki. 2023, 42: 1, 123-139. EDN: INZPHJ. https://doi.org/10.26117/2079-6641-2023-42-1-123-139.

**Funding.** Not applicable.

**Competing interests.** The authors declare that there are no conflicts of interest regarding authorship and publication.

**Contribution and Responsibility.** All authors contributed to this article. Authors are solely responsible for providing

the final version of the article in print. The final version of the manuscript was approved by all authors.

^**Correspondence: E-mail: urinovak@mail.ru; usmonov-doniyor@inbox.ru*

*The content is published under the terms of the Creative Commons Attribution 4.0 International License*

© Urinov A. K.∗, Usmonov D. A., 2023

© Institute of Cosmophysical Research and Radio Wave Propagation, 2023 (original layout, design, compilation)

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**Information about authors**

**Urinov Akhmadzhon Kushakovich** – D. Sci. (Phys. & Math.), Professor, Professor of the Department of Mathematical Analysis and Differential Equations, Fergana State University, Fergana, Uzbekistan, https://orcid.org/0000-0002-9586-1799.

**Usmonov Doniyor Abdumutolib ugli** – Researcher, Department of Mathematical Analysis and Differential Equations, Fergana State University, Fergana, Uzbekistan,

https://orcid.org/0000-0002-3574-075X.