Vestnik КRAUNC. Fiz.-Mat. Nauki. 2025. vol. 51. no. 2. P. 45 – 56. ISSN 2079-6641

MATHEMATICS
https://doi.org/10.26117/2079-6641-2025-51-2-45-56
Research Article
Full text in English
MSC 35R11

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Non-Local Initial-Boundary Value Problem for a Degenerate Second Order Equation with Fractional Caputo Derivative and Riemann-Liouville Integral

D. A. Usmonov^{\ast}, A. N. Omonova

Fergana State University, 150100, Fergana, Murrabbiylar str., 19, Republic Uzbekistan

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 initial boundary 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 second-order equation degenerating on two sides of the rectangular and contains the with fractional Caputo derivative and Riemann-Liouville integro operators 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. 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: initial-boundary problem, Caputo fractional derivative, degenerate differential equation, Mittag-Leffler-type functions of two variables.

Received: 23.06.2025; Revised: 09.09.2025; Accepted: 10.09.2025; First online: 22.09.2025

For citation. Usmonov D. A., Omonova A. N. Non-Local initial-Boundary value problem for a degenerate second order equation with fractional Caputo derivative and Riemann-Liouville integral. Vestnik KRAUNC. Fiz.-mat. nauki. 2025, 51: 2, 45-56. EDN: IEXHAM. https://doi.org/10.26117/2079-6641-2025-51-2-45-56.

Funding. The study was conducted without the support of foundations.

Competing interests. 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.

^{\ast}Correspondence: E-mail: usmonov-doniyor@inbox.ru

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

© Usmonov D. A., Omonova A. N., 2025

© Institute of Cosmophysical Research and Radio Wave Propagation, 2025 (original layout, design, compilation)

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