Vestnik КRAUNC. Fiz.-Mat. Nauki. 2025. vol. 51. no. 2. P. 28 – 44. ISSN 2079-6641

MATHEMATICS
https://doi.org/10.26117/2079-6641-2025-51-2-28-44
Research Article
Full text in Russian
MSC 35R30

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On Solvability of One Inverse Problem for a Fourth Order Equation in the Rectangular Domain

A. B. Bekiev^{\ast}

Karakalpak State University named after Berdakh, 230112, Nukus city, Ch.Abdirov St. 1, Uzbekistan

Abstract. In this paper, for the fourth order equation in the rectangular domain, the inverse problem of finding the unknown right-hand side is considered. The solution of the problem is constructed as a sum of series of eigenfunctions and its associated functions of the corresponding spectral problem. The eigenfunctions of the corresponding spectral problem and its associated functions are complete system and form a Riesz basis in the space L_2(0,1). The uniqueness of the solution to the inverse problem follows from the completeness of the system of eigen and associated functions. Sufficient conditions are established for the given initial functions, which guarantee existence and stability theorems for the solution of the problem. In a closed region, the absolute and uniform convergence of the found solution to the inverse problem is shown in the form of a series, as well as series obtained by term-by-term differentiation with respect to t and x, three and four times, respectively. It has also been proven that the solution to the inverse problem is stable according to the norms of spaces L_2(0,1) , W^n_2(0, 1) and C(\Omega), with respect to changes in input data.

Key words: fourth order equation, inverse problem, method of separation of variables, uniqueness, existence, stability.

Received: 29.03.2025; Revised: 23.06.2025; Accepted: 29.06.2025; First online: 17.09.2025

For citation. Bekiev A. B. On solvability of one inverse problem for a fourth order equation in the rectangular domain. Vestnik KRAUNC. Fiz.-mat. nauki. 2025, 51: 2, 28-44. EDN: TNKVGB. https://doi.org/10.26117/2079-6641-2025-51-2-28-44.

Funding. The work was carried out without the support of funds.

Competing interests. There are no conflicts of interest regarding authorship and publication.

Contribution and Responsibility. The author participated in the writing of the article and is fully responsible for the submission of the final version of the article to print.

^{\ast}Correspondence: E-mail: ashir1976@mail.ru

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

© Bekiev A. B., 2025

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

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Information about the author

Bekiev Ashirmet Bekievich – PhD (Phy. & Math.), Dosent, Department of Differential Equation, Karakalpak State University, Nukus, Uzbekistan, ORCID 0000-0001-8630-4360.

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