Vestnik КRAUNC. Fiz.-Mat. Nauki. 2026. vol. 54. no. 2. P. 48 – 56. ISSN 2079-6641

MATHEMATICS
https://doi.org/10.26117/2079-6641-2026-54-2-48-56
Research Article
Full text in English
35G15, 35R25, 35C10, 11J25

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On Periodic Solutions of a Boundary Value Problem for an Anisotropic Fourth-Order Equation

A. V. Yuldasheva¹^{\ast}, S. A. Sharipova²^{\ast}

¹Branch of MSU in Tashkent, 22 Amir Temur Street, Tashkent, 100060, Uzbekistan
²National University of Uzbekistan, 4 University Street, Tashkent, 100174, Uzbekistan

Abstract. The paper considers a boundary value problem for the fourth-order equation U_{xxxx} +U_{yyyy} = 0, defined in a rectangular domain under the condition of periodicity in one of the spatial variables. The studied model arises in describing stationary deformation processes of anisotropic elastic media, where material properties differ by direction. A feature of the formulation is the combination of rigid boundary conditions on one boundary of the domain and the specification of the bending moment distribution on the opposite boundary. It is shown that when expanding the solution into a Fourier series in the periodic coordinate, the problem reduces to a fourth-order differential equation. Small denominators arise, due to the spectral properties of the corresponding differential operator. The presence of small denominators can lead to instability of the solution and ill-posedness of the problem in the classical sense. The paper discusses the conditions for the existence of a periodic solution. A connection is established between the algebraic nature of the dimensionless parameter, given by the ratio of the domain thickness to a fundamental geometric constant, and the smoothness of the boundary function. It is shown that the regularity of the solution is significantly determined by the Diophantine properties of this parameter, which influence the behavior of small denominators in the spectral expansion. It is proved that the problem is solvable for almost all values of the specified parameter in the sense of Lebesgue measure, except for a set of exceptional cases corresponding to strong algebraic dependence. The obtained results can be applied in modeling bending processes in layered and orthotropic structures, as well as in the study of ill-posed problems in mathematical physics.

Key words: boundary value problem, fourth-order equation, periodic solution, anisotropic medium, small denominators, ill-posed problem, plate bending, Fourier series.

Received: 18.05.2026; Revised: 18.06.2026; Accepted: 20.06.2026; First online: 14.09.2026

For citation. Yuldasheva A. V., Sharipova S. A. On periodic solutions of a boundary value problem for an anisotropic fourth-order equation. Vestnik KRAUNC. Fiz.-mat. nauki. 2026, 54: 2, 48-56. EDN: BYRGNS. https://doi.org/10.26117/2079-6641-2026-54-2-48-56.

Funding. This work was not supported by any external funding.

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: a_v_yuldasheva@mail.ru

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

© Yuldasheva A. V., Sharipova S. A., 2026

© Institute of Cosmophysical Research and Radio Wave Propagation, 2026 (original layout, design, compilation)

References

1. Modern Methods in the Theory of Boundary Value Problems. M.: MAKS Press, 2018. 296 pp. (In Russian)
2. Ptashnik B. I. Ill-Posed Boundary Value Problems for Partial Differential Equations. Kyiv: Naukova Dumka, 1984. 264 pp. (In Russian)
3. Yakubov S. and Yakubov Ya. Differential-Operator Equations: Ordinary and Partial Differential Equations. N.Y.: Chapman & Hall/CRC, 2000. 700 pp.
4. Mikhailets V. A. and Molyboga N. M.On boundary value problems for differential-operator equations, Ukrainian Mathematical Journal, 2016. vol. 68, no. 6, pp. 846— 861.
5. Ntouyas S.K. Nonlocal Initial and Boundary Value Problems: A Survey / Handbook of Differential Equations: Ordinary Differential Equations. N.Y., Elsevier B.V., 2008, pp. 461-557.
6. Strikwerda John C. Well-Posed and Stable Initial-Boundary Value Problems / Finite Difference Schemes and Partial Differential Equations, Second Edition. e-book, SIAM, 2004, pp. 275-310.
7. Bourgin D. G., Duffin R. The Dirichlet problem for the vibrating string equation, Bull. Amer. Math. Soc., 1939. vol. 45, no. 12, pp. 851–858.
8. Alimov Sh. A.On Lp-solutions of a boundary value problem, Uzb. Math. J., 1999. №1, С. 3–9 (In Russian).
9. Alimov Sh. A.On solvability of an ill-posed problem, Uzb. Math. J., 1999. no. 3, pp. 19–29 (In Russian).
10. Yuldasheva A. V.On solvability of the boundary value problem for one even-order equation, Vestnik KRAUNC. Fiz.-Mat. Nauki, 2016. vol. 12, no. 1, pp. 35–40 DOI:10.18454/2079-6641-2016-12-1-41-47.
11. Yuldasheva A. V.On stability of a boundary value problem for an even-order equation, Vestnik KRAUNC. Fiz.-Mat. nauki, 2015. vol. 10, no. 1, pp. 5-11 DOI:10.18454/2079-6641-2015-10-1-5-11 (In Russian).
12. Yuldasheva A. V., Sharipova S. A. Analysis of boundary value problems for partial differential equation of the fourth order / Differential and difference equations, Abstracts of Russian-Chinese Conference (Novosibirsk, 2025), Abstracts. Novosibirsk, Novosibirsk State University, 2025, pp. 155.
13. Shidlovsky A. B. Transcendental Numbers. M.: Nauka, 1987. 447 pp. (In Russian)
14. Roth K. F. Rational approximation to algebraic numbers, Mathematika, 1955. vol. 2, pp. 1–20.
15. Khintschin A. Zur metrischen Theorie der diophantischen Approximationen, Math. Zeit., 1926. vol. 24, no. 4, pp. 706–714.

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