# Vestnik КRAUNC. Fiz.-Mat. Nauki. 2023. vol. 42. no. 1. P. 9-26. ISSN 2079-6641

**MATHEMATICS **

https://doi.org/10.26117/2079-6641-2023-42-1-9-26

Research Article

Full text in Russian

MSC 35K35

**Solvability of a Nonlocal Inverse Problem for a Fourth-Order Equation**

**A. B. Bekiev^***

Karakalpak State University named after Berdakh, Uzbekistan, Republic of Karakalpakstan, 230112, Nukus, Ch. Abdirov St. 1.

**Abstract.** In this paper, we consider a nonlocal inverse problem of finding an unknown right-hand side with

one variable for a fourth-order partial differential equation in a rectangular domain. The eigenfunctions and

associated functions of the corresponding spectral problem and its biorthogonal functions are complete and

form a Riesz basis in the space L_2\left(0,1\right) . Criteria for the uniqueness and existence of a solution to the considered nonlocal inverse problem for a fourth-order equation are established. The uniqueness of the solution of the nonlocal inverse problem follows from the completeness of the system of biorthogonal functions. The solution of the problem is constructed as the sum of a series in terms of eigenfunctions and associated functions of the corresponding spectral problem. Sufficient conditions are established for the boundary functions that guarantee existence and stability theorems for the solution of the problem under consideration. In a closed domain, absolute and uniform convergence of the found solution of 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 two and four times, respectively, depending on the smoothness of the function given the initial conditions. In this case, small denominators arise, which hinder the convergence of these series. It is proved that, depending on the size of the domain, the set of non-zero solutions of the expression in the denominator is not empty. And also, it is shown that if this denominator is equal to zero, then this problem will have a non-trivial solution under homogeneous conditions. It is also proved that the solution of the inverse problem is stable in the norms of the spaces L_2, W^n_2 and C\left(\Omega\pm\right) with respect to changes in the input data.

*Key words: fourth order equation, inverse nonlocal problem, uniqueness, existence, stability.*

Received: 02.01.2023; Revised: 01.02.2023; Accepted: 08.04.2023; First online: 15.04.2023

**For citation.** Bekiev A. B. Solvability of a nonlocal inverse problem for a fourth-order equation. Vestnik KRAUNC. Fiz.-mat. nauki. 2023, 42: 1, 9-26. EDN: BJBNHI. https://doi.org/10.26117/2079-6641-2023-42-1-9-26.

**Funding.** The study was carried out without financial support from foundations.

**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 submitting the final version of the article to print.

**^* ***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., 2023

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

**References**

- Amanov D. Razreshimost i spektralnye svoistva kraevykh zadach dlia uravnenii chetnogo poriadka [Solvability and spectral properties of boundary value problems for equations of even order]. avtoref. dis. d-ra fiz.mat. Tashkent: AN RUz, 2019, P. 64. (In Russian)
- Amirov Sh., Kokhanov A. I. Global solvability of initial boundary-value problems for nonlinear analogs of the Boussinesq equation, Matem. zametki, 2016, 99, 2, 171-180. DOI: 10.4213/mzm10617(In Russian)
- Denisov A.M. Vvedenie d teoriyu obratnykh zadach [Elements of the Theory of inverse Problems]. Moskva-Izd-vo MGU, 1994, 208. (In Russian)
- Dzhuraev T. D., Sopuev A. K teorii differentsialnykh uravnenii v chastnykh proizvodnykh chetvertogo poriadka [To the theory of partial differential equations of the fourth order], Tashkent, Fan, 2000, 144. (In Russian)
- Kavanikhin S. I. Obratye i nekorrektnye zadachi [Inverse and ill-posed problems], Novosibirsk, Sibirskoe nauchnoe izdatelstvo, 2009, 457. (In Russian)
- Kaliev I. A., Mugafarov M. F., Fattakhova O.V Inverse problem for forward-backward parabolic equation with generalized conjugation conditions, Ufimsk. matem/zhurn., 2011, 3, 97, 368-381. mi.mathnet.ru/ufa92(In Russian)
- Kamynin V. L. The inverse problem of the simultaneous determination of the right-hand side and the lowest coeffcient in a parabolic equation with many space variables, Mat. zametki, 2015, 97, 3, 368-381. DOI: 10.4213/mzm10499 (In Russian)
- Kamynin V. L. The Inverse Problem of Simultaneous Determination of the Two Lower Space-Dependent Coeffcients in a Parabolic Equation, Mat. zametki, 2019, 106, 2, 248-261, DOI: 10.4213/mzm12164(In Russian)
- Kozhanov A. I. Inverse problems of recovering the right-hand side of a special type of parabolic equations. Mathematical Notes, Mat. zametki SVFU, 2016, 23, 4, 31-45, (In Russian).
- Kokhanov A. I. Parabolic equations with unknown time-dependent coeffcients, Zh. vychisl. matem. i matem. fiz., 2017, 57, 6, 961-972, (In Russian).
- Korotkii A. I. Starodubtseva Yu.V. Modelirovanie priamykh i obratnykh granichnykh zadach dlia stratsionarnykh modelei teplomassoperenosa [Modeling direct and inverse boundary value problems for stationary models of heat and mass transfer]. Ekaterinburg, Izd-vo Ural. un-ta, 2015, 168,(In Russian)
- Lavrentev M. M. Ob odnoi obratnoi zadache dlia volnovogo uravnenia [On an inverse problem for the wave equation]. // Dokl. AN SSSR, 1964, 57, 2,520-521,(In Russian)
- Megraliev Ia. Inverse boundary value problem for the equation of bending of thin plates with an additional integral condition, Dalnevostochnyi matematicheskii zhurnal, 2013, 13, 1, 83-101,(In Russian)
- Megraliev Ia. T. Inverse boundary value problem for a Boussinesq type equation of fourth order with nonlocal time integral conditions of the second kind, Vestnik Udmurtskogo universiteta. Matematika. Mekhanika. Kompyuternym nauki, 2016, 26, 4, 503-514, (In Russian)
- Piatkov S. G., Kvich E. S. Recovering of Lower order coeffcients in forwardbackward parabolic equations, Vestnik YuUrGU. Seriia Matematika. Mekhanika. Fizika., 2018, 10, 4, 23-29, DOI: 10.14529/mmph180403, (In Russian)
- Romanov V. G. Obratnye zadachi matematicheskoi fiziki [Inverse Problems of Mathematical Physics], Moskva, Nauka, 1984, 264 (In Russian)
- Sabitov K. B., Khadzhi I. A. Boundary value problem for Lavrentyev-Biczadze’s Equ ation with unknown right-hand part, Izv. vuzov. Matem. 2011, 5, 44-52 (In Russian)
- Sabitov K. B., Martemianova N. V. Nonlocal inverse problem for a mixed type equation, Izv. vizov. Matem., 2011, 2, 71-85, (In Russian)
- Sabitov K. B. Martemianova N. V. An inverse problem for an elliptic-hyperbolic equation with nonlocal boundary conditions, Sib. mate. zhurn., 2012, 52, 3, 633-647, (In Russian)
- Sabitov K. B., Sidorov S. N. Inverse problem for degenerate parabolic-hyperbolic equation with nonlocal boundary condition, Izvestiia vuzov. Matematika, 2015, 1, 46-59 (In Russian)
- Teleshova L. A. obratnye zadachi dlia parabolicheskikh uravnenii vysokogo poriadka [Inverse problems for high-order parabolic equations]. Dis. kand. fiz.-matem. nauk. Ulan-Ude, 2017, 155.
- Yuldashev T. K. On one mixed differential equation of the fourth order, Izvestiia Instituta matematiki i informatiki UdGU, 2016, 1(47), 119-128, (In Russian)
- Yuldashev T. K. Mixed differential equation of Boussinesq type, Vest. Volgogr. gos. un-ta. Ser. 1, Mat. Fiz. 2016, 2(33), 13-23, (In Russian)
- Berdyshev A. S., Cabada A., Kadirkulov B. J. The Samarskii-Ionkin type problem for the fourth order parabolic equation with fractional differential operator, Computers and Mathematics with Applications, 2011, 67, 3884-3893.
- Jiang D., Liu Y., Yamamoto M. Inverse source problem for the hyperbolic equation witha time-dependent principal part, J. Differential Equations. 2017, 262, 1, 653-681. DOI: 10.1016/j.jde.2016.09.036
- Prilepko A. I., Orlovsky D.G. and Vasin I. A. Methods for Solving Inverse Problems in Mathematical Physics, New York-Basel, Global Express Ltd., 1999, 709.

### Information about the author

**Bekiev Ashirmet Bekievich** – Ph.D. (Phys.& Math.), Assistant professor, Department of Differensial Equation, Karakalpak State University, Nukus, Uzbekistan,

https://orcid.org/0000-0001-8630-4360.