On the solvability of one inverse problem for a fourth-order equation
DOI:
https://doi.org/10.31489/2025m3/75-84Keywords:
fourth-order equation, inverse problem, classical solution, method of separation of variables, uniform convergence of the solution, uniqueness, existence, stability of the solutionAbstract
In this paper, for a fourth-order equation in a rectangular domain, an inverse problem of finding the unknown right-hand side, which depends on one variable, is considered. Criteria for the uniqueness and existence of a solution to the inverse problem under consideration for a fourth-order equation are established. The solution to the problem is constructed as the sum of a series in eigenfunctions of the corresponding spectral problem. The uniqueness of the solution to the inverse problem follows from the completeness of the system of eigenfunctions. Sufficient conditions are established for the boundary functions that guarantee theorems of existence and stability of the solution to the problem. In a closed domain, absolute and uniform convergence of the found solution to the inverse problem in the form of a series in the class of regular solutions is shown, as well as series obtained by term-by-term differentiation with respect to t and x three and four times, respectively. The stability of the solution of the inverse problem in the norms of the space of square-summable functions and in the space of continuous functions with respect to changes in the input data has also been proven.
References
Smirnov, M.M. (1972). Modelnoe uravnenie smeshannogo tipa chetvertogo poriadka [Model equation of mixed type of fourth order]. Leningrad: Izd-vo LGU [in Russian].
Amirov, S.H., & Khojanov, A.I. (2016). Globalnaia razreshimost nachalno–kraevykh zadach dlia nekotorykh nelineinykh analogov uravneniia Bussineska [Global solvability of initial boundaryvalue problems for nonlinear analogs of the Boussinesq equation]. Matematicheskie Zametki. — Mathematical Notes, 99(2), 183–191 [in Russian]. https://doi.org/10.1134/S0001434616010211
Imanbetova, A.B., Sarsenbi, A.A., & Seilbekov, B. (2024). On solvability of the inverse problem for a fourth-order parabolic equation with a complex-valued coefficient. Bulletin of the Karaganda University. Mathematics Series, 1(113), 60–72. https://doi.org/10.31489/2024m1/60-72
Apakov, Yu.P., & Melikuzieva, D.M. (2024). On the solution of a boundary problem for a fourth order equation containing a third time derivative in semi-bounded domains. Uzbek Mathematical Journal, 68(3), 5–11. https://doi.org/10.29229/uzmj.2024-3-1
Amanov, D., & Murzambetova, M.B. (2013). Kraevaia zadacha dlia uravneniia chetvertogo poriadka s mladshim chlenom [A boundary value problem for a fourth order partial differential equation with the lowest term]. Vestnik Udmurtskogo universiteta. Matematika. Mekhanika. Kompiuternye nauki — Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, (1), 1–10 [in Russian]. https://doi.org/10.20537/vm130101
Bekiev, A.B., & Shixiev, R.M. (2022). Razreshimost kraevoi zadachi dlia smeshannogo uravneniia chetvertogo poriadka [Resolution of the boundary value problem for a mixed equation of the fourth order]. Doklady AMAN — Reports of the AIAS, 22(2), 11–20 [in Russian].
Megraliev, Ya.T., & Velieva, B.K. (2019). Obratnaia kraevaia zadacha dlia linearizovannogo uravneniia Benni–Liuka s nelokalnymi usloviiami [Inverse boundary value problem for the linearized Benney-Luke equation with nonlocal conditions]. Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Kompiuternye nauki — Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 29(2), 166–182 [in Russian]. https://doi.org/10.20537/vm190203
Berdyshev, A.S., Cabada, A., & Kadirkulov, B.J. (2011). The Samarskii–Ionkin type problem for the fourth order parabolic equation with fractional diferential operator. Computers and Mathematics with Applications, 62(10), 3884–3893.
Yuldashev, T.K. (2016). Ob odnom smeshannom differentsialnom uravnenii chetvertogo poriadka [On a mixed type fourth-order differential equation]. Izvestiia Instituta matematiki i informatiki Udmurtskii Gosudarstvennyi Universitet — News of the Institute of Mathematics and Informatics of the Udmurt State University, 1(47), 119–128 [in Russian].
Amanov, D., & Otarova, J.A. (2008). Boundary value problem for a fourth-order mixed-type equation. Uzbek Mathematical Journal, (3), 13–22.
Sabitov, K.B.(2021). Nachalno-granichnye zadachi dlia uravneniia kolebanii balki s uchetom ee vrashchatelnogo dvizheniia pri izgibe [Initial-boundary value problems for the beam vibration equation with allowance for its rotational motion under bending]. Differentsialnye uravneniia — Differential equations, 57(3), 364–374 [in Russian]. https://doi.org/10.1134/S0012266121030071
Sabitov, K.B., & Fadeeva, O.V. (2021). Kolebaniia konsolnoi balki [Console beam vibrations]. Prikladnaia matematika i Fizika — Applied Mathematics and Physics, 53(1), 5–12 [in Russian]. https://doi.org/10.52575/2687-0959-2021-53-1-5-12
Sabitov, K.B. (2022). Kolebaniia plastiny s granichnymi usloviiami “sharnirzadelka” [Vibrations of plate with boundary hinged attachment conditions]. Vestnik Samarskogo gosudarstvennogo tekhnicheskogo Universiteta. Seriia «Fiziko-matematicheskie nauki» — Bulletin of Samara State Technical University. Series “Physical and Mathematical Sciences”, 26(4), 650–671 [in Russian]. https://doi.org/10.14498/vsgtu1950
Karachik, V., & Turmetov, B. (2017). Solvability of some neumann-type boundary value problems for biharmonic equations. Electronic Journal of Differential Equations, 2017(218), 1–17.
Urinov, A.K., & Azizov, M.S. (2021). Boundary value problems for a fourth order partial differential equation with an unknown right-hand part. Lobachevskii Journal of Mathematics, 42(3), 632–640. https://doi.org/10.1134/S1995080221030203
Dzhuraev, T.D., & Sopuev, A.K. (2000). Teorii differentsialnykh uravnenii v chastnykh proizvodnykh chetvertogo poriadka [To the theory of partial differential equations of the fourth order]. Tashkent: Fan [in Russian].
Sabitov, K.B., & Martemyanova, N.V. (2011). Nelokalnaia obratnaia zadacha dlia uravneniia smeshannogo tipa [A nonlocal inverse problem for a mixed-type equation]. Izvestiia vuzov. Matematika — News of universities. Mathematics, (2), 71–85 [in Russian]. https://doi.org/10.3103/S1066369X11020083
Khojanov, A.I. (2016). Obratnye zadachi vosstanovleniia pravoi chasti spetsialnogo vida v parabolicheskom uravnenii [Inverse problems of recovering the right-hand side of a special type of parabolic equations]. Matematicheskie zametki SVFU — Mathematical notes of NEFU, 23(4), 31–45 [in Russian].
Khojanov, A.I. (2004). Nelineinye nagruzhennye uravneniia i obratnye zadachi [Nonlinear loaded equations and inverse problems]. Zhurnal vychislitelnoi matematiki i fiziki — Journal of Computational Mathematics and Physics, 44(4), 694–716 [in Russian].
Denisov, A.M. (1994). Vvedenie v teoriiu obratnykh zadach [Introduction to the theory of inverse problems]. Moscow: MGU [in Russian].
Kabanixin, S.I. (2009). Obratnye i nekorrektnye zadachi [Inverse and ill-posed problems]. Novosibirsk: Sibirskoe nauchnoe izdatelstvo [in Russian].
Romanov, V.G. (1984). Obratnye zadachi matematicheskoi fiziki [Inverse problems of mathematical physics]. Moscow: Nauka [in Russian].
