+7 (3412) 91 60 92

## Archive of Issues

Azerbaijan Baku; Ganja
Year
2019
Volume
29
Issue
2
Pages
166-182
 Section Mathematics Title Inverse boundary value problem for the linearized Benney-Luke equation with nonlocal conditions Author(-s) Megraliev Ya.T.a, Velieva B.K.b Affiliations Baku State Universitya, Ganja State Universityb Abstract The paper investigates the solvability of an inverse boundary-value problem with an unknown coefficient and the right-hand side, depending on the time variable, for the linearized Benney-Luke equation with non-self-adjoint boundary and additional integral conditions. The problem is considered in a rectangular domain. A definition of the classical solution of the problem is given. First, we consider an auxiliary inverse boundary-value problem and prove its equivalence (in a certain sense) to the original problem. To investigate the auxiliary inverse boundary-value problem, the method of separation of variables is used. By applying the formal scheme of the variable separation method, the solution of the direct boundary problem (for a given unknown function) is reduced to solving the problem with unknown coefficients. Then, the solution of the problem is reduced to solving a certain countable system of integro-differential equations for the unknown coefficients. In turn, the latter system of relatively unknown coefficients is written as a single integro-differential equation for the desired solution. Next, using the corresponding additional conditions of the inverse auxiliary boundary-value problem, to determine the unknown functions, we obtain a system of two nonlinear integral equations. Thus, the solution of an auxiliary inverse boundary-value problem is reduced to a system of three nonlinear integro-differential equations with respect to unknown functions. A special type of Banach space is constructed. Further, in a ball from a constructed Banach space, with the help of contracted mappings, we prove the solvability of a system of nonlinear integro-differential equations, which is also the unique solution to the auxiliary inverse boundary-value problem. Finally, using the equivalence of these problems the existence and uniqueness of the classical solution of the original problem are proved. Keywords inverse boundary value problem, Benney-Luke equation, existence, uniqueness of classical solution UDC 517.95 MSC 35-XX DOI 10.20537/vm190203 Received 24 May 2019 Language Russian Citation Megraliev Ya.T., Velieva B.K. Inverse boundary value problem for the linearized Benney-Luke equation with nonlocal conditions, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2019, vol. 29, issue 2, pp. 166-182. References Algazin S.D., Kiiko I.A. Flatter plastin i obolochek (Flutter of plates and shells), Moscow: Nauka, 2006. Shabrov S.A. About the estimates of the function influence of a mathematical model of fourth order, Vestnik Voronezhskogo Gosudarstvennogo Universiteta. Seriya: Fizika. Matematika, 2015, no. 2, pp. 168-179 (in Russian). https://elibrary.ru/item.asp?id=23478452 Benney D.J., Luke J.C. On the interactions of permanent waves of finite amplitude, Journal of Mathematical Physics, 1964, vol. 43, pp. 309–313. https://doi.org/10.1002/sapm1964431309 Tikhonov A.N. On the solution of ill-posed problems and the method of regularization, Doklady Akademii Nauk SSSR, 1963, vol. 151, no. 3, pp. 501-504 (in Russian). http://mi.mathnet.ru/eng/dan28329 Lavrent'ev M.M., Romanov V.G., Shishatskii S.T. Nekorrektnye zadachi matematicheskoi fiziki i analiza (Ill-posed problems of mathematical physics and analysis), Moscow: Nauka, 1980. Eskin G. Inverse problems for general second order hyperbolic equations with time-dependent coefficients, Bulletin of Mathematical Sciences, 2017, vol. 7, issue 2, pp. 247-307. https://doi.org/10.1007/s13373-017-0100-2 Jiang D.J., Liu Y.K., Yamamoto M. Inverse source problem for the hyperbolic equation with a time-dependent principal part, Journal of Differential Equations, 2017, vol. 262, issue 1, pp. 653-681. https://doi.org/10.1016/j.jde.2016.09.036 Nakamura G., Watanabe M., Kaltenbacher B. On the identification of a coefficient function in a nonlinear wave, Inverse Problems, 2009, vol. 25, issue 3, 035007. https://doi.org/10.1088/0266-5611/25/3/035007 Shcheglov A.Y. Inverse coefficient problem for a quasilinear hyperbolic equation with final overdetermination, Computational Mathematics and Mathematical Physics, 2006, vol. 46, issue 4, pp. 616-635. https://doi.org/10.1134/S0965542506040099 Janno J., Seletski A. Reconstruction of coefficients of higher order nonlinear wave equations by measuring solitary waves, Wave Motion, 2015, vol. 52, pp. 15-25. https://doi.org/10.1016/j.wavemoti.2014.08.005 Kozhanov A.I., Namsaraeva G.V. Linear inverse problems for a class of equations of Sobolev type, Chelyabinsk Physical and Mathematical Journal, 2018, vol. 3, issue 2, pp. 153-171 (in Russian). https://doi.org/10.24411/2500-0101-2018-13203 Yuldashev T.K. On a nonlocal inverse problem for a Benney-Luke type integro-differential equation with degenerate kernel, Vestnik TVGU. Seriya: Prikladnaya Matematika, 2018, issue 3, pp. 19-41 (in Russian). https://doi.org/10.26456/vtpmk500 Mehraliev Ya.T. Inverse problem of the Boussinesq-Love equation with an extra integral condition, Sibirskii Zhurnal Industrial'noi Matematiki, 2013, vol. 16, no. 1, pp. 75-83 (in Russian). http://mi.mathnet.ru/eng/sjim768 Megraliev Ya.T., Alizade F.Kh. 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. Komp'yuternye Nauki, 2016, vol. 26, issue 4, pp. 503-514 (in Russian). https://doi.org/10.20537/vm160405 Sabitov K.B., Martem'yanova N.V. A nonlocal inverse problem for a mixed-type equation, Russian Mathematics, 2011, vol. 55, issue 2, pp. 61-74. https://doi.org/10.3103/S1066369X11020083 Mehraliev Ya.T. On an inverse boundary value problem for the second order elliptic equation with additional integral condition, Vladikavkaz. Mat. Zh., 2013, vol. 15, no. 4, pp. 30-43 (in Russian). http://mi.mathnet.ru/eng/vmj482 Full text