phone +7 (3412) 91 60 92

Archive of Issues

Azerbaijan Baku; Ganja
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
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.
  1. Algazin S.D., Kiiko I.A. Flatter plastin i obolochek (Flutter of plates and shells), Moscow: Nauka, 2006.
  2. 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).
  3. 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.
  4. 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).
  5. 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.
  6. 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.
  7. 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.
  8. 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.
  9. 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.
  10. 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.
  11. 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).
  12. 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).
  13. 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).
  14. 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).
  15. 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.
  16. 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).
Full text
<< Previous article
Next article >>