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Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2019, Volume 29, Issue 2, Pages 166–182 (Mi vuu674)  

This article is cited in 1 scientific paper (total in 1 paper)

MATHEMATICS

Inverse boundary value problem for the linearized Benney-Luke equation with nonlocal conditions

Ya. T. Megralieva, B. K. Velievab

a Baku State University, ul. Z. Khalilova, 23, Baku, AZ1148, Azerbaijan
b Ganja State University, ul. Khatai, 187, Ganja, AZ2000, Azerbaijan

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.

DOI: https://doi.org/10.20537/vm190203

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Bibliographic databases:

UDC: 517.95
MSC: 35-XX
Received: 24.05.2019

Citation: Ya. T. Megraliev, B. K. Velieva, “Inverse boundary value problem for the linearized Benney-Luke equation with nonlocal conditions”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 29:2 (2019), 166–182

Citation in format AMSBIB
\Bibitem{MegVel19}
\by Ya.~T.~Megraliev, B.~K.~Velieva
\paper Inverse boundary value problem for the linearized Benney-Luke equation with nonlocal conditions
\jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki
\yr 2019
\vol 29
\issue 2
\pages 166--182
\mathnet{http://mi.mathnet.ru/vuu674}
\crossref{https://doi.org/10.20537/vm190203}
\elib{https://elibrary.ru/item.asp?id=39136242}


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    This publication is cited in the following articles:
    1. L. I. Mammadova, I. M. Nabiev, “Spektralnye svoistva operatora Shturma–Liuvillya so spektralnym parametrom, kvadratichno vkhodyaschim v granichnoe uslovie”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 30:2 (2020), 237–248  mathnet  crossref
  • Вестник Удмуртского университета. Математика. Механика. Компьютерные науки
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