An inverse boundary value problem for a linearized Benny–Luc equation with nonlocal boundary conditions

Abstract: The work is devoted to the study of the solvability of an inverse boundary value problem with an unknown time-dependent coefficient for the linearized Benney–Luke equation with non-conjugate boundary conditions and integral conditions. The goal of the paper consists of the determination of the unknown coefficient together with the solution. The problem is considered in a rectangular domain. The definition of the classical solution of the problem is given. First, the given problem is reduced to an equivalent problem in a certain sense. Then, using the Fourier method the equivalent problem is reduced to solving the system of integral equations. Thus, the solution of an auxiliary inverse boundary value problem reduces to a system of three nonlinear integro-differential equations for unknown functions. Concrete Banach space is constructed. Further, in the ball from the constructed Banach space by the contraction mapping principle, the solvability of the system of nonlinear integro-differential equations is proved. This solution is also a unique solution to the equivalent problem. Finally, by equivalence, the theorem of existence and uniqueness of a classical solution to the given problem is proved.


PUBLIC INTEREST STATEMENT
Many problems of mathematical physics, continuum mechanics are boundary problems that reduce to the integration of a differential equation or a system of partial differential equations for given boundary and initial conditions. Problems in which, together with the solution of a differential equation, it is also required to determine the coefficient of the equation itself, or the right-hand side of the equation, in mathematics and mathematical modeling are called inverse problems. The theory of inverse problems for differential equations is an actively developing area of modern mathematics. The goal of the paper consists of the determination of the unknown coefficient together with the solution. Our paper establishes existence and uniqueness of the solution to an inverse boundary value problem for the Benny-Luc equation with integral conditions.

Introduction
There are many cases where the needs of the practice bring about the problems of determining coefficients or the right-hand side of differential equations from some knowledge of its solutions. Such problems are called inverse boundary value problems of mathematical physics. Inverse boundary value problems arise in various areas of human activity such as seismology, mineral exploration, biology, medicine, and quality control in industry, which makes them an active field of contemporary mathematics. Inverse problems for various types of have been studied in many papers. Many problems of gas dynamics, theory of elasticity, theory of plates, and shells are reduced to the consideration of differential equations in high-order partial derivatives (Algazin & Kiyko, 2006). Of particular interest from the point of view of applications are differential equations of the fourth order (Shabrov, 2015), (Benney & Luke, 1964). Partial differential equations of the Benney-Luke type have applications in mathematical physics (Benney & Luke, 1964). Problems in which, together with the solution of a differential equation, it is also required to determine the coefficient of the equation itself, or the right-hand side of the equation, in mathematics and mathematical modeling are called inverse problems. The theory of inverse problems for differential equations is an actively developing area of modern mathematics. Various inverse problems for individual types of partial differential equations have been studied in many papers (Eskin, 2017;Janno & Seletski, 2015;Jiang, Liu, & Yamamoto, 2017;Lavrentyev, Romanov, & Shishatskii, 1980;Nakamura, Watanabe, & Kaltenbacher, 2009;Shcheglov, 2006;Tikhonov, 1963) . The theory of inverse boundary value problems for fourth-order equations remains poorly understood. The papers (Kozhanov & Namsaraeva, 2018) and others are devoted to inverse boundary value problems for equations of the fourth order. In (Yuldashev, 2018), the unique solvability of a nonlocal inverse problem for a fourth-order Benney-Luke integro-differential equation with a degenerate kernel is considered. In contrast to Yuldashev (2018), this paper studies the inverse boundary value problem for the fourth-order Benney-Luke equation with integral conditions of the first kind.
Proof. It is known that the boundary value problem (7), (8) is equivalent to the integral equation Having denoted and we write (10) in the form of an operator equation: yðtÞ ¼ AyðtÞ: Equation (11) will be studied in the space C½0; T.
It is easy to see that the operator A is continuous in the space C½0; T.
Let us show that A is a contraction mapping in C½0; T. Indeed, for any yðtÞ; yðtÞ from C½0; Twe have: AyðtÞ À A yðtÞ k k C½0;T pðtÞ k k C½0;T þ 2RT T yðtÞ À yðtÞ k k C½0;T : Then, using (9) in (12), we obtain A is contraction mapping in the space C½0; T. Therefore, in the space C½0; T, the operator A has a single fixed point yðtÞ which is a solution of Equation (11). Thus, integral equation (10) has a unique solution in C½0; T and consequently, boundary value problem (7), (8) also has a unique solution in C½0; T. Since yðtÞ ¼ 0 is the solution of boundary value problem (7), (8), then it has only trivial solution.
The theorem is proved. □
The following theorem is proved by means of Theorem 3.2 be satisfied. Then in the sphere K ¼ K R ð z k k E 5 T R ¼ AðTÞ þ 2Þ of the space E 5 T ,problem (1)-(6) has a unique classical solution.