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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.
1. 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, Citation2006). Of particular interest from the point of view of applications are differential equations of the fourth order (Shabrov, Citation2015), (Benney & Luke, Citation1964). Partial differential equations of the Benney–Luke type have applications in mathematical physics (Benney & Luke, Citation1964). 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, Citation2017; Janno & Seletski, Citation2015; Jiang, Liu, & Yamamoto, Citation2017; Lavrentyev, Romanov, & Shishatskii, Citation1980; Nakamura, Watanabe, & Kaltenbacher, Citation2009; Shcheglov, Citation2006; Tikhonov, Citation1963) . The theory of inverse boundary value problems for fourth-order equations remains poorly understood. The papers (Kozhanov & Namsaraeva, Citation2018) and others are devoted to inverse boundary value problems for equations of the fourth order. In (Yuldashev, Citation2018), the unique solvability of a non-local inverse problem for a fourth-order Benney–Luke integro-differential equation with a degenerate kernel is considered. In contrast to Yuldashev (Citation2018), this paper studies the inverse boundary value problem for the fourth-order Benney–Luke equation with integral conditions of the first kind.
2. Problem statement and its reduction to an equivalent problem
Let . Consider the following inverse problem. It is required to find a trio
of functions
,
,
connected by Equation [3]:
in the domain , with the non-local initial conditions
the boundary conditions
integral conditions
and with the overdetermination conditions
where are fixed numbers,
,
,
,
,
,
are the given functions, and
,
,
are the desired functions. We introduce the notation
Definition 2.1. Under the classic solution of inverse boundary value problem, we understand the trio of functions
,
,
satisfying Equation (1) and conditions (2)–(6) in the ordinary sense.
In order to investigate problem (1)–(5), we first consider the following problem:
where is a given number,
,
are the given functions,
is a desired function, under the solution of problem (7),(8) we understand the function
from
and satisfying conditions (6),(7) in the ordinary sense. The following lemma is proved:
Lemma 2.2. Let ,
and
Then problem (7),(8) has only a trivial solution.
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:
Equation (11) will be studied in the space .
It is easy to see that the operator is continuous in the space
.
Let us show that is a contraction mapping in
. Indeed, for any
from
we have:
Then, using (9) in (12), we obtain is contraction mapping in the space
. Therefore, in the space
, the operator
has a single fixed point
which is a solution of Equation (11). Thus, integral equation (10) has a unique solution in
and consequently, boundary value problem (7), (8) also has a unique solution in
. Since
is the solution of boundary value problem (7), (8), then it has only trivial solution.
The lemma is proved. □
Along with problem (1)–(6), we consider the following auxiliary inverse boundary value problem. It is required to determine a triple functions
,
,
from relations (1)–(3) .
where
The following theorem is valid.
Theorem 2.3. Let
,
,
and the consistency conditions
be satisfied. Then the following statements are valid:
• Each classical solution of problem (1)–(6) is the solution of problem (1)–(3), (13)–(15)
• Each solution of problem (1)–(3), (13)–(15), is a classical solution of the problem (1)-(3),if
Proof. Let be a solution of problem (1)–(6). Integrating Equation (1) over
from 0 to 1, we have:
Assuming that ,
, in view of (3),(4), we arrive at fulfillment (13).
Substituting and
in Equation (1), respectively, we find:
Under the assumption and differentiating two times (6) we have:
Considering these relations, from (18) and (19), taking into account (6), the fulfillment of (14) and (15) follows, respectively.
Now, suppose that is a solution to problem (1)–(3), (13)–(15), and (16) is satisfied. Then from (17) and (13), we find:
From (22) and we have:
Since, by Lemma 2.2, problem (20), (21) has only a trivial solution,
, i.e. conditions (4) are satisfied.
Further, from (14) and (18), (15) and (19) we get:
From (2) and consistency conditions
, we have:
From (22), (24), and also from (23), (25) by virtue of Lemma 2.2, we conclude that the conditions (5) and (6) is obtained.
The theorem is proved. □
3. Existence and uniqueness of the classical solution of the inverse problem
The first component of the solution
to problem (1)–(3), (13)–(15) will be sought in the form:
where
and
Then, applying the formal scheme of the Fourier method, from (1), (2), we get:
where
Solving the problem (27), (28) we find:
where
After substituting expressions from (29),
from (30) into (26), to determine the component
of the solution of problem (1)—(3), (13)—(15), we get:
Now from (14) and (15), taking into account (30), we obtain:
or considering that
we have:
Assume that
Then from (32) and (33) we find:
Further, after substituting the expression
from (30) into (34), (35), respectively, we have:
Thus, the solution of the problem (1)–(3), (13)–(15) was reduced to the solution of the system (31), (36), (37) with respect to the unknown functions ,
and
.
To study the question of the uniqueness of the solution of problem (1)–(3), (13)–(15), the following Lemma is important:
Lemma 3.1. If is any solution of problem (1)–(3), (13)–(15),then the functions
, defined by
satisfy system (29) and (30) on .
Proof. Let be any solution of (1)–(3), (13)–(15). Then, multiplying both sides of Equation (1) by the function
, integrating the obtained equality over
from
to
and using the relations,
we obtain that (27) are satisfied.
Similarly, from (2) we obtain that condition (28) is satisfied.
Thus, is a solution to problem (27), (28). Hence, it immediately follows that the functions
satisfy the system (29), (30) on
. The lemma is proved. □
Obviously, if is a solution to system (29)–(30) then the triple
of functions,
,
and
is a solution to system (31), (36), (37). From Lemma 3.1 it follows that:
Remark 1. Let system (31), (36), (37) have a unique solution. Then the problem (1)–(3), (12), (13) cannot have more than one solution, i.e. if problem (1)–(3), (13)–(15) has a solution, then it is unique.
Now, in order to study the problem (1)–(3), (13)–(15) we consider the following spaces:
• We denote by , a consisting of all functions
of the form
considered in , where each of the functions form
is continuous on
and
The norm in this set is defined as follows:
• The space can be described the space consisting of a topological product
The norm of element is determined by the formula:
It is obvious that and
are Banach spaces.
Now, in the space consider the operator
where
,
and
are equal to the right-hand sides of (29), (30), (36) and (37).
It is easy to see that
Taking into account these relations and , with the help of simple transformations we find:
Suppose that the data of the problem (1)–(3), (13)–(15) satisfy the following conditions:
(1)
(2)
(3)
(4)
(5)
(6)
Then, considering (38)—(39), (40) and (41) we get:
where
From inequalities (45)–(48) we deduce:
where
So, we can prove the following theorem:
Theorem 3.2. Let conditions be satisfied, and
then problem (1)–(3), (13)–(15) has a unique solution in the sphere of the space
Proof.In the space consider the equation
where the components
of the operator
are determined by the right-hand sides of Equations (31), (36), and (37).
Consider the operator in the sphere
from
. Similar to (45) we get that for any
the following estimates are valid:
Then, using (46), from (48) and (49), it follows that the operator acts in the sphere
and it is contraction mapping. Therefore, in the sphere
, the operator
has a unique fixed point
, that is a solution of Equation (47).
The function , as the element of the space
, has continuous derivatives
in
.
Now, differentiating two times (29), (30), we get:
It is clear that . Further, from (50) and (51), we obtained:
It is seen that .
From (26) it is easy to see that and the validity of the estimates:
Then, it follows that .
It is easy to verify that Equation (1) and conditions (2), (3), (13)–(15) are satisfied in the ordinary sense. Consequently, is a solution of problem (1)–(3), (13)–(15) and by Lemma3 it is unique in the sphere
. Theorem is proved.
The following theorem is proved by means of Theorem 3.2
Theorem 3.3. Let all the conditions of theorem2,
and consistency conditions
be satisfied. Then in the sphere of the space
,problem (1)–(6) has a unique classical solution.
Additional information
Funding
Notes on contributors
Yashar T. Mehraliyev
Yashar T. Mehraliyev He is working at the Baku State University and he is chief of the Chairprofessor. His research interests include Direct and Inverse Problems for Partial Differential Equations, The Spectral Theory of Differential Equations, Nonlinear Functional Analysis.
Bahar K. Valiyeva
Bahar K. Valiyeva She is currently Ph.D. appliicant and the main research interests of him include Inverse Problems for Partial Differential Equations.
Aysel T. Ramazanova
Aysel T. Ramazanova She works and doing Postdoc Research at the University Duisburg-Essen. Her research interests include Inverse Problems for Partial Differential Equations, optimal control problem in the processes described by elliptic and hyperbolic equations.
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