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Advances in Mathematical Physics
Volume 2018, Article ID 8301656, 8 pages
https://doi.org/10.1155/2018/8301656
Research Article

On an Inverse Problem of Reconstructing a Heat Conduction Process from Nonlocal Data

1Institute of Mathematics and Mathematical Modeling, 125 Pushkin Str., 050010 Almaty, Kazakhstan
2Al-Farabi Kazakh National University, 71 Al-Farabi Ave., 050040 Almaty, Kazakhstan
3South-Kazakhstan State Pharmaceutical Academy, 1 al-Farabi Sq., 160019 Shymkent, Kazakhstan

Correspondence should be addressed to Makhmud A. Sadybekov; zk.htam@vokebydas

Received 28 February 2018; Accepted 21 May 2018; Published 12 June 2018

Academic Editor: Pavel Kurasov

Copyright © 2018 Makhmud A. Sadybekov et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider an inverse problem for a one-dimensional heat equation with involution and with periodic boundary conditions with respect to a space variable. This problem simulates the process of heat propagation in a thin closed wire wrapped around a weakly permeable insulation. The inverse problem consists in the restoration (simultaneously with the solution) of an unknown right-hand side of the equation, which depends only on the spatial variable. The conditions for redefinition are initial and final states. Existence and uniqueness results for the given problem are obtained via the method of separation of variables.

1. Introduction and Statement of the Problem

The problems that imply the determination of coefficients or the right-hand side of a differential equation (together with its solution) are commonly referred to as inverse problems of mathematical physics. In this paper we consider one family of problems implying the determination of the density distribution and of heat sources from given values of initial and final distributions. This problem simulates the process of heat propagation in a thin closed wire wrapped around a weakly permeable insulation. The mathematical statement of such problems leads to an inverse problem for the heat equation, where it is required to find not only a solution of the problem, but also its right-hand side that depends only on a spatial variable.

In this paper, we will consider the inverse problem close to that investigated in [1, 2]. Together with the solution it is necessary to find an unknown right-hand side of the equation. The equation contains the usual time derivative and an involution with respect to the spatial variable. In contrast to [1], we investigate the problem under nonlocal boundary conditions with respect to the spatial variable. The conditions for overdetermination are initial and final states.

The second of the main differences in the investigated inverse problem being studied is that the unknown function enters, both in the right-hand side of the equation and in the conditions of the initial and final overdetermination.

Let us consider a problem of modeling the thermal diffusion process which is close to that described in the report of Cabada and Tojo [2], where the example that describes a concrete situation in physics is given. Consider a closed metal wire (length ) wrapped around a thin sheet of insulation material in the manner shown in Figure 1.

Figure 1: The closed metal wire wrapped around a thin sheet of insulation material.

Assume that the position is the lowest of the wire, and the insulation goes up to the left at and to the right up to . Since the wire is closed, points and coincide.

The layer of insulation is assumed to be slightly permeable. Therefore, the temperature value from one side affects the diffusion process on the other side. For this reason, the standard heat equation is modified and to its right-hand side a third term (where ) is added. Here is the temperature at point of the wire at time .

Thus, this process is described by the following:in the domain . Here is the influence of an internal source that does not change with time; is an initial time point and is a final one.

As the additional information we take values of the initial and final conditions of the temperature:

Since the wire is closed, it is natural to assume that the temperature at the ends of the wire is the same at all times:

Since is the temperature at point of the wire at time , average temperature along the entire length of the wire at time can be calculated from formula . If additional heating (cooling) is applied through the boundary point , then the average temperature of the wire increases (decreases). The greater the importance of this heating (cooling) is, the faster the average temperature of the wire changes.

Consider a process in which the temperature at one end at every time point is proportional to the change speed of the average value of the temperature throughout the wire. Then,Here is a proportionality coefficient.

Thus the investigated process is reduced to the following mathematical inverse problem: Find the right-hand side of the heat equation (1) and its solution subject to the initial and final conditions (2), the boundary condition (3), and condition (4).

2. Reduction to a Mathematical Problem

Condition (4) is significantly nonlocal (i.e., values of the unknown function over the entire domain are used). The integral along inner lines of the domain is present in this condition. Using the idea of A.A. Samarskii, we transform this condition. Taking into account (1) from (4), we getHence

Let us introduce the following notations:

Then in terms of a new function we get the following inverse problem: In the domain find the right-hand side of the heat equation with involution,and its solution satisfying the initial conditions,and the final conditions,and the boundary ones,where and are given sufficiently smooth functions; is a nonzero real number such that ; and .

In the physical sense, the second of conditions (11) means the equality of the distribution densities at the ends of the interval. And the first of conditions (11) means the proportionality of the difference of fluxes across opposite boundaries to the density value at the boundary. We note that in [1] the Dirichlet boundary conditions were used instead of condition (11).

The well-posedness of direct and inverse problems for parabolic equations with involution was considered in [35].

The solvability of various inverse problems for parabolic equations was studied in papers of Anikonov Yu.E. and Belov Yu.Ya., Bubnov B.A., Prilepko A.I. and Kostin A.B., Monakhov V.N., Kozhanov A.I., Kaliev I. A., Sabitov K.B., and many others. These citations can be seen in our papers [6, 7]. In [1] there are good references to publications on related issues. We note [3, 828] from recent papers close to the theme of our article. In these papers different variants of direct and inverse initial-boundary value problems for evolutionary equations are considered, including problems with nonlocal boundary conditions and problems for equations with fractional derivatives.

In this paper we shall use a spectral problem for an ordinary differential operators with involution. Such and similar spectral problems were considered in [2940].

Definition 1. By a regular solution of the inverse problem (8)-(11) we mean a pair of functions of the class , that inverts (8) and conditions (9)-(11) into an identity.

Definition 2. By a generalized solution of the inverse problem (8)-(11) we mean a pair of functions of the class , that satisfies (8) and conditions (9)-(11) almost everywhere.

When one uses the method of separation of variables to solve the problem, a spectral problem appears, which is mentioned in the next section.

3. Spectral Problem

The use of the Fourier method for solving problem (8)–(11) leads to a spectral problem for an operator given by the differential expression,and the boundary conditions,where is a spectral parameter.

The spectral problems for (12) were first considered, apparently in [37]. There was considered a case of Dirichlet and Neumann boundary conditions and cases of conditions in the form (13) for . Here we consider the case . We assume that .

We represent a general solution of (12) in the following form:where and are arbitrary complex numbers. Satisfying the boundary conditions (13) for finding eigenvalues, we obtain the following:Therefore, the spectral problem (12)-(13) has two series of eigenvalues:, , where = , as , with corresponding normalized eigenfunctions given byHere is a normalization coefficient:

Lemma 3. The system of functions (17) is complete and orthonormal in .

Proof. We note that the system of eigenfunctions (17) does not depend on the parameter . Only the eigenvalues of problem (12)-(13) depend on .
In the case when , system (17) is a system of eigenfunctions of the classical Strum-Liouville problem:with the self-adjoint boundary conditions (13).
Consequently, system (17) forms the complete orthonormal system in ; that is, it is the orthonormal basis of the space.

4. Uniqueness of the Solution of the Problem

Let the pair of functions be a solution of the inverse problem (8)-(11). Let us introduce notations:

We apply the operator to . Then, using (8), by integrating by parts, we obtain the following problem:

Here we use the following notations:

It is easy to see that the function is a partial solution of the inhomogeneous equation (21). Therefore, we obtain a unique solution of the Cauchy problem (21)-(22) in the following form:

Since , , we have for all . Consequently,

Therefore, using the condition of "final overdetermination" (23), we get

Lemma 4. If , then the generalized solution of the inverse problem (8)-(11) is unique.

Proof. Suppose that there are two generalized solutions of the inverse problem (8)-(11): and . DenoteThen the functions and satisfy (8), the boundary conditions (11), and the homogeneous conditions (9) and (10): Therefore, by using the notations (20) from (27), we findSincethis givesSince , then for from (32), we have .
For the generalized solution of the problem (8)-(11) it is necessary that . From this, by virtue of Parseval’s equality for orthogonal bases, we have thatis necessary.
Since ,then (32) and inequality (33) are possible only at . Hence we obtain .
Therefore, using this result, from (25) and (27), we findfor all values of the indexes for and for . Further, by the completeness of system (17) in , we obtainThe uniqueness of the generalized solution of the inverse problem (8)-(11) is proved.

Corollary 5. If , then the regular solution of the inverse problem (8)-(11) is unique.

5. Construction of a Formal Solution of the Problem

As the eigenfunctions system (17) forms the orthonormal basis in , then the unknown functions and can be represented aswhere and are unknown functions; and are unknown constants.

Substituting (37) and (38) into (8), we obtain the inverse problems (21)-(23). If the constants are assumed to be given, then the solutions of these inverse problems exist, are unique, and are represented by formulas (25) and (27). Substituting (25) and (27) into series (37) and (38), we obtain a formal solution of the inverse problem (8)-(11).

Let and satisfy the boundary conditions (13). Since , then

Therefore, from the analysis of formula (27), it is easy to see that, in order for the formal solution (37) of the problem (8)-(11) to be a generalized solution, it is necessary thatholds.

As above, we calculatewhere and

Thus, (40) holds if and only if for all values of the indexes . This is possible only in the case

Thus, condition (43) is a necessary condition for the existence of the generalized solution of the inverse problem (8)-(11). In this case, problems (8)-(11) and (1)-(4) coincide. Indeed, from (43) and (1) we have

For the first integral we apply condition (4) and calculate the second integral. Then we obtainThis means that the boundary conditions (4) and (11) coincide. Hence, the problems (8)-(11) and (1)-(4) also coincide.

Thus, in what follows, we shall consider problem (1)-(3) with the boundary condition:

Thus, in what follows, we will consider the inverse problem (1)-(3), (46).

Similarly, as before, the formal solution of this problem can be constructed in the form of serieswhere

In order to complete our study, it is necessary, as in the Fourier method, to justify the smoothness of the resulting formal solutions and the convergence of all appearing series.

6. Main Results

Here we present the existence and uniqueness results for our inverse problem.

Theorem 6 (let ). (A) Let and let the functions satisfy the boundary conditions (13). Then for a real number such that the inverse problem (1)-(3), (46) has a unique generalized solution, which is stable in norm:where the constant does not depend on and .
(B) Let and the functions and , satisfy the boundary conditions (13), then for a real number such that the inverse problem (1)-(3), (46) has a unique regular solution.

Proof. Since , , we have for all . Consequently, there exists a constant , such that for all the values of the indexes we haveSince , then . Therefore, from representations (49) and (50), we get estimateswhere constant does not depend on the indexes and on the functions and .
As proved in Lemma 3, the eigenfunctions system (17) forms the orthonormal basis in . Therefore, we have an expansion:Taking into account (12), we obtainFrom this, by Parseval’s equality, it is easy to obtain an estimate:Similarly, for the coefficients of the expansion,we have the inequalityNow, from (53), (57), and (59), we see that the series (48) converges and the estimateholds.
Taking into account (12), from (47), we obtainSubstituting the value instead of , we findHere we use the properties , obtained from the explicit form (12) of the eigenfunctions.
Multiplying (62) by and summing with (61), we haveSince , from this, by Parseval’s equality, we findfor each . Integrating this equation with respect to , taking (54) into account, we haveHence, using (57) and (59), we obtainNow we can easily obtain an estimate for from (1). This fact together with (60) and (66) gives the necessary estimate (51) for the solution.
From the obtained estimates it also follows that in the constructed by us formal solution of the inverse problem all the series converge, they can be term-by-term differentiated, and the series obtained during differentiation also converge in sense of the metrics .
From (47) and (54), by using the Holder’s inequality, it is easy to justify the inequalitywhich justifies the continuity of in the closed domain .
From the representation of the solution in the form of series (37), (38) and inequalities (53), (54) it is easy to justify the estimatesLet and let the functions , and , satisfy the boundary conditions (13), then the number series in the right-hand side of (68) converges. Therefore, in such case the constructed by us formal solution gives the regular solution of the inverse problem (1)-(3), (46).
The uniqueness of this constructed solution follows from Lemma 4.
The theorem is completely proved.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This paper was published with the support of the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan under Project AP05133271 and Target Program BR05236656.

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