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

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.


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 2) wrapped around a thin sheet of insulation material in the manner shown in Figure 1.
Assume that the position  = 0 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 ( 2 Φ/ 2 )(, ) a third term ( 2 Φ/ 2 )(−, ) (where || < 1) is added.Here Φ(, ) is the temperature at point  of the wire at time .
Here () is the influence of an internal source that does not 2 Advances in Mathematical Physics change with time;  = 0 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 Φ(, ) 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 Φ() = ∫  − Φ(, ).If additional heating (cooling) is applied through the boundary point  = −, then the average temperature of the wire Φ() increases (decreases).The greater the importance of this heating (cooling) is, the faster the average temperature of the wire Φ() 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 Φ(, ) subject to the initial and final conditions (2), the boundary condition (3), and condition (4).

Reduction to a Mathematical Problem
Then in terms of a new function (, ) we get the following inverse problem: In the domain Ω = {(, ) : − <  < , 0 <  < } find the right-hand side () of the heat equation with involution, and its solution (, ) satisfying the initial conditions, where () and () are given sufficiently smooth functions;  is a nonzero real number such that || < 1; and  = 1/( − 1).
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 (−, ) = (, ) = 0 were used instead of condition (11).
The well-posedness of direct and inverse problems for parabolic equations with involution was considered in [3][4][5].
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, 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.
When one uses the method of separation of variables to solve the problem, a spectral problem appears, which is mentioned in the next section.

Spectral Problem
The use of the Fourier method for solving problem ( 8)-( 11) leads to a spectral problem for an operator L 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  = 0.Here we consider the case  ̸ = 0. We assume that  > 0.
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   = (/( + 1)) > 0, as  → ∞, with corresponding normalized eigenfunctions given by Here ]  is a normalization coefficient: Lemma 3. The system of functions ( 17) is complete and orthonormal in  2 (−, ).
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  = 0, 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  2 (−, ); that is, it is the orthonormal basis of the space.

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:
Similarly, as before, the formal solution of this problem can be constructed in the form of series where 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.

Main Results
Here we present the existence and uniqueness results for our inverse problem.
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  2 .
From ( 47) and (54), by using the Holder's inequality, it is easy to justify the inequality max From the representation of the solution in the form of series (37), (38) Let (), () ∈  4 [−, ] 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.

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