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An Inverse Coefficient Problem for a Parabolic Equation under Nonlocal Boundary and Integral Overdetermination Conditions

Oussaeif Taki-Eddine*, Bouziani Abdelfatah

Department of Mathematics and Informatics, the Larbi Ben M.hidi University, Oum El Bouaghi

Abstract: This paper investigates the inverse problem of simultaneously determining the time-dependent thermal diffusivity and the temperature distribution in a parabolic equation in the case of nonlocal boundary conditions containing a real parameter and integral overdetermination conditions. Under some consistency conditions on the input data the existence, uniqueness and continuously dependence upon the data of the classical solution are shown by using the generalized Fourier method.

Keywords: heat equation, inverse problem, nonlocal boundary condition, integral overdetermination condition, Fourier method

Received June 03, 2014; Revised June 16, 2014; Accepted June 19, 2014

Citation: Oussaeif Taki-Eddine, Bouziani Abdelfatah, An Inverse Coefficient Problem for a Parabolic Equation under Nonlocal Boundary and Integral Overdetermination Conditions. International Journal of Partial Differential Equations and Applications, 2 (3). 38-43. January 2014. doi:10.12691/ijpdea-2-3-1

1. Introduction

Suppose that one needs to determine simultaneously the temperature distributionu as well as thermal diffusivity coefficient satisfying the heat equation

[picture] | (1.1)

with initial conditon

[picture] | (1.2)

the boundary conditions

[picture] | (1.3)

and the energy condition

[picture] | (1.4)

Where the parameter in an arbitrary real number and , , are given functions. The nonlocal second boundary condition in (1.3) is the main specific feature of this problem; for it acquires the form

[picture] | (1.5)

and was comprehensively studied in [1], are well-known as the Samarskii-Ionkin conditions, whilst (1.4) speci.es an integral additional specification of the energy. The problem of finding a pair will be called an inverse problem.

Denote the domain by

Definition 1. The pair from the class for which conditions (1.1)-(1.4) are satisfied and on the interval is called the classical solution of the inverse problem (1.1)-(1.4):

Nonlocal boundary conditions like (1:4) arise from many important applications in heat transfer, thermoelasticity, contro theory, life sciences and plays an important role in engineering and physics [8-17]. For example, for heat propagation in a thin rod in which the law of variation E(t) of the total quantit of heat in the rod is given in [1]. Various statements of inverse problems on determination of thermal coefficient in onedimensional heat equation were studied in [15,16,17]. It is important to note that in the papers [15,16] the time-dependent thermal coefficient is determined from nonlocal overdetermination condition data. Besides, in [12,15] the coefficients of the heat equations are determined in the case of nonlocal boundary conditions.

In this paper the source control parameter a(t) needs to be determined by thermal energy E(t), and the existence and uniqueness of the classical solution of the problem (1.1) - (1.4) is reduced to fixed point principles by applying the Fourier method. The boundary conditions (1.3) admit the expansions by the system of eigenfunctions and associated functions corresponding to the spectral problem.

The paper is organized as follows. In Section 2, the eigenvalues and eigenfunctions of the auxiliary spectral problem and some of their properties are introduced by applying the Fourier method to the problem (1.1) - (1.4). In Section 3, the existence and uniqueness of the solution of the inverse problem (1.1) - (1.4) is proved. Finally, the continuous dependence upon the data of the solution of the inverse problem is shown in Section 4.

2. The Auxiliary Spectral Problem

The use of the Fourier method for solving problem (1.1)-(1.3) leads to the spectral problem for the operatorlgiven by the differential expression and boundary conditions

[picture] | (2.1)

The boundary conditions in (2.1) are regular, but not strongly regular ([2]; pp: 66-67). The system of root functions of the operatorl is complete, but it does not form even a regular basis in [3]. However, as it was shown in [4], on the base of these eigenfunctions one can construct a basis allowing one to use the method of separation of variables for solving the initial-boundary value problem subject to the boundary condition (1.3).

In recent times a special attention has been paid to the study of direct and inverse problems for various classes of partial differential equations in particular cases of boundary conditions which are not strongly regular. Let us cite only those of them that are most close to the subject under consideration [5,6,7].

In this paper we propose to use the results described in [4] for solving the inverse problem (1.1)-(1.4). So, for construction of the basis of eigenfunctions of problem (2.1), we cite the necessary results from [4].

Consider the case in which We seek eigenvalues in the set of real numbers. Note that is not an eigenvalue, since problem (2.1) for this value of has only the trivial solution. Suppose that Then the eigenfunction should have the form By taking into account the nonlocal boundary condition, we obtain two equations

Solutions of the first equation form a series of eigenvalues and eigenfunctions of the operator (1.4) of the form

[picture] | (2.2)

The second equation can be represented as

with this equation has countably many solutions satisfying the inequalities for positive :

for sufficiently large So, that there exists a second series of eigenvalues and eigenfunctions of the form

[picture] | (2.3)

[picture] | (2.4)

Form a Riesz basis in and to find the biorthogonal system of consider the adjoint differential operator of in the sense of the inner product of the space

So, for two arbitrary twice continuously differentiable function and on such that we have

this implies

[picture] | (2.5)

we obtain by using we have since is arbitrary, we get Now by using (2.5) and we obtain

[picture] | (2.6)

It is easy to verify that the eigenvalues of this problem are same as for the problem (2.1). The system of eigenfunctions and associated functions of the problem (2.5) is denoted by

where

For the function system (2.4), there exists a biorthogonal normalized system, given by

[picture] | (2.7)

It is shown in [4] that the systems (2.4) and (2.7) form biorthogonal systems on the interval , i.e.

3. Existence and Uniqueness of the Solution of the Inverse Problem

We have the following assumptions on the data of the problem (1.1) - (1.4):

where

The main result is presented as follows.

Theorem 1. Let the assumptions (A₁)-(A₃) be satisfied. Then the inverse problem (1)- (4) has a unique classical solution

Proof. By applying the standard procedure of the Fourier method, Any solution of equation (1.1) can represented as:

[picture] | (3.1)

So, by replacing in equation (1.1) by the representation (3.1), we get

[picture] | (3.2)

multiplying the equation (3:2) by and integrating over we get the following system of equations:

Substituting the solution of this system of equations and initial condition (1.2) in (3.1), we obtain the solution of the problem (1.1) - (1.3) in the following form:

[picture] | (3.3)

Under the conditions and the series (3.3) and its x-partial derivative are uniformly convergent in since their majorizing sums are absolutely convergent. Therefore, their sums and are continuous in In addition, the t-partial derivative and the xx-second-order partial derivative series are uniformly convergent in Thus, we have In addition, is continuous in Differentiating (1.4) under the assumption we obtain

[picture] | (3.4)

using (3.3) and (3.4), yield

[picture] | (3.5)

where

[picture] | (3.6)

Let us denote

It is easy to verify that under conditions and

Let us show that is a contraction mapping in Then, we have for

[picture] | (3.7)

where

Since the estimates

are true by using the mean value theorem, where From the last inequalities, we obtain

[picture] | (3.8)

and

[picture] | (3.9)

Substituting (3.8), (3.9) and the above estimates in (3.7), we get

[picture] | (3.10)

where

In the case Equation (3.10) has a unique solution by the Banach fixed point theorem.

Now, let us show that the solution obtained for (1)-(4), is unique. Suppose that is also a solution pair of (1:1)-(1:4). Then from the representation (3.3) of the solution, we have:

[picture] | (3.11)

From the equation (3.5), and (3.10), we obtain

since implies that By substituting into (3.11), we get

Theorem 1 has been proved.

4. Continuous Dependence of (a,u) upon the Data

Theorem 2. Under assumption (A₁) - (A₃); the solution depends continuously upon the data.

Proof. Let and be two sets of the data, which satisfy the assumptions (A1) - (A3): Then there exist positive constants such that

[picture] | (4.1)

[picture] | (4.2)

Let and be solutions of the inverse problem (1.1)-(1.4) corresponding to the data and respectively. According to (3.5), we have

First, let us estimate the difference It is easy to see that by using (4.1) and (4.2), then

by using the previous inequality, we obtain

this implies

where

From (3.3), a similar estimate is also obtained for the difference as

5. Conclusion

The inverse problem regarding the simultaneously identification of the timedependent thermal diffusivity and the temperature distribution in one-dimensional heat equation with nonlocal boundary condition and integral overdetermination conditions has been considered. Where the nonlocal boundary conditions containing a real parameter. For the zero value of the parameter, this conditions is well known as the Samarskii-Ionkin conditions and has been comprehensively studiedand. This inverse problem has been investigated from both theoretical. In this study the conditions for the existence, uniqueness and continuous dependence upon the data of the problem have been established.

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