An Inverse Coefficient Problem for a Parabolic Equation under Nonlocal Boundary and Integral Overdetermination Conditions
1Department of Mathematics and Informatics, the Larbi Ben M.hidi University, Oum El Bouaghi
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
International Journal of Partial Differential Equations and Applications, 2014 2 (3),
Received June 03, 2014; Revised June 16, 2014; Accepted June 19, 2014Copyright © 2014 Science and Education Publishing. All Rights Reserved.
Cite this article:
- Taki-Eddine, Oussaeif, and 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 (2014): 38-43.
- Taki-Eddine, O. , & Abdelfatah, B. (2014). 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.
- Taki-Eddine, Oussaeif, and 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, no. 3 (2014): 38-43.
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Suppose that one needs to determine simultaneously the temperature distributionu as well as thermal diffusivity coefficient satisfying the heat equation
with initial conditon
the boundary conditions
and the energy condition
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
and was comprehensively studied in , 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 . 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
The boundary conditions in (2.1) are regular, but not strongly regular (; pp: 66-67). The system of root functions of the operatorl is complete, but it does not form even a regular basis in . However, as it was shown in , 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  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 .
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
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
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
we obtain by using we have since is arbitrary, we get Now by using (2.5) and we obtain
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
For the function system (2.4), there exists a biorthogonal normalized system, given by
It is shown in  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):
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:
So, by replacing in equation (1.1) by the representation (3.1), we get
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:
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
using (3.3) and (3.4), yield
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
Since the estimates
are true by using the mean value theorem, where From the last inequalities, we obtain
Substituting (3.8), (3.9) and the above estimates in (3.7), we get
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:
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
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
From (3.3), a similar estimate is also obtained for the difference as
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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