The inverse problem of the heat equation with periodic boundary and integral overdetermination conditions
© Kanca; licensee Springer. 2013
Received: 26 November 2012
Accepted: 24 February 2013
Published: 18 March 2013
In this paper the inverse problem of finding the time-dependent coefficient of heat capacity together with the solution of a heat equation with periodic boundary and integral overdetermination conditions is considered. The conditions for the existence and uniqueness of a classical solution of the problem under consideration are established. A numerical example using the Crank-Nicolson finite-difference scheme combined with the iteration method is presented.
Definition 1 The pair from the class , for which conditions (1)-(4) are satisfied and on the interval , is called a classical solution of the inverse problem (1)-(4).
The parameter identification in a parabolic differential equation from the data of integral overdetermination condition plays an important role in engineering and physics [1–6]. This integral condition in parabolic problems is also called heat moments are analyzed in .
Boundary value problems for parabolic equations in which one or two local classical conditions are replaced by heat moments [5–9]. In , a physical-mechanical interpretation of the integral conditions was also given. Various statements of inverse problems on determination of this coefficient in a one-dimensional heat equation were studied in [1–3, 5, 6, 10, 11]. In the papers [1, 3, 5], the coefficient is determined from heat moment. Boundary value problems and inverse problems for parabolic equations with periodic boundary conditions are investigated in [10, 12].
In the present work, one heat moment is used with a periodic boundary condition for the determination of a source coefficient.The existence and uniqueness of the classical solution of the problem (1)-(4) is reduced to fixed point principles by applying the Fourier method.
The paper organized as follows. In Section 2, the existence and uniqueness of the solution of the inverse problem (1)-(4) is proved by using the Fourier method. In Section 3, the continuous dependence upon the data of the inverse problem is shown. In Section 4, the numerical procedure for the solution of the inverse problem using the Crank-Nicolson scheme combined with the iteration method is given. Finally, in Section 5, numerical experiments are presented and discussed.
2 Existence and uniqueness of the solution of the inverse problem
We have the following assumptions on the data of the problem (1)-(4).
(A1) , , , ;
- (1), , , ;
- (2), ;
- (1), , ;
- (2), ;
- (3), ;
where , , .
The inverse problem (1)-(4) has a solution in ;
The solution of the inverse problem (1)-(4) is unique in , where the number () is determined by the data of the problem.
which implies that . By substituting in (9), we have . □
3 Continuous dependence of upon the data
Theorem 3 Under assumptions (A1)-(A3), the solution of the problem (1)-(4) depends continuously upon the data for small T.
The proof of the theorem is verified by analogy to .
4 Numerical method
We use the finite difference method with a predictor-corrector type approach that is suggested in . Apply this method to the problem (1)-(4).
where and are the indices for the spatial and time steps respectively, , , , , . At the level, adjustment should be made according to the initial condition and the compatibility requirements.
where , , .
The system of equations (17)-(19) can be solved by the Gauss elimination method, and is determined. If the difference of values between two iterations reaches the prescribed tolerance, the iteration is stopped, and we accept the corresponding values , () as , (), on the th time step, respectively. By virtue of this iteration, we can move from level j to level .
5 Numerical example and discussions
In this section, we present examples to illustrate the efficiency of the numerical method described in the previous section.
Let us apply the scheme which was explained in the previous section for the step sizes , .
The inverse problem regarding the simultaneous identification of the time-dependent coefficient of heat capacity together with the temperature distribution in a one-dimensional heat equation with periodic boundary and integral overdetermination conditions has been considered. This inverse problem has been investigated from both theoretical and numerical points of view. In the theoretical part of the article, the conditions for the existence, uniqueness and continuous dependence upon the data of the problem have been established. In the numerical part, a numerical example using the Crank-Nicolson finite-difference scheme combined with the iteration method is presented.
Dedicated to Professor Hari M Srivastava.
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