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Inverse Coefficient Problem of the Parabolic Equation with Periodic Boundary and Integral Overdetermination Conditions
This paper investigates the inverse problem of finding a time-dependent diffusion coefficient in a parabolic equation with the periodic boundary and integral overdetermination conditions. Under some assumption on the data, the existence, uniqueness, and continuous dependence on the data of the solution are shown by using the generalized Fourier method. The accuracy and computational efficiency of the proposed method are verified with the help of the numerical examples.
Denote the domain by
The parameter identification in a parabolic differential equation from the data of integral overdetermination condition plays an important role in engineering and physics [1–7]. This integral condition in parabolic problems is also called heat moments .
Boundary value problems for parabolic equations in one or two local classical conditions are replaced by heat moments [8–13]. These kinds of conditions such as (5) arise from many important applications in heat transfer, thermoelasticity, control theory, life sciences, and so forth. For example, in heat propagation in a thin rod, the law of variation of the total quantity of heat in the rod is given in . In , a physical-mechanical interpretation of the integral conditions was also given.
Various statements of inverse problems on determination of thermal coefficient in one-dimensional heat equation were studied in [4, 5, 7, 14]. In papers [4, 5, 7], the time-dependent thermal coefficient is determined from the heat moment.
In the present work, one heat moment is used with periodic boundary condition for the determination of thermal coefficient. The existence and uniqueness of the classical solution of the problem (2)–(5) is reduced to fixed point principles by applying the Fourier method.
This paper organized as follows. In Section 2, the existence and uniqueness of the solution of inverse problem (2)–(5) are proved by using the Fourier method. In Section 3, the continuous dependence on the solution 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 an 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
where , , .
Theorem 2. Let the assumptions ()–() be satisfied. Then the following statements are true. (1) The inverse problem (2)–(5) has a solution in . (2) The solution of inverse problem (2)–(5) is unique in , where the number () is determined by the data of the problem.
Proof. By applying the standard procedure of the Fourier method, we obtain the following representation for the solution of (2)–(4) for arbitrary :
The assumptions , , , and are consistent conditions for the representation (2) of the solution to be valid. Furthermore, under the smoothness assumptions , , and for all , the series (6) and its -partial derivative converge uniformly in since their majorizing sums are absolutely convergent. Therefore, their sums and are continuous in . In addition, the -partial derivative and the -second-order partial derivative series are uniformly convergent for ( is an arbitrary positive number). Thus, and satisfies the conditions (2)–(4). In addition, is continuous in because the majorizing sum of -partial derivative series is absolutely convergent under the condition and in . Equation (6) can be differentiated under the condition () to obtain and this yields where Denote Using the representation (8), the following estimate is true: Introduce the set as It is easy to see that Compactness of is verified by analogy to . By virtue of Schauder’s fixed-point theorem, we have a solution of (8).
Now let us show that there exists () for which the solution of the problem (2)–(5) is unique in . Suppose that is also a solution pair of the problem (2)–(5). Then from the representations (6) and (8) of the solution, we have where The following estimate is true: Using the estimates we obtain Let be arbitrary fixed number. Fix a number , , such that Then from the equality (10), we obtain which implies that . By substituting in (9), we have .
3. Continuous Dependence of on the Data
Proof. Let and be two sets of the data, which satisfy the assumptions ()–(). Then there exist positive constants , such that
Let and be solutions of the inverse problem (2)–(5) corresponding to the data and , respectively. According to (8), First let us estimate the difference . It is easy to compute that where , , are some constants.
If we consider these estimates in , we obtain The inequality holds for small . Finally, we obtain where .
From (6), a similar estimate is also obtained for the difference as
4. Numerical Method
We subdivide the intervals and into and subintervals of equal lengths and , respectively. Then we add two lines and to generate the fictitious points needed for dealing with the second boundary condition. We choose the Crank-Nicolson scheme, which is absolutely stable and has a second-order accuracy in both and . The Crank-Nicolson scheme for (2)–(5) is as follows: where and are the indices for the spatial and time steps, respectively, , , , and , . At the level, adjustment should be made according to the initial condition and the compatibility requirements.
Equation (27) form an linear system of equations where
Now, let us construct the predicting-correcting mechanism. First, multiplying (2) by from 0 to 1 and using (4) and (5), we obtain The finite difference approximation of (30) is where , , . For , and the values of help us to start our computation. We denote the values of , at the th iteration step , , respectively. In numerical computation, since the time step is very small, we can take , , , . At each th iteration step, we first determine from the formula
Then from (27) we obtain The system of (34) 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. In virtue of this iteration, we can move from level to level .
5. Numerical Examples and Discussions
Let us apply the scheme which was explained in the previous section for the step sizes , .
Example 2. Consider the inverse problem (2)–(5), with
It is easy to check that the analytical solution of the problem (2)–(5) is
Let us apply the scheme which was explained in the previous section for the step sizes , .
In the case when , the comparisons between the analytical solution (38) and the numerical finite difference solution are shown in Figures 3 and 4.
The inverse problem regarding the simultaneously identification of the time-dependent thermal diffusivity and the temperature distribution in 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 paper, the conditions for the existence, uniqueness, and continuous dependence on the data of the problem have been established. In the numerical part, the sensitivity of the Crank-Nicolson finite-difference scheme combined with an iteration method with the examples has been illustrated.
- J. R. Cannon, Y. P. Lin, and S. Wang, “Determination of a control parameter in a parabolic partial differential equation,” The Journal of the Australian Mathematical Society B, vol. 33, no. 2, pp. 149–163, 1991.
- J. R. Cannon, Y. Lin, and S. Wang, “Determination of source parameter in parabolic equations,” Meccanica, vol. 27, no. 2, pp. 85–94, 1992.
- A. G. Fatullayev, N. Gasilov, and I. Yusubov, “Simultaneous determination of unknown coefficients in a parabolic equation,” Applicable Analysis, vol. 87, no. 10-11, pp. 1167–1177, 2008.
- M. I. Ivanchov, “Inverse problems for the heat-conduction equation with nonlocal boundary condition,” Ukrainian Mathematical Journal, vol. 45, no. 8, pp. 1186–1192, 1993.
- M. I. Ivanchov and N. V. Pabyrivska, “Simultaneous determination of two coefficients in a parabolic equation in the case of nonlocal and integral conditions,” Ukrainian Mathematical Journal, vol. 53, no. 5, pp. 674–684, 2001.
- M. I. Ismailov and F. Kanca, “An inverse coefficient problem for a parabolic equation in the case of nonlocal boundary and overdetermination conditions,” Mathematical Methods in the Applied Sciences, vol. 34, no. 6, pp. 692–702, 2011.
- F. Kanca and M. I. Ismailov, “The inverse problem of finding the time-dependent diffusion coefficient of the heat equation from integral overdetermination data,” Inverse Problems in Science and Engineering, vol. 20, no. 4, pp. 463–476, 2012.
- J. R. Cannon, “The solution of the heat equation subject to the specification of energy,” Quarterly of Applied Mathematics, vol. 21, pp. 155–160, 1963.
- L. I. Kamynin, “A boundary value problem in the theory of heat conduction with a nonclassical boundary condition,” Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki, vol. 4, no. 6, pp. 1006–1024, 1964.
- N. I. Ionkin, “Solution of a boundary-value problem in heat conduction with a nonclassical boundary condition,” Differential Equations, vol. 13, pp. 204–211, 1977.
- N. I. Yurchuk, “Mixed problem with an integral condition for certain parabolic equations,” Differential Equations, vol. 22, pp. 1457–1463, 1986.
- V. M. Vīgak, “Construction of a solution of the heat conduction problem with integral conditions,” Doklady Akademii Nauk Ukrainy, vol. 8, pp. 57–60, 1994.
- N. I. Ivanchov, “Boundary value problems for a parabolic equation with integral conditions,” Differential Equations, vol. 40, no. 4, pp. 591–609, 2004.
- W. Liao, M. Dehghan, and A. Mohebbi, “Direct numerical method for an inverse problem of a parabolic partial differential equation,” Journal of Computational and Applied Mathematics, vol. 232, no. 2, pp. 351–360, 2009.
- I. Sakinc, “Numerical solution of a quasilinear parabolic problem with periodic boundary condition,” Hacettepe Journal of Mathematics and Statistics, vol. 39, no. 2, pp. 183–189, 2010.
- J. Choi, “Inverse problem for a parabolic equation with space-periodic boundary conditions by a Carleman estimate,” Journal of Inverse and Ill-Posed Problems, vol. 11, no. 2, pp. 111–135, 2003.
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