Identification of the unknown coefficient in a quasi-linear parabolic equation by a semigroup approach
© Ozbilge and Demir; licensee Springer 2013
Received: 26 November 2012
Accepted: 13 April 2013
Published: 26 April 2013
This article presents a semigroup approach to the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient in the quasi-linear parabolic equation with Dirichlet boundary conditions , . It is shown that the unknown coefficient can be approximately determined via the semigroup approach.
where . The left and right boundary values , are assumed to be constants. The functions and satisfy the following conditions:
(C2) , , .
Here represents the solution of the parabolic problem (1). It is assumed that the function is noisy free measured output data. In this context the parabolic problem (1) will be referred to as a direct (forward) problem with the inputs and . The function is assumed to belong to and satisfy the consistency conditions . Moreover, it is assumed that the function satisfies the consistency conditions , .
In inverse problems, mostly numerical methods are employed to determine the unknown function. Even using one or more measured output data in analytic methods does not help in determination of the unknown function exactly [4, 5].
The purpose of this paper is to determine the analytic representation of the approximate unknown coefficient via a semigroup approach. We determine the approximate solution of the boundary value problem (1) without determining the unknown coefficient . In the determination of it, we just use the values , which are obtained and the over measured data . However, this cannot be possible for any kind of a boundary value problem with different type of boundary conditions. The semigroup approach for inverse problems for the identification of unknown coefficient in a quasi-linear parabolic equations was studied by Demir and Ozbilge [6, 7]. Moreover, the identification of the unknown diffusion coefficient in a linear parabolic equation was studied by Demir and Hasanov . The study in this paper is based on similar philosophy to that in [6–9].
Recent studies [6–9] on the identification of the unknown function via a semigroup approach  show that the semigroup approach is the most effective analytical method which sheds more light on the unknown function. The weak point of this method is that you cannot apply this method to the initial boundary value problems with boundary conditions other than constant or zero. Overcoming this problem in some way makes this method more effective and valuable.
The paper is organized as follows. In Section 2, determination of the unknown coefficient is given via the semigroup approach. Some concluding remarks are given in Section 3.
2 Determination of the unknown coefficient via the semigroup approach
We can represent the solution of this problem in terms of a semigroup of linear operators. Here, is a second-order differential operator and its domain is , where and are Sobolev spaces. Obviously, by completion since the initial value function belongs to . Hence is dense in , which is a necessary condition for being an infinitesimal generator.
which leads us to the determination of the unknown coefficient analytically.
for each .
does not hold. □
This lemma leads us to the solution of the parabolic problem (1) without knowing the unknown coefficient , but knowing and . Then the solution enables us to determine the unknown coefficient approximately.
Substituting the integration constant into the identity (15), we obtain the approximate solution of the parabolic problem (1). Hence, by using this solution and the parabolic equation, we are able to determine the unknown coefficient approximately.
The aim of this study was to determine the unknown coefficient via the semigroup approach. Lemma in Section 2 plays a crucial role in the identification of the unknown coefficient . This lemma leads us to new conclusions about the identification problems via the semigroup approach. We will present these results in our future studies….
Dedicated to Professor Hari M Srivastava.
The research was supported by parts by the Scientific and Technical Research Council (TUBITAK) of Turkey and Izmir University of Economics.
- Showalter R: Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations. Am. Math. Soc., Providence; 1997.Google Scholar
- Hasanov A, Demir A, Erdem A: Monotonicity of input-output mappings in inverse coefficient and source problem for parabolic equations. J. Math. Anal. Appl. 2007, 335: 1434–1451. 10.1016/j.jmaa.2007.01.097MathSciNetView ArticleGoogle Scholar
- Hasanov A, DuChateau P, Pektas B: An adjoint approach and coarse-fine mesh method for identification of the diffusion coefficient in a linear parabolic equations. J. Inverse Ill-Posed Probl. 2006, 14: 435–463. 10.1515/156939406778247615MathSciNetView ArticleGoogle Scholar
- DuChateau P, Gottlieb J: Introduction to Inverse Problems in Partial Differential Equations for Engineers, Physicists and Mathematicians. Kluwer Academic, Amsterdam; 1996.Google Scholar
- Renardy M, Rogers R: An Introduction to Partial Differential Equations. Springer, New York; 2004.Google Scholar
- Demir A, Ozbilge E: Analysis of a semigroup approach in the inverse problem of identifying an unknown coefficient. Math. Methods Appl. Sci. 2008, 31: 1635–1645. 10.1002/mma.989MathSciNetView ArticleGoogle Scholar
- Demir A, Ozbilge E: Semigroup approach for identification of the unknown diffusion coefficient in a quasi-linear parabolic equation. Math. Methods Appl. Sci. 2007, 30: 1283–1294. 10.1002/mma.837MathSciNetView ArticleGoogle Scholar
- Demir A, Hasanov A: Identification of the unknown diffusion coefficient in a linear parabolic equation by the semigroup approach. J. Math. Anal. Appl. 2008, 340: 5–15. 10.1016/j.jmaa.2007.08.004MathSciNetView ArticleGoogle Scholar
- Ozbilge E: Identification of the unknown diffusion coefficient in a quasi-linear parabolic equation by semigroup approach with mixed boundary conditions. Math. Methods Appl. Sci. 2008, 31: 1333–1344. 10.1002/mma.974MathSciNetView ArticleGoogle Scholar
- Ashyralyev, A, San, ME: An approximation of semigroup method for stochastic parabolic equations. Abstr. Appl. Anal. (in press)Google Scholar
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