- Open Access
Identification of the unknown boundary condition in a linear parabolic equation
© Demir and Ozbilge; licensee Springer 2013
- Received: 27 December 2012
- Accepted: 18 February 2013
- Published: 7 March 2013
In this article, a semigroup approach is presented for the mathematical analysis of the inverse problems of identifying the unknown boundary condition in a linear parabolic equation with Dirichlet boundary conditions , by making use of the over measured data and separately.
- Integral Equation
- Inverse Problem
- Integral Representation
- Null Space
- Initial Boundary
where . The left boundary value is assumed to be constant. The functions and satisfy the following conditions:
(C2) , , .
In physics, many applications of this problem can be found. The simple model of flame propagation and the spread of a biological populations, where , denote the temperature and density respectively, are given by the equation in problem (1). Especially represents the density-dependent coefficient in the problems of the spread of biological populations [5–9].
Here the solution of the parabolic problem (1) is denoted by . In this context, the parabolic problem (1) will be referred to as a direct (forward) problem with the inputs , and . Notice that and . Therefore it is assumed that the functions and respectively satisfy the consistency conditions and .
The paper is organized as follows. In Section 2, an analysis of the semigroup approach is given for the inverse problem with the single measured output data given at . A similar analysis is applied to the inverse problem with the single measured output data given at the point in Section 3. Some concluding remarks are given in Section 4.
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.
In the following, despite doing the calculations in the smooth function space, by completion they are valid in the Sobolev space.
From the construction of the semigroup , it can be concluded that the null space of it consists of only zero function, i.e., . This result implies the uniqueness of the unknown boundary condition .
This is the integral representation of a solution of the initial-boundary value problem (5) on . It is obvious from the eigenfunctions , the domain of eigenfunctions can be extended to a closed interval . Moreover, they are continuous on . Under this extension, the uniqueness of the initial-boundary value problems (4) and (5) imply that the integral representation (11) becomes the integral representation of a solution of the initial-boundary value problem (4) on .
which implies that can be determined analytically. The right-hand side of identity (12) defines the semigroup representation of the unknown boundary condition at .
which is the initial condition we have.
Here is a second-order differential operator, its domain is . It is clear from the definition of that .
As in the previous section, the construction of the semigroup implies that the null space of it consists of only zero function, i.e., . The uniqueness of the unknown boundary condition follows from this result.
This is the integral representation of a solution of the initial-boundary value problem (15) on . It is obvious from the eigenfunctions , the domain of eigenfunctions can be extended to a closed interval . Moreover, they are continuous on . Under this extension, the uniqueness of the initial-boundary value problems (15) and (16) imply that integral representation (22) becomes the integral representation of a solution of the initial-boundary value problem (16) on .
which implies that can be determined analytically.
The right-hand side of identity (23) defines the semigroup representation of the unknown boundary condition at .
The purpose of this study is to identify the unknown boundary condition at via the semigroup approach by using the over measured data and . The crucial point here is the unique extensions of the solutions on to which are implied by the uniqueness of the solutions. This key leads to the integral representation of the unknown boundary condition at obtained analytically.
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.
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