Abstract
This article presents a semigroup approach for the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient in the linear parabolic equation with mixed boundary conditions , . The aim of this paper is to investigate the distinguishability of the inputoutput mappings , via semigroup theory. In this paper, we show that if the null space of the semigroup consists of only zero function, then the inputoutput mappings and have the distinguishability property. It is shown that the types of the boundary conditions and the region on which the problem is defined have a significant impact on the distinguishability property of these mappings. Moreover, in the light of measured output data (boundary observations) or/and , the values and of the unknown diffusion coefficient at and , respectively, can be determined explicitly. In addition to these, the values and of the unknown coefficient at and , respectively, are also determined via the input data. Furthermore, it is shown that measured output data and can be determined analytically by an integral representation. Hence the inputoutput mappings , are given explicitly in terms of the semigroup.
1 Introduction
Consider the following initial boundary value problem:
where . The left flux and the right boundary condition are assumed to be constants. The functions and satisfy the following conditions:
Under these conditions, the initial boundary value problem (1) has the unique solution [14].
Consider the inverse problem of determining the unknown coefficient [59] from the following observations at the boundaries and :
Here is the solution of the parabolic problem (1). The functions , are assumed to be 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 . It is assumed that the functions and belong to and satisfy the consistency conditions , .
We denote by , the set of admissible coefficients and introduce the inputoutput mappings , , where
Then the inverse problem [10] with the measured data and can be formulated as the following operator equations:
We denote by , the set of admissible coefficients . The monotonicity, continuity and hence invertibility of the inputoutput mappings and are given in [3,4].
The aim of this paper is to study a distinguishability of the unknown coefficient via the above inputoutput mappings. We say that the mapping (or ) has the distinguishability property if () implies . This, in particular, means injectivity of the inverse mappings and .
The purpose of this paper is to study the distinguishability of the unknown coefficient via the above inputoutput mappings. The results presented here are the first ones, to the knowledge of authors, from the point of view of semigroup approach [11] to inverse problems. This approach sheds more light on the identifiability of the unknown coefficient [12] and shows how much information can be extracted from the measured output data, in particular in the case of constant flux and boundary data [1215].
The paper is organized as follows. In Section 2, the analysis of the semigroup approach is given for the inverse problem with the measured data . A similar analysis is applied to the inverse problem with the single measured output data given at the point in Section 3. The inverse problem with two Neumann measured data and is discussed in Section 4. Finally, some concluding remarks are given in Section 5.
2 Analysis of the inverse problem with measured output data
Consider now the inverse problem with one measured output data at . In order to formulate the solution of the parabolic problem (1) in terms of a semigroup, let us first arrange the parabolic equation as follows:
Then the initial boundary value problem (1) can be rewritten in the following form:
Here we assume that was known. Later we will determine the value . In order to formulate the solution of the parabolic problem (5) in terms of a semigroup, we need to define the following function:
which satisfies the following parabolic problem:
Here is a secondorder differential operator, its domain is . Since the initial value function belongs to , it is obvious that .
Denote by the semigroup of linear operators generated by the operator −A[5,6]. Note that we can easily find the eigenvalues and eigenfunctions of the differential operator A. Furthermore, the semigroup can be easily constructed by using the eigenvalues and eigenfunctions of a differential operator A. For this reason, we first consider the following eigenvalue problem:
This problem is called a SturmLiouville problem. We can easily determine that the eigenvalues are for all and the corresponding eigenfunctions are . In this case, the semigroup can be represented in the following way:
where . The null space of the semigroup of the linear operators can be defined as follows:
From the definition of the semigroup , we can say that the null space of it is an empty set, i.e., . This result is very important for the uniqueness of the unknown coefficient .
The unique solution of the initial value problem (7) in terms of a semigroup can be represented in the following form:
Hence, by using identity (6), the solution of the parabolic problem (5) in terms of a semigroup can be written in the following form:
In order to arrange the above solution representation, let us define the following:
Then we can rewrite the solution representation in terms of and in the following form:
Substituting into this solution representation yields
Taking into account the overmeasured data , we get
which implies that can be determined analytically.
Differentiating both sides of the above identity with respect to x and using semigroup properties at yield
Using the boundary condition , we can write for all which can be rewritten in terms of a semigroup in the following form:
Taking limit as in the above identity, we obtain the following explicit formula for the value of the unknown coefficient :
The righthand side of identity (11) defines explicitly the semigroup representation of the inputoutput mapping on the set of admissible unknown diffusion coefficients :
Let us differentiate now both sides of identity (8) with respect to t:
Using the semigroup property , we obtain
Taking in the above identity, we get
Since , we have . Taking into account this and substituting yield
Solving this equation for and substituting , we obtain the following explicit formula for the value of the first derivative of the unknown coefficient at :
Under the determined values and , the set of admissible coefficients can be defined as follows:
The following lemma implies the relationship between the diffusion coefficients at and the corresponding outputs , .
Lemma 2.1Letandbe solutions of the direct problem (5) corresponding to the admissible coefficients. Suppose that, , are the corresponding outputs and denote by, . If the condition
holds, then the outputs, , satisfy the following integral identity:
Proof The solutions of the direct problem (5) corresponding to the admissible coefficients can be written at as follows:
respectively, by using representation (11). From identity (9) it is obvious that for each . Hence the difference of these formulas implies the desired result. □
This lemma with identity (14) implies the following.
Corollary 2.1Let conditions of Lemma 2.1 hold. Then, , if and only if
Since the StrumLiouville problem generates a complete orthogonal family of eigenfunctions, the null space of a semigroup contains only zero function, i.e., . Thus Corollary 2.1 states that if and only if for all . The definition of implies that for all .
The combination of the conclusions of Lemma 2.1 and Corollary 2.1 can be given by the following theorem which states the distinguishability of the inputoutput mapping .
Theorem 2.1Let conditions (C1) and (C2) hold. Assume thatis the inputoutput mapping defined by (3) and corresponding to the measured output. Then the mappinghas the distinguishability property in the class of admissible coefficients, i.e.,
3 Analysis of the inverse problem with measured output data
Consider now the inverse problem with one measured output data at . As in the previous section, let us arrange the parabolic equation as follows:
Then the initial boundary value problem (1) can be rewritten in the following form:
In order to formulate the solution of the above parabolic problem in terms of a semigroup, let us use the same variable in identity (6), which satisfies the following parabolic problem:
Here is a secondorder differential operator, its domain is . Since the initial value function belongs to , it is obvious that .
Denote by the semigroup of linear operators generated by the operator −A[5,6]. As in the previous section, we can easily find the eigenvalues and eigenfunctions of the differential operator B. Furthermore, the semigroup can be easily constructed by using the eigenvalues and eigenfunctions of the differential operator B. For this reason, we first consider the following eigenvalue problem:
This problem is called a SturmLiouville problem. We can easily determine that the eigenvalues are for all and the corresponding eigenfunctions become . Hence the semigroup can be represented in the following form:
The null space of the semigroup of the linear operators can be defined as follows:
Since the SturmLiouville problem generates a complete orthogonal family of eigenfunctions, we can say that the null space of the semigroup is an empty set, i.e., . This result is very important for the uniqueness of the unknown coefficient .
The unique solution of the initial value problem (16) in terms of a semigroup can be represented in the following form:
Hence, by using identity (6), the solution of the parabolic problem (15) in terms of a semigroup can be written in the following form:
Defining the following:
The solution representation of the parabolic problem (17) can be rewritten in the following form:
Differentiating both sides of the above identity with respect to x and substituting yield
Taking into account the overmeasured data , we get
Now we can determine the value . From the overmeasured data , the identity for all can be rewritten in terms of a semigroup in the following form:
Taking limit as in the above identity yields
The righthand side of the above identity defines the semigroup representation of the inputoutput mapping on the set of admissible unknown diffusion coefficient :
Differentiating both sides of identity (17) with respect to t, we get
Using semigroup properties, we obtain
Taking in the above identity, we get
Since , we have . Taking into account this and substituting , we get
Solving this equation for and substituting , we reach the following result:
Then we can define the admissible set of diffusion coefficients as follows:
The following lemma implies the relation between the coefficients at and the corresponding outputs , .
Lemma 3.1Letandbe solutions of the direct problem (16) corresponding to the admissible coefficients. Suppose that, , are the corresponding outputs and denote by, . If the condition
holds, then the outputs, , satisfy the following integral identity:
Proof The solutions of the direct problem (15) corresponding to the admissible coefficients can be written at as follows:
respectively, by using formula (20). From definition (18), it is obvious that for each . Hence the difference of these formulas implies the desired result. □
This lemma with identity (23) implies the following conclusion.
Corollary 3.1Let the conditions of Lemma 3.1 hold. Then, , if and only if
hold.
Since the null space of it consists of only zero function, i.e., , Corollary 3.1 states that if and only if for all . The definition of implies that for all .
Theorem 3.1Let conditions (C1) and (C2) hold. Assume thatis the inputoutput mapping defined by (3) and corresponding to the measured output. Then the mappinghas the distinguishability property in the class of admissible coefficients, i.e.,
4 The inverse problem with mixed output data
Consider now the inverse problem (1)(2) with two measured output data and . As shown before, having these two data, the values as well as can be defined by the above explicit formulas. Based on this result, let us define now the set of admissible coefficients as an intersection:
On this set, both inputoutput mappings and have distinguishability property.
Corollary 4.1The inputoutput mappingsanddistinguish any two functionsfrom the set, i.e.,
5 Conclusion
The aim of this study was to analyze distinguishability properties of the inputoutput mappings and which are naturally determined by the measured output data. In this paper we show that if the null spaces of the semigroups and include only zero function then the corresponding inputoutput mappings and have distinguishability property.
This study shows that boundary conditions and the region on which the problem is defined have a significant impact on the distinguishability of the inputoutput mappings and since these key elements determine the structure of the semigroups and of linear operators and their null spaces.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Acknowledgements
Dedicated to Professor Hari M Srivastava.
The research was supported in part by the Scientific and Technical Research Council (TUBITAK) and Izmir University of Economics.
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