Abstract
This article deals with the mathematical analysis of the inverse coefficient problem of identifying the unknown coefficient in the linear time fractional parabolic equation , , with mixed boundary conditions , . By defining the inputoutput mappings and , the inverse problem is reduced to the problem of their invertibility. Hence the main purpose of this study is to investigate the distinguishability of the inputoutput mappings and . This work shows that the inputoutput mappings and have the distinguishability property. Moreover, the value of the unknown diffusion coefficient at can be determined explicitly by making use of measured output data (boundary observation) , which brings greater restriction on the set of admissible coefficients. It is also shown that the measured output data and can be determined analytically by a series representation, which implies that the inputoutput mappings and can be described explicitly.
1 Introduction
The inverse problem of determining an unknown coefficient in a linear parabolic equation by using overmeasured data has generated an increasing amount of interest from engineers and scientist during the last few decades. This kind of problem plays a crucial role in engineering, physics and applied mathematics. The problem of recovering an unknown coefficient or coefficients in the mathematical model of physical phenomena is frequently encountered. Intensive study has been carried out on this kind of problem, and various numerical methods have been developed in order to overcome the problem of determining an unknown coefficient or coefficients [19]. The inverse problem of unknown coefficients in a quasilinear parabolic equations was studied by Demir and Ozbilge [5,6]. Moreover, the identification of the unknown diffusion coefficient in a linear parabolic equation was studied by Demir and Hasanov [7].
Fractional differential equations are generalizations of ordinary and partial differential equations to an arbitrary fractional order. By linear timefractional parabolic equation, we mean a certain paraboliclike partial differential equation governed by master equations containing fractional derivatives in time [10,11]. The research areas of fractional differential equations range from theoretical to applied aspects. The main goal of this study is to investigate the inverse problem of determining an unknown coefficient in a onedimensional time fractional parabolic equation. We first obtain the unique solution of this problem using the Fourier method of separation of variables with respect to the eigenfunctions of the corresponding SturmLiouville eigenvalue problem under certain conditions [12]. As the next step, the noisefree measured output data are used to introduce the inputoutput mappings and . Finally, we investigate the distinguishability of the unknown coefficient via the above inputoutput mappings and .
Consider now the following initial boundary value problem:
where and the fractional derivative is defined in the CaputoDzherbashyan sense , , being the RiemannLiouville fractional integral
The left and right boundary value functions and belong to . The functions and satisfy the following conditions:
Under these conditions, the initial boundary value problem (1) has the unique solution defined in the domain which belongs to the space . Moreover, it satisfies the equation, initial and boundary conditions. The space contains the functions such that .
This kind of problem plays a crucial role in engineering, physics and applied mathematics since it is used successfully to model complex phenomena in various fields such as fluid mechanics, viscoelasticity, physics, chemistry and engineering. The problem of recovering an unknown coefficient or coefficients in the mathematical model of physical phenomena is frequently encountered.
Consider the inverse problem of determining the unknown coefficient from the Neumanntype measured output data at the boundary :
and the Dirichlettype measured output data at the boundary :
Here is the solution of parabolic problem (1). The functions and are assumed to be noisefree measured output data. In this context, 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 and .
By denoting , the set of admissible coefficients , let us introduce the inputoutput mappings and , where
and
Then the inverse problem with the measured output data and can be formulated as follows:
which reduces the inverse problem of determining the unknown coefficient to the problem of invertibility of the inputoutput mappings and . Hence this leads us to investigate the distinguishability of the unknown coefficient via the above inputoutput mappings. We say that the mappings and have the distinguishability property if implies and the same holds for . This, in particular, means the injectivity of inverse mappings and . In this paper, measured output data of Neumann type at the boundary and measured output data of Dirichlet type at the boundary are used in the identification of the unknown coefficient. In addition, in the determination of the unknown parameter, analytical results are obtained.
The paper is organized as follows. In Section 2, an analysis of the inverse problem with the single measured output data at the boundary is given. An analysis of the inverse problem with the single measured output data at the boundary is considered in Section 3. Finally, some concluding remarks are given in the last section.
2 An analysis of the inverse problem with given measured data
Consider now the inverse problem with one measured output data at . In order to formulate the solution of parabolic problem (1) by using the Fourier method of the separation of variables, let us first introduce an auxiliary function as follows:
by which we transform problem (1) into a problem with homogeneous boundary conditions. Hence the initial boundary value problem (1) can be rewritten in terms of in the following form:
The unique solution of the initialboundary value problem can be represented in the following form [12]:
where
Moreover, , being the generalized MittagLeffler function defined by
Assume that is the solution of the following SturmLiouville problem:
The Neumanntype measured output data at the boundary in terms of can be written in the following form:
In order to arrange the above solution, let us define the following:
The solution in terms of and can then be rewritten in the following form:
Differentiating both sides of the above identity with respect to x and substituting yields
Taking into account the overmeasured data ,
is obtained, which implies that can be determined analytically. Substituting into this yields
Hence we obtain the following explicit formula for the value of the unknown coefficient
Under the determined value , the set of admissible coefficients can be defined as follows:
The righthand side of identity (4) defines the inputoutput mapping on the set of admissible source functions
The following lemma implies the relation between the parameters at and the corresponding outputs , .
Lemma 1Letandbe the solutions of direct problem (2), corresponding to the admissible parameters. If, , are the corresponding outputs. If the condition, then the outputs, , satisfy the following integral identity:
Proof By using identity (4), the measured output data , , can be written as follows:
respectively. Note that the definition of implies that . Hence, the difference of these formulas implies the desired result. □
The lemma and the definitions of and given above enable us to reach the following conclusion.
Corollary 1Let the conditions of Lemma 1 hold. If in addition
holds, where
Proof If , , then . If , then . Since depends on , then from the uniqueness of solution .
Since , form a basis for the space and , , then implies that at least for some . Hence by Lemma 1 we conclude that , which leads us to the following consequence: implies that . □
Theorem 1Let conditions (C1), (C2) hold. Assume thatis the inputoutput mapping defined by (4) and corresponding to the measured output. In this case the mappinghas the distinguishability property in the class of admissible parameters, i.e.,
3 An analysis of the inverse problem with given measured data
Consider now the inverse problem with one measured output data at . Taking into account the overmeasured data ,
is obtained, which implies that can be determined analytically.
The set of admissible coefficients can be defined as follows:
The righthand side of identity (5) defines the inputoutput mapping on the set of admissible parameters :
The following lemma implies the relation between the parameters at and the corresponding outputs , .
Lemma 2Letandbe the solutions of direct problem (2), corresponding to the admissible parameters. If, , are the corresponding outputs. The outputs, , satisfy the following integral identity:
Proof By using identity (5), the measured output data , , can be written as follows:
respectively. Note that the definition of implies that . Hence, the difference of these formulas implies the desired result. □
The lemma and the definitions given above enable us to reach the following conclusion.
Corollary 2Let the conditions of Lemma 2 hold. If, in addition,
holds, where
Proof If , , then . If , then . Since depends on , then from the uniqueness of solution .
Since , form a basis for the space and , , then implies that at least for some . Hence by Lemma 2 we conclude that , which leads us to the following consequence: implies that . □
Theorem 2Let conditions (C1), (C2) hold. Assume thatis the inputoutput mapping defined by (6) and corresponding to the measured output. In this case the mappinghas the distinguishability property in the class of admissible parameters, i.e.,
4 Conclusion
The aim of this study was to investigate the distinguishability properties of the inputoutput mappings and , which are determined by the measured output data at and , respectively. In this study, we conclude that the distinguishability of the inputoutput mappings and holds, which implies the injectivity of the inverse mappings and . This provides the insight that compared to the Dirichlet type, the Neumanntype measured output data is more effective for the inverse problems of determining unknown coefficients. Moreover, the measured output data and are obtained analytically by a series representation, which leads to the explicit form of the inputoutput mappings and . We also show that the value of the unknown coefficient at is determined by using the Neumanntype measured output data at , which brings more restrictions on the set of admissible coefficients. However, is not obtained by the Dirichlettype measured output data at . This provides the insight that the Neumanntype measured output data is more effective than that of Dirichlet type for the inverse problems of determining an unknown coefficient. This work advances our understanding of the use of the Fourier method of separation of variables and the inputoutput mapping in the investigation of inverse problems for fractional parabolic equations. The author plans to consider various fractional inverse problems in future studies, since the method discussed has a wide range of applications.
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
The research was supported in part by the Scientific and Technical Research Council (TUBITAK) of Turkey and Izmir University of Economics.
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