Inverse problem for a time-fractional parabolic equation
- Ebru Ozbilge^{1}Email author and
- Ali Demir^{2}
https://doi.org/10.1186/s13660-015-0602-y
© Ozbilge and Demir; licensee Springer. 2015
Received: 6 October 2014
Accepted: 18 February 2015
Published: 4 March 2015
Abstract
This article deals with the mathematical analysis of the inverse coefficient problem of identifying the unknown coefficient \(k(x)\) in the linear time-fractional parabolic equation \(D_{t}^{\alpha}u(x,t)=(k(x)u_{x})_{x}+qu_{x}(x,t)+p(t)u(x,t)\), \(0<\alpha\leq1\), with mixed boundary conditions \(k(0)u_{x}(0,t)=\psi_{0}(t)\), \(u(1,t)=\psi_{1}(t)\). By defining the input-output mappings \(\Phi[\cdot]:\mathcal {K}\rightarrow C[0,T]\) and \(\Psi[\cdot]:\mathcal{K}\rightarrow C^{1}[0,T]\) 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 input-output mappings \(\Phi[\cdot]\) and \(\Psi[\cdot]\). This work shows that the input-output mappings \(\Phi[\cdot]\) and \(\Psi [\cdot]\) have distinguishability property. Moreover, the value \(k(1)\) of the unknown diffusion coefficient \(k(x)\) at \(x=1\) can be determined explicitly by making use of measured output data (boundary observation) \(k(1)u_{x}(1,t)=h(t)\), which brings about a greater restriction on the set of admissible coefficients. It is also shown that the measured output data \(f(t)\) and \(h(t)\) can be determined analytically by a series representation. Hence the input-output mappings \(\Phi [\cdot]: \mathcal{K}\rightarrow C[0,T]\) and \(\Psi[\cdot]:\mathcal{K}\rightarrow C^{1}[0,T]\) can be described explicitly, where \(\Phi[k]=u(x,t;k)|_{x=0}\) and \(\Psi[k]=k(x)u_{x}(x,t;k)|_{x=1}\).
Keywords
1 Introduction
The inverse problem of determining unknown coefficient in a linear parabolic equation by using over-measured data has generated increasing interest from engineers and scientists during the last few decades. This kind of problem plays a crucial role in engineering, physics, and applied mathematics. The problem of recovering unknown coefficient(s) in the mathematical model of a physical phenomenon is frequently encountered. Intensive study has been carried out on this kind of problem, and various numerical methods were developed in order to overcome the problem of determining unknown coefficients [1–27]. The inverse problem of an unknown coefficient in a quasi-linear parabolic equation has been studied by Demir and Ozbilge [1, 2]. Moreover, the existence and uniqueness of solutions for fractional differential equations with nonlocal and integral boundary conditions have been studied by Ashyralyev and Sharifov. Also, finite difference methods for fractional parabolic and hyperbolic differential equations with various conditions have been studied by Ashyralyev et al. [3–8]. Second order implicit finite difference schemes have been applied to the right-hand side of the identification problem by Erdogan and Ashyralyev [9].
Fractional differential equations are generalizations of ordinary and partial differential equations to an arbitrary fractional order. By linear time-fractional parabolic equation, we mean certain parabolic-like 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 unknown coefficient \(k(x)\) in a one-dimensional time fractional parabolic equation. We first obtain the unique solution of this problem using Fourier method of separation of variables with respect to the eigenfunctions of a corresponding Sturm-Liouville eigenvalue problem under certain conditions [12]. As the next step, the noisy free measured output data are used to introduce the input-output mappings \(\Phi[\cdot]:\mathcal{K}\rightarrow C[0,T]\) and \(\Psi[\cdot]:\mathcal {K}\rightarrow C^{1}[0,T]\). Finally we investigate the distinguishability of the unknown coefficient via the above input-output mappings \(\Phi[\cdot]\) and \(\Psi[\cdot]\).
- (C1)
\(k(x)\in C^{1}[0,1]\).
- (C2)
\(g(x)\in C^{2}[0,1]\), \(g^{\prime}(0)=\frac{\psi _{0}(0)}{k(0)}\), \(g(1)=\psi_{1}(0)\).
- (C3)
\(p(t)\in C[0,1]\).
We say that the mappings \(\Phi[\cdot]:\mathcal{K}\rightarrow C[0,T]\) and \(\Psi[\cdot]:\mathcal{K}\rightarrow C^{1}[0,T]\) have the distinguishability property if \(\Phi[k_{1}]\neq\Phi[k_{2}]\) implies \(k_{1}(x)\neq k_{2}(x)\) and the same holds for \(\Psi[\cdot]\). This, in particular, means injectivity of the inverse mappings \(\Phi ^{-1}\) and \(\Psi^{-1}\). In this paper, measured output data of Neumann type at the boundary \(x=1\) 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 \(f(t)\) at the boundary \(x=0\) is given. An analysis of the inverse problem with the single measured output data \(h(t)\) at the boundary \(x=1\) 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 \(f(t)\)
Lemma 1
Proof
The lemma and the definitions of \(w_{n}(t)\) and \(z_{n}(t)\) given above enable us to reach the following conclusion:
Corollary 1
Note that \(\langle \xi^{1}(x,t)-\xi^{2}(x,t),\phi_{n}(x)\rangle \neq0\) for some \(n \in N\) implies that \(k_{1}(x)\neq k_{2}(x)\). Hence by Lemma 1 we conclude that \(k_{1}(x)\neq k_{2}(x)\). Moreover, it leads us to the following important consequence that the input-output mapping \(\Phi[k]\) is distinguishable, i.e. \(k_{1}(x) \neq k_{2}(x)\) implies \(\Phi [k_{1}]\neq\Phi[k_{2}]\).
Theorem 1
3 An analysis of the inverse problem with given measured data \(h(t)\)
Lemma 2
Proof
The lemma and the definitions given above enable us to reach the following conclusion:
Corollary 2
Note that \(\langle \xi^{1}(x,t)-\xi^{2}(x,t),\phi_{n}(x)\rangle \neq0\) for some \(n \in N\) implies \(k_{1}(x)\neq k_{2}(x)\). Hence by Lemma 2 we conclude that \(k_{1}(x) \neq k_{2}(x)\). Moreover, it leads us to the important consequence that the input-output mapping \(\Psi[k]\) is distinguishable i.e. \(k_{l}(x) \neq k_{2}(x)\) implies \(\Psi[k_{1}] \neq\Psi[k_{2}]\).
Theorem 2
4 Conclusion
The aim of this study was to investigate the distinguishability properties of the input-output mappings \(\Phi [\cdot]:\mathcal{K}_{0}\rightarrow C[0,T]\) and \(\Psi[\cdot]:\mathcal {K}_{1}\rightarrow C^{1}[0,T]\), which are determined by the measured output data at \(x=0\) and \(x=1\), respectively. In this study, we conclude that the distinguishability of the input-output mappings \(\Phi[\cdot]\) and \(\Psi [\cdot]\) hold which implies the injectivity of the inverse mappings \(\Phi^{-1}\) and \(\Psi^{-1}\). This provides the insight that compared to the Dirichlet type, the Neumann type of measured output data is more effective for the inverse problems of determining unknown coefficients. Moreover, the measured output data \(f(t)\) and \(h(t)\) are obtained analytically by a series representation, which leads to the explicit form of the input-output mappings \(\Phi[\cdot]\) and \(\Psi[\cdot]\). We also show that the value of the unknown coefficient \(k(x)\) at \(x=1\) is determined by using the Neumann type of measured output data at \(x=1\) which brings about more restrictions on the set of admissible coefficients. However, \(k(0)\) is not obtained by Dirichlet type of measured output data at \(x=0\). This provides the insight that Neumann type of measured output data is more effective than Dirichlet type for the inverse problem of determining an unknown coefficient. This work advances our understanding of the use of the Fourier method of separation of variables and the input-output mapping in the investigation of inverse problems for fractional parabolic equations. The authors plan to consider various fractional inverse problems in future studies, since the method discussed has a wide range of applications.
Declarations
Acknowledgements
We would like to thank to the referees for their valuable comments and corrections. The research was supported partly by the Scientific and Technical Research Council (TUBITAK) of Turkey and Izmir University of Economics.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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