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Numerical solution and distinguishability in time fractional parabolic equation
Boundary Value Problems volumeÂ 2015, ArticleÂ number:Â 142 (2015)
Abstract
This article deals with the mathematical analysis of the inverse problem of identifying the distinguishability of inputoutput mappings in the linear time fractional inhomogeneous parabolic equation \(D_{t}^{\alpha}u(x,t)=(k(x)u_{x})_{x}+r(t)F(x,t)\), \(0<\alpha\leq 1\), with mixed boundary conditions \(u(0,t)=\psi_{0}(t)\), \(u_{x}(1,t)=\psi_{1}(t)\). By defining the inputoutput mappings \(\Phi[\cdot]:\mathcal{K}\rightarrow C^{1}[0,T]\) and \(\Psi[\cdot]:\mathcal{K}\rightarrow C[0,T]\) 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 inputoutput mappings \(\Phi[\cdot]\) and \(\Psi[\cdot]\). Moreover, the measured output data \(f(t)\) and \(h(t)\) can be determined analytically by a series representation, which implies that the inputoutput mappings \(\Phi[\cdot]:\mathcal{K}\rightarrow C^{1}[0,T]\) and \(\Psi[\cdot]:\mathcal{K}\rightarrow C[0,T]\) can be described explicitly, where \(\Phi[r]=k(x)u_{x}(x,t;r)_{x=0}\) and \(\Psi [r]=u(x,t;r)_{x=1}\). Also, numerical tests using finite difference scheme combined with an iterative method are presented.
1 Introduction
The inverse problem of unknown source function in a linear inhomogeneous 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 problems play a crucial role in engineering, physics and applied mathematics. The problem of recovering an unknown source function in the mathematical model of a physical phenomenon is frequently encountered. Intensive study has been carried out on this kind of problem, and various inverse problems and many numerical methods developed. In papers [1â€“3] a coupled method for inverse source problem of spatial fractional anomalous diffusion equations and a boundarytype collocation method for inverse Cauchy inhomogeneous potential problems were considered. The inverse problem of an unknown coefficient in quasilinear parabolic equations was studied by Demir and Ozbilge [4, 5]. Moreover, the existence and uniqueness of solutions for fractional differential equations with nonlocal and integral boundary conditions were studied by Ashyralyev and Sharifov [6]. Initial boundaryvalue problems for the onedimensional time fractional diffusion equation were studied by Amanov and Ashyralyev in [7], the finite difference method for fractional parabolic equations with Neumann boundary conditions was studied by Ashyralyev and Cakir in [8], numerical solution of a fractional Schrodinger differential equation with the Dirichlet boundary condition was studied by Ashyralyev and Hicdurmaz in [9]. Moreover, Ashyralyev and Dal studied finite difference methods and iteration methods for fractional hyperbolic partial differential equations with the Neumann condition in [10]. Second order implicit finite difference schemes were applied to the righthand side of the identification problem by Erdogan and Ashyralyev [11].
Fractional differential equations are generalizations of ordinary and partial differential equations to an arbitrary fractional order. By a linear timefractional parabolic equation we mean certain paraboliclike partial differential equation governed by master equations containing fractional derivatives in time [12, 13]. The research areas of fractional differential equations range from theoretical to applied aspects.
An inverse time independent source problem for a fractional diffusion equation is studied and analytical solution can be obtained based on the method of the eigenfunction expansion. Moreover, the uniqueness of the inverse problem is established by analytic continuation and Laplace transform. The efficiency and accuracy of the proposed computational method are supported by the numerical examples [14]. A time dependent inverse source problem with additional measurement data at an inner point for the fractional diffusion equation is investigated and stable and accurate numerical approximation is obtained by means of the boundary element method and the first order Tikhonov regularization. Results are verified by the numerical examples [15].
In this paper, the mathematical analysis of a time dependent inverse source problem with additional measurement data at a boundary point for the fractional diffusion equation is done. The distinguishability of the inputoutput mappings is investigated and the measured output data \(f(t)\) and \(h(t)\) can be constructed, which leads to the explicit form of the inputoutput mappings. It is shown that the distinguishability of the inputoutput mappings holds, which implies the injectivity of the inverse mappings \(\Phi^{1}\) and \(\Psi^{1}\).
Since fractional derivatives are necessarily nonlocal, the sensible models of nonlocal phenomena are made by means of them. In the modeling of the anomalous diffusion in porous media, the best approach was made by the use of the fractional derivatives since it includes the nonlocal phenomena and the anomalous behaviors can be governed by the derivative order. The analysis of the timefractional porous medium equation was given in [16, 17], and a sensible model for the anomalous diffusion in porous medium equation was shown.
The main goal of this study is to investigate the distinguishability of the unknown source function via inputoutput mappings in a onedimensional time fractional inhomogeneous 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 [18]. As the next step, the noisy free measured output data \(f(t)\) and \(h(t)\) are used to introduce the inputoutput mappings \(\Phi[\cdot]:\mathcal{K}\rightarrow C^{1}[0,T]\) and \(\Psi[\cdot]:\mathcal{K}\rightarrow C[0,T]\), where \(\mathcal{K}\) represents the set of admissible source functions. The set of admissible source functions \(\mathcal{K}\) includes all functions \(r(t)\) such that problem (1) has a solution. Finally, we investigate the distinguishability of the unknown function \(r(t)\) via the above inputoutput mappings \(\Phi[\cdot]\) and \(\Psi[\cdot]\).
Consider now the following initial boundary value problem:
where \(\Omega_{T}=\{(x,t)\in R^{2}: 0< x<1, 0<t\leq T\}\) and the fractional derivative \(D_{t}^{\alpha}u(x,t)\) is defined in the Caputo sense \(D_{t}^{\alpha}u(x,t)=(I^{1\alpha}u^{\prime})(t)\), \(0<\alpha \leq1\), \(I^{\alpha}\) being the RiemannLiouville fractional integral,
The left and right boundary value functions \(\psi_{0}(t)\) and \(\psi_{1}(t)\) belong to \(C[0,T]\). The functions \(0< c_{0}\leq k(x)< c_{1}\) and \(g(x)\) satisfy the following conditions:

(C1)
\(k(x)\in C^{1}[0,1]\),

(C2)
\(g(x)\in C^{2}[0,1]\), \(g(0)=\psi_{0}(0)\), \(g^{\prime}(1)=\psi _{1}(0)\).
Under these conditions, the initial boundary value problem (1) has the unique solution \(u(x,t)\) defined in the domain \(\overline{\Omega }_{T}=\{(x,t)\in R^{2}: 0\leq x\leq1, 0\leq t\leq T\}\) which belongs to the space \(C(\overline{\Omega}_{T})\cap W_{t}^{1}(0,T]\cap C_{x}^{2}(0,1)\). Moreover, it satisfies the equation, initial and boundary conditions. Note that the space \(W_{t}^{1}(0,T]\) contains the functions \(f\in C^{1}(0,T]\) such that \(f^{\prime}(x)\in L(0,T)\).
Consider the inverse problem of determining the distinguishability of the unknown function \(r(t)\) from the mixed type of measured output data at the boundaries \(x=0\) and \(x=1\), respectively,
Then, the inverse problem with the measured output data \(f(t)\) and \(h(t)\) can be formulated as follows:
These formulations reduce the inverse problem of determining an unknown function \(r(t)\) to the problem of invertibility of the inputoutput mappings \(\Phi[\cdot]\) and \(\Psi[\cdot]\). This leads us to investigating the distinguishability of the source function via the above inputoutput mappings. We say that the mappings \(\Phi[\cdot ]:\mathcal{K}\rightarrow C^{1}[0,T]\) and \(\Psi[\cdot]:\mathcal {K}\rightarrow C[0,T]\) have the distinguishability property if \(\Phi[r_{1}]\neq\Phi[r_{2}]\) implies \(r_{1}(t)\neq r_{2}(t)\) and \(\Psi[r_{1}]\neq\Psi[r_{2}]\) implies \(r_{1}(t)\neq r_{2}(t)\). 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 \(x=0\) and measured output data of Dirichlet type at the boundary \(x=1\) are used in the determination of the distinguishability of the unknown function \(r(t)\). In addition, in the distinguishability of the unknown function \(r(t)\), analytical results are obtained.
The paper is organized as follows. In SectionÂ 2, analysis of the inverse problem with the single measured output data \(f(t)\) at the boundary \(x=0\) is given. Analysis of the inverse problem with the single measured output data \(h(t)\) at the boundary \(x=1\) is considered in SectionÂ 3. Numerical procedure and an example are given in SectionÂ 4. Finally, some concluding remarks are given in the last section.
2 Analysis of the inverse problem with given measured data \(f(t)\)
Consider now the inverse problem with one measured output data \(f(t)\) at \(x=0\). In order to formulate the solution of parabolic problem (1) by using the Fourier method of separation of variables, let us first introduce an auxiliary function \(v(x,t)\) 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 \(v(x,t)\) in the following form:
The unique solution of the initialboundary value problem can be represented in the following form [18]:
where
Moreover, \(\langle\zeta(\theta),\phi_{n}(\theta)\rangle=\int_{0}^{1}\phi _{n}(\theta )\zeta(\theta)\,d\theta\), \(E_{\alpha,\beta}\) being the generalized MittagLeffler function defined by
Assume that \(\phi_{n}(x)\) is the solution of the following SturmLiouville problem:
The Neumann type of measured output data at the boundary \(x=0\) can be written in terms of \(v(x,t)\) in the following form:
In order to arrange the above solution, let us define the following:
The solution in terms of \(z_{n}(t)\), \(w_{n}(t)\) and \(y_{n}(t)\) can then be rewritten in the following form:
Differentiating both sides of the above identity with respect to x and substituting \(x=0\) yields
Taking into account the overmeasured data \(k(0)(v_{x}(0,t)+\psi_{1}(t))=f(t)\),
is obtained, which implies that \(f(t)\) can be determined analytically. The righthand side of identity (5) defines the inputoutput mapping \(\Phi[r]\) on the set of admissible source functionÂ \(\mathcal{K}\):
The following lemma implies the relation between the source functions \(r_{1}(t),r_{2}(t)\in\mathcal{K}\) at \(x=0\) and the corresponding outputs \(f_{j}(t):=k(0)u_{x}(0,t;r_{j})\), \(j=1,2\).
Lemma 1
Let \(\upsilon_{1}(x,t)=\upsilon(x,t;r_{1})\) and \(\mathit{\upsilon}_{2}(x,t)=\upsilon(x,t;r_{2})\) be the solutions of the direct problem (2), corresponding to the admissible parameters \(r_{1}(t),r_{2}(t)\in\mathcal{K}\). If \(f_{j}(t)=k(0)(v_{x}(0,t;r_{j})+\psi_{1}(t))\), \(j=1,2\), are the corresponding outputs, the outputs \(f_{j}(t)\), \(j=1,2\), satisfy the following integral identity:
for each \(t\in(0,T]\), where \(\Delta f(t)=f_{1}(t)f_{2}(t)\), \(\Delta w_{n}(t)=w_{n}^{1}(t)w_{n}^{2}(t)\), \(\Delta r(t)=r_{1}(t)r_{2}(t)\) and \(\Delta y_{n}(t)=y_{n}^{1}(t)y_{n}^{2}(t)=\int_{0}^{t} s^{\alpha 1}E_{\alpha,\alpha}(\lambda_{n} s^{\alpha})\langle[\Delta r(ts)]F(ts),\phi _{n}(\theta)\rangle\,ds\).
Proof
By using identity (5), the measured output data \(f_{j}(t):=k(0)(v_{x}(0,t)+\psi_{1}(t))\), \(j=1,2\), can be written as follows:
respectively. Hence the difference of these formulas implies the desired result.â€ƒâ–¡
The lemma and the definitions enable us to reach the following conclusion.
Corollary 1
Let the conditions of LemmaÂ 1 hold. If in addition
\(\forall t\in(0,T]\), \(\forall n=0,1,\ldots\) holds, then \(f_{1}(t)=f_{2}(t)\), \(\forall t\in[0,T]\).
Since \(\phi_{n}(x)\), \(\forall n=0,1,2,\ldots\)â€‰, form a basis for the space and \(\phi_{n}^{\prime}(0) \neq0\), \(\forall n=0,1,2,\ldots \)â€‰, then \(r_{1}(t)\neq r_{2}(t)\) implies that \(\langle r_{1}(t)r_{2}(t),\phi _{n}(x)\rangle\neq0\) at least for some \(n \in\mathcal{N}\). Hence by LemmaÂ 1 we conclude that \(f_{1}(t)\neq f_{2}(t)\), which leads us to the following consequence: \(\Phi[r_{1}]\neq\Phi[r_{2}]\) implies that \(r_{1}(t)\neq r_{2}(t)\).
Theorem 1
Let conditions (C1), (C2) hold. Assume that \(\Phi[\cdot]:\mathcal{K}\rightarrow C^{1}[0,T]\) is the inputoutput mapping defined by (6) and corresponding to the measured output \(f(t):=k(0)u_{x}(0,t)\). In this case the mapping \(\Phi[r]\) has the distinguishability property in the class of admissible parameters \(\mathcal{K}\), i.e.,
Proof
From the above explanations the proof of the theorem is clear.â€ƒâ–¡
3 Analysis of the inverse problem with given measured data \(h(t)\)
Consider now the inverse problem with one measured output data \(h(t)\) at \(x=1\). Taking into account the overmeasured data \(h(t)=(v(1,t)+\psi _{0}(t)+\psi_{1}(t))\),
is obtained, which implies that \(h(t)\) can be determined analytically. The righthand side of identity (7) defines the inputoutput mapping \(\Psi[r]\) on the set of admissible source functions \(\mathcal{K}\):
The following lemma implies the relation between the parameters \(r_{1}(t),r_{2}(t) \in\mathcal{K}\) at \(x=1\) and the corresponding outputs \(h_{j}(t):=u(1,t;r_{j})\), \(j=1,2\).
Lemma 2
Let \(\upsilon_{1}(x,t)=\upsilon(x,t;r_{1})\) and \(\mathit{\upsilon}_{2}(x,t)=\upsilon(x,t;r_{2})\) be the solutions of the direct problem (2), corresponding to the admissible parameters \(r_{1}(t),r_{2}(t)\in\mathcal{K}\). If \(h_{j}(t)=v(1,t;r_{j})+\psi_{1}(t)+\psi_{0}(t)\), \(j=1,2\), are the corresponding outputs, the outputs \(h_{j}(t)\), \(j=1,2\), satisfy the following integral identity:
for each \(t\in(0,T]\), where \(\Delta h(t)=h_{1}(t)h_{2}(t)\), \(\Delta w_{n}(t)=w_{n}^{1} (t)w_{n}^{2} (t)\), \(\Delta r(t)=r_{1}(t)r_{2}(t)\).
Proof
By using identity (7), the measured output data \(h_{j}(t):=v(1,t)+\psi_{0}(t)+\psi_{1}(t)\), \(j=1,2\), can be written as follows:
respectively. Since \(z_{n}^{1}(t)=z_{n}^{2}(t)\) from the definition then the difference of these formulas implies the desired result.â€ƒâ–¡
Corollary 2
Let the conditions of LemmaÂ 2 hold. If in addition
holds, then \(h_{1}(t)=h_{2}(t)\), \(\forall t\in(0,T]\).
Since \(\phi_{n}(x)\), \(\forall n=0,1,2,\ldots\)â€‰, form a basis for the space and \(\phi_{n}^{\prime}(0) \neq0\), \(\forall n=0,1,2,\ldots \)â€‰, then \(r_{1}(t)\neq r_{2}(t)\) implies that \(\langle r_{1}(t)r_{2}(t),\phi _{n}(x)\rangle\neq0\) at least for some \(n \in\mathcal{N}\). Hence by LemmaÂ 2 we conclude that \(h_{1}(t)\neq h_{2}(t)\), which leads us to the following consequence: \(\Psi[r_{1}]\neq\Psi[r_{2}]\) implies that \(r_{1}(t)\neq r_{2}(t)\).
Theorem 2
Let conditions (C1), (C2) hold. Assume that \(\Psi[\cdot]:\mathcal{K}\rightarrow C[0,T]\) is the inputoutput mapping defined by (8) and corresponding to the measured output \(h(t):=u(1,t)\). In this case the mapping \(\Psi[r]\) has the distinguishability property in the class of admissible parameters \(\mathcal{K}\), i.e.,
Proof
From the above explanations the proof of the theorem is clear.â€ƒâ–¡
4 Numerical procedure
We use finite difference method to problem (1). We subdivide the intervals \([ 0,1 ] \) and \([ 0,T ] \) into M and N subintervals of equal lengths \(h=\frac{1}{M}\) and \(\tau=\frac{T}{N}\), respectively. The first order implicit scheme of problem (1) is as follows [19]:
where \(1\leq i\leq M\) and \(0\leq j\leq N\) are the indices for the spatial and time steps, respectively, \(u_{i}^{j}=u(x_{i},t_{j})\), \(r^{j}=r(t_{j})\), \(g_{i}=g(x_{i})\), \(F_{i}^{j}=F(x_{i},t_{j})\), \(\psi_{0}^{j}=\psi_{0}(t_{j})\), \(\psi_{1}^{j}=\psi_{1}(t_{j})\), \(x_{i}=ih\), \(t_{j}=j\tau\). At the \(t=0\) level, adjustment should be made according to the initial condition and the compatibility requirements.
Now, let us construct the predictingcorrecting mechanism. Firstly, if we use the measured output data is \(u(1,t)=h(t)\), we obtain
The finite difference approximation of \(r(t)\) is
where \(H^{j}=D_{t}^{\alpha}h(t_{j})\), \(j=0,1,\ldots,N\).
In numerical computation, since the time step is very small, we can take \(r^{j(0)}=r^{j1}\), \(u_{i}^{j(0)}=u_{i}^{j1}\), \(j=0,1,2,\ldots,N\), \(i=1,2,\ldots,M\). At each sth iteration step we first determine \(r^{j(s)}\) from the formula
The system of equations (14)(17) can be solved by the Gauss elimination method and \(u_{i}^{j(s)}\) is determined. If the difference of values between two iterations reaches the prescribed tolerance, the iteration is stopped and we accept the corresponding values \(r^{j(s)}\), \(u_{i}^{j(s)}\) (\(i=1,2,\ldots,N_{x}\)) as \(r^{j}\), \(u_{i}^{j}\) (\(i=1,2,\ldots ,N_{x}\)), on the \((j)\)th time step, respectively. In virtue of this iteration, we can move from level j to level \(j+1\).
Example 1
Consider the following problem for \(\alpha=1/2\):
and the measured output data is \(h(t)=t^{3}\).
The exact solution of this problem is \(\{ r(t),u(x,t) \} = \{ t^{2},t^{3}\sin\frac{\pi}{2}x \} \).
Let us apply the scheme above for the step sizes \(h=0.05\), \(\tau =0.05\). FiguresÂ 1, 2 show the exact and the numerical solutions of \(\{ r(t),u(x,t) \} \) when \(T=1/2\).
From these figures it can be seen that the agreement between the numerical and exact solutions for \(r(t)\) and \(u(x,T)\) is excellent.
Next, we will illustrate the stability of the numerical solution with respect to the noisy overdetermination data, defined by the function
where Î³ is the percentage of noise and Î¸ are random variables generated from a uniform distribution in the interval \([1,1]\).
In the case when \(T=1/2\), the illustrations of the sensitivity of the scheme with respect to noisy overdetermination data are shown in FiguresÂ 3, 4 and 5.
Example 2
In the previous Example 1, a smooth function given by \(r(t)= t^{2}\) is considered. In Example 2, a more severe discontinuous test function is given:
Let us apply the scheme above for the step sizes \(h=0.05\), \(\tau =0.05\). FigureÂ 6 shows the exact and the numerical solutions of \(r(t)\) when \(T=1/2\).
5 Some discussions
In the previous section, in ExampleÂ 1, the manmade noise in the measured output data is added to show the stability of the numerical method. From FiguresÂ 3, 4 and 5 it can be seen that the results are quite stable for small noise in the input data. Also in ExampleÂ 2, a discontinuous source function is given to show the efficiency of the present method. From FigureÂ 6 it can be seen that the agreement between the numerical and exact solutions for \(r(t)\) is excellent.
6 Conclusion
The aim of this study was to investigate the distinguishability properties of the inputoutput mappings \(\Phi [\cdot]:\mathcal{K}\rightarrow C[0,T]\) and \(\Psi[\cdot]:\mathcal {K}\rightarrow C^{1}[0,T]\), 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 inputoutput mappings holds, which implies the injectivity of the inverse mappings \(\Phi^{1}\) and \(\Psi^{1}\). The measured output data \(f(t)\) and \(h(t)\) are obtained analytically by a series representation, which leads to the explicit form of the inputoutput mappings \(\Phi[\cdot]\) and \(\Psi[\cdot]\). 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.
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Acknowledgements
The research was supported in parts by the Scientific and Technical Research Council (TUBITAK) of Turkey, Izmir University of Economics, Kocaeli University and Kadir Has University.
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Demir, A., Kanca, F. & Ozbilge, E. Numerical solution and distinguishability in time fractional parabolic equation. Bound Value Probl 2015, 142 (2015). https://doi.org/10.1186/s1366101504056
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DOI: https://doi.org/10.1186/s1366101504056