Analysis of the inverse problem in a time fractional parabolic equation with mixed boundary conditions

The inverse problem of determining an unknown coefficient in a linear parabolic equation by using over-measured 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 [1–9]. The inverse problem of unknown coefficients in a quasi-linear 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 time-fractional parabolic equation, we mean a 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 an unknown coefficient $k(x)$ in a one-dimensional 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 Sturm-Liouville eigenvalue problem under certain conditions [12]. As the next step, the noise-free measured output data are used to introduce the input-output mappings $\mathrm{\Phi }[\cdot ]:\mathcal{K}\to C^1[0,T]$ and $\mathrm{\Psi }[\cdot ]:\mathcal{K}\to C[0,T]$. Finally, we investigate the distinguishability of the unknown coefficient via the above input-output mappings $\mathrm{\Phi }[\cdot ]$ and $\mathrm{\Psi }[\cdot ]$.

Under these conditions, the initial boundary value problem (1) has the unique solution $u(x,t)$ defined in the domain ${\overline{\mathrm{\Omega }}}_T=\{(x,t)\in R^2:0\le x\le 1,0\le t\le T\}$ which belongs to the space $C({\overline{\mathrm{\Omega }}}_T)\cap W_t^1(0,T]\cap C_x^2(0,1)$. Moreover, it satisfies the equation, initial and boundary conditions. The space $W_t^1(0,T]$ contains the functions $f\in C^1(0,T]$ such that $f^{\prime }(t)\in L(0,T)$.

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.

Here $u=u(x,t)$ is the solution of parabolic problem (1). The functions $f(t)$ and $h(t)$ are assumed to be noise-free measured output data. In this context, parabolic problem (1) will be referred to as a direct (forward) problem with the inputs $g(x)$ and $k(x)$. It is assumed that the functions $f(t)$ and $h(t)$ belong to $C[0,T]$ and satisfy the consistency conditions $f(0)=k(0)g^{\prime }(0)$ and $g(1)=h(0)$.

which reduces the inverse problem of determining the unknown coefficient $k(x)$ to the problem of invertibility of the input-output mappings $\mathrm{\Phi }[\cdot ]$ and $\mathrm{\Psi }[\cdot ]$. Hence this leads us to investigate the distinguishability of the unknown coefficient via the above input-output mappings. We say that the mappings $\mathrm{\Phi }[\cdot ]:\mathcal{K}\to C^1[0,T]$ and $\mathrm{\Psi }[\cdot ]:\mathcal{K}\to C[0,T]$ have the distinguishability property if $\mathrm{\Phi }[k_1]\ne \mathrm{\Phi }[k_2]$ implies $k_1(x)\ne k_2(x)$ and the same holds for $\mathrm{\Psi }[\cdot ]$. This, in particular, means the injectivity of inverse mappings ${\mathrm{\Phi }}^{-1}$ and ${\mathrm{\Psi }}^{-1}$. 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 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.

The following lemma implies the relation between the parameters $k_1(x),k_2(x)\in {\mathcal{K}}_0$ at $x=0$ and the corresponding outputs $f_j(t):=k(0)u_x(0,t;p_j)$, $j=1,2$.

for each $t\in (0,T]$, where $\mathrm{\Delta }f(t)=f_1(t)-f_2(t)$, $\mathrm{\Delta }{w}_n(t)=w^1(t)-w^2(t)$.

respectively. Note that the definition of $z_n(t)$ implies that $z_n^1(t)=z_n^2(t)$. Hence, the difference of these formulas implies the desired result. □

The lemma and the definitions of $w_n(t)$ and $z_n(t)$ given above enable us to reach the following conclusion.

then $f_1(t)=f_2(t)$, $\mathrm{\forall }t\in [0,T]$.

Proof If $〈{\xi }_1(x,t)-{\xi }_2(x,t),{\varphi }_n(x)〉=0$, $\mathrm{\forall }n=0,1,\dots$ , then $k_1(x)=k_2(x)$. If $k_1(x)=k_2(x)$, then $u_1(x,t)=u_2(x,t)$. Since $f(t)$ depends on $u(x,t)$, then from the uniqueness of solution $f_1(t)=f_2(t)$.

Since ${\varphi }_n(x)$, $\mathrm{\forall }n=0,1,2,\dots$ form a basis for the space and ${\varphi }_n^{\prime }(0)\ne 0$, $\mathrm{\forall }n=0,1,2,\dots$ , then $k_1(x)\ne k_2(x)$ implies that $〈{\xi }_1(x,t)-{\xi }_2(x,t),{\varphi }_n(x)〉\ne 0$ at least for some $n\in \mathcal{N}$. Hence by Lemma 1 we conclude that $f_1(t)\ne f_2(t)$, which leads us to the following consequence: $k_1(x)\ne k_2(x)$ implies that $\mathrm{\Phi }[k_1]\ne \mathrm{\Phi }[k_2]$. □

is obtained, which implies that $h(t)$ can be determined analytically.

The following lemma implies the relation between the parameters $k_1(x),k_2(x)\in {\mathcal{K}}_1$ at $x=1$ and the corresponding outputs $h_j(t):=u(1,t;k_j)$, $j=1,2$.

for each $t\in (0,T]$, where $\mathrm{\Delta }h(t)=h_1(t)-h_2(t)$, $\mathrm{\Delta }{w}_n(t)=w^1(t)-w^2(t)$.

respectively. Note that the definition of $z_n(t)$ implies that $z_n^1(t)=z_n^2(t)$. Hence, the difference of these formulas implies the desired result. □

The lemma and the definitions given above enable us to reach the following conclusion.

then $h_1(t)=h_2(t)$, $\mathrm{\forall }t\in [0,T]$.

Proof If $〈{\xi }_1(x,t)-{\xi }_2(x,t),{\varphi }_n(x)〉=0$, $\mathrm{\forall }n=0,1,\dots$ , then $k_1(x)=k_2(x)$. If $k_1(x)=k_2(x)$, then $u_1(x,t)=u_2(x,t)$. Since $h(t)$ depends on $u(x,t)$, then from the uniqueness of solution $h_1(t)=h_2(t)$.

Since ${\varphi }_n(x)$, $\mathrm{\forall }n=0,1,2,\dots$ form a basis for the space and ${\varphi }_n^{\prime }(0)\ne 0$, $\mathrm{\forall }n=0,1,2,\dots$ , then $k_1(x)\ne k_2(x)$ implies that $〈{\xi }_1(x,t)-{\xi }_2(x,t),{\varphi }_n(x)〉\ne 0$ at least for some $n\in \mathcal{N}$. Hence by Lemma 2 we conclude that $h_1(t)\ne h_2(t)$, which leads us to the following consequence: $k_1(x)\ne k_2(x)$ implies that $\mathrm{\Psi }[k_1]\ne \mathrm{\Psi }[k_2]$. □

The aim of this study was to investigate the distinguishability properties of the input-output mappings $\mathrm{\Phi }[\cdot ]:\mathcal{K}\to C^1[0,T]$ and $\mathrm{\Psi }[\cdot ]:\mathcal{K}\to C[0,T]$, 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 $\mathrm{\Phi }[\cdot ]$ and $\mathrm{\Psi }[\cdot ]$ holds, which implies the injectivity of the inverse mappings ${\mathrm{\Phi }}^{-1}$ and ${\mathrm{\Psi }}^{-1}$. This provides the insight that compared to the Dirichlet type, the Neumann-type 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 $\mathrm{\Phi }[\cdot ]$ and $\mathrm{\Psi }[\cdot ]$. We also show that the value of the unknown coefficient $k(x)$ at $x=0$ is determined by using the Neumann-type measured output data at $x=0$, which brings more restrictions on the set of admissible coefficients. However, $k(1)$ is not obtained by the Dirichlet-type measured output data at $x=1$. This provides the insight that the Neumann-type 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 input-output 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.