Determining a time-dependent coefficient in a time-fractional diffusion-wave equation with the Caputo derivative by an additional integral condition

Abstract

This paper is devoted to recovering a time-dependent zeroth-order coefficient in a time-fractional diffusion-wave equation with the time Caputo derivative from an additional integral condition. The uniqueness and a conditional stability for such an inverse problem are proved. Then the two-point gradient method is used to solve the inverse zeroth-order coefficient problem numerically. Some properties of the forward operator are obtained, such as the Fréchet differentiability, the Lipschitz continuity and the tangential cone condition to guarantee the convergence of the proposed algorithm. Four numerical examples in one-dimensional and two-dimensional spaces are provided to show the effectiveness and stability of the suggested algorithm.

Introduction

Let $\Omega \subset {\mathbb{R}}^d$ $(d\le 3)$ be a bounded domain with sufficiently smooth boundary. In this paper, we consider the following initial–boundary value problem for a time-fractional diffusion-wave equation as follows

$$\left\{\begin{array}{cc}\hfill & {\partial }_{0+}^{\alpha }u(x,t)+\mathcal{A}u(x,t)+p\left(t\right)u(x,t)=f(x,t),x\in \Omega ,t\in (0,T],\hfill \\ \hfill & u(x,0)=\phi \left(x\right),x\in \Omega ,\hfill \\ \hfill & u_t(x,0)=\psi \left(x\right),x\in \Omega ,\hfill \\ \hfill & {\partial }_{\nu }u(x,t)=0,x\in \partial \Omega ,t\in (0,T],\hfill \end{array}\right.$$

where $1<\alpha <2$ and ${\partial }_{0+}^{\alpha }$ denotes the Caputo fractional left-sided derivative of order $\alpha$ defined by

$${\partial }_{0+}^{\alpha }u(x,t)=\frac{1}{\Gamma (2-\alpha )}{\int }_0^t\frac{{\partial }^{2}u(x,s)}{\partial s^2}\frac{ds}{{(t-s)}^{\alpha -1}},t>0,$$

in which $\Gamma \left(\cdot \right)$ is the Gamma function, and the second order elliptic operator $\mathcal{A}$ is defined by

$$\mathcal{A}u(x,t)=-\sum _{i,j=1}^d{\partial }_{x_i}\left(a_{ij}\left(x\right){\partial }_{x_j}u(x,t)\right),x\in \Omega ,$$

in which the coefficients satisfy

$$\left\{\begin{array}{c}{a}_{ij}\left(x\right)=a_{ji}\left(x\right)\in C^1\left(\overline{\Omega }\right),i,j=1,2,\dots ,d,\hfill \\ \sum _{i,j=1}^d{a}_{ij}\left(x\right){\xi }_i{\xi }_j\ge a_0\sum _{i=1}^d{\xi }_i^2,x\in \overline{\Omega },\xi =({\xi }_1,\dots ,{\xi }_d)\in {\mathbb{R}}^d,a_0>0,\hfill \end{array}\right.$$

and

$${\partial }_{\nu }u(x,t)=\sum _{i,j=1}^d{a}_{ij}\left(x\right){\partial }_{x_j}u(x,t){\nu }^i\left(x\right),x\in \partial \Omega ,$$

where $\nu \left(x\right)=({\nu }^1\left(x\right),\dots ,{\nu }^d\left(x\right))$ is the unit outward normal vector at $x\in \partial \Omega$.

For the last few decades, time-fractional diffusion $(0<\alpha <1)$ and diffusion-wave $(1<\alpha <2)$ equations have been widely studied because of the advantages of the fractional derivatives to describe the sub-diffusion and super-diffusion phenomena [1], [2]. In addition, due to the memory property, fractional calculus has been extensively used in many other areas, such as biology, physics, hydrology, chemistry and biochemistry, medicine and finance [3], [4], [5], [6], [7], [8].

If the fractional order $\alpha$, the zeroth-order coefficient $p\left(t\right)$, the source term $f(x,t)$, the initial value $\phi \left(x\right)$ and the initial speed $\psi \left(x\right)$ are all given, then problem (1.1) is a direct problem for the time-fractional diffusion-wave equation. Direct problems for time-fractional diffusion-wave equations have been studied extensively in recent years [9], [10], [11], [12], [13], [14], [15], [16], [17] for examples. For nonlinear time-fractional partial differential equations, there are also some works on detail studies, refer to [18], [19].

Inverse problems for time-fractional diffusion-wave equations are to recover the initial data or the source function or the zeroth-order coefficient and so on by some additional data. There are not many papers to consider them. In [20], Šišková et al. provided a numerical algorithm based on the Rothe method to deal with an inverse time-dependent source problem for a time-fractional diffusion-wave equation, and gave the convergence of iterated solution. In [21], Wei et al. used the Tikhonov regularization method to solve the backward problem for a time-fractional diffusion-wave equation. In [22], Wei et al. considered an inverse space-dependent source problem in a time-fractional diffusion-wave equation from final time measured data combined with the conjugate gradient algorithm. In [23], Xian et al. used a variational method to solve an inverse initial value problem for a time-fractional diffusion-wave equation. In [24], Liao et al. identified the fractional order and the space source term simultaneously in a time-fractional diffusion-wave equation, and provided the uniqueness and stability of the inverse problem. In [25], Yan et al. determined a space-dependent source term in a time-fractional diffusion-wave equation from a part of noisy boundary data, and prove the uniqueness for the inverse problem by the Titchmarsh convolution theorem and the Duhamel principle. The above works are focused on the linear ill-posed problems.

To our best knowledge, there are just a few papers on nonlinear inverse problems for time-fractional diffusion-wave equations. In [26], Yan et al. applied a Bayes method to identify the fractional order and diffusion coefficient in a time-fractional diffusion-wave equation. In [27], the authors used the measured data on a single boundary point to identify a time-dependent zeroth-order coefficient in a time-fractional diffusion-wave equation and the Levenberg–Marquardt method was applied to solve the inverse problem numerically. However, the convergence of the algorithm could not be obtained.

The main goal in this paper is to determine the zeroth-order coefficient $p\left(t\right)$ by the following additional integral condition

$$g\left(t\right)={\int }_{\Omega }\omega \left(x\right)u(x,t)dx,t\in (0,T),$$

where $\omega \left(x\right)\in L^2\left(\Omega \right)$ is a non-zero weight function if no other specified.

Time-fractional diffusion-wave equations can be used to formulate the anomalous super-diffusion phenomena of particles in heterogeneous porous media, refer to [1], [4], [28] for comprehending physical backgrounds. However in real applications, the time-dependent zeroth-order coefficient $p\left(t\right)$ may be unknown, for example in the transport process of contamination in underground soil, if the pollutants have a degradation reaction such that the amount of pollutants decreases with respective to time, then the zeroth-order term in Eq. (1.1) is involved where the coefficient $p\left(t\right)$ describes the rate of degraded amount to pollutants in unit volume and unit time, see [29] for a reference where $p$ is depending only on spatial variable.

As we know, for the inverse problems to determine coefficients under the additional condition (1.3) in time fractional diffusion equations, there are only a few works to consider them. For examples, Lopushanska et al. in [30] considered an inverse coefficient problem in a semi-linear fractional telegraph equation, and proved the existence and uniqueness for the inverse problem. In [31], El-Borai studied an inverse Cauchy problem in a Hilbert space for fractional abstract differential equations, and gave the existence and uniqueness result for the inverse problem. In [32], Lopushansky et al. identified the diffusivity parameter in a space–time fractional diffusion equation, the authors gave the existence and uniqueness result for the inverse problem, and used an iterative algorithm to solve the inverse problem numerically. For a related problem to a time-fractional diffusion equation by using additional data on a point, Fujishiro et al. in [33] gave a stability of recovering a time-dependent zeroth-order coefficient without any numerical methods. Because of the difficulty to obtain a tangential cone condition for forward operators, there are few works to give the convergence analysis of gradient-type regularization algorithms, such as the Levenberg–Marquardt method, the Gauss–Newton method, etc while applying on nonlinear inverse problems for partial differential equations. Based on the above motivations, in this paper we will prove the convergence of the two-point gradient method while solving this inverse coefficient problem by using such integral condition.

We always assume $p\left(t\right)\in AC[0,T]$, $f(x,t)\in AC([0,T];L^2\left(\Omega \right))\cap L^{\infty }(0,T;D\left(A\right))$, $\phi \left(x\right)\in D\left(A\right)$, $\psi \left(x\right)\in D\left(A^{1-\frac{1}{\alpha }}\right)$ and $\omega \left(x\right)\in L^2\left(\Omega \right)$ unless other specified, where $AC[0,T]$ is the space of absolutely continuous functions on $[0,T]$, and $D\left(A^{\gamma }\right)$ $(\gamma \ge 0)$ is the Hilbert scale space defined in Section 2.

The rest of this paper is organized as follows. In Section 2, we present some preliminaries used in this paper. We prove the uniqueness and a conditional stability for the considered inverse problem in Section 3. In Section 4, we show the ill-posedness for the inverse problem and derive some properties of the forward operator. In Section 5, we present the two-point gradient method and give the convergence result for the proposed algorithm. Numerical results for four examples in one-dimensional and two-dimensional spaces are investigated in Section 6. Finally, we give a conclusion in Section 7.