Simultaneous inversion of time-dependent source term and fractional order for a time-fractional diffusion equation (Q2292028)

In this paper, an inverse problem for the time fractional diffusion equation \begin{align*} \frac{\partial^\alpha u(x,t)}{\partial t^\alpha} = Lu(x,t) + F(x,t), \quad x \in \Omega, \ t > 0, \end{align*} with initial and boundary conditions \begin{align*} & u(x,0) = u_0(x), \ x \in \overline{\Omega}, \qquad \frac{\partial u(x,t)}{\partial \textbf{n}} = 0, \ x \in \partial \Omega, \ t > 0, \end{align*} is considered on \( \Omega \subset \mathbb{R}^d, \, 1 \le d \le 3 \), where \( \Omega \) is an interval, if \( d = 1 \), and for \( d \in \{2, 3 \} \) it is assumed to have a smooth boundary. In addition, \( -L \) is a symmetric, uniformly elliptic operator defined by \begin{align*} -Lu(x) = - \sum_{i=1}^d \frac{\partial}{\partial x_i} \Big( \sum_{j=1}^d \theta_{i,j}(x) \frac{\partial}{\partial x_j} u(x) \Big) + c(x) u(x), \end{align*} where \( u: \overline{\Omega} \to \mathbb{R} \), and \( \theta_{i,j} \in C^1(\overline{\Omega}) \) and \( c \in C(\overline{\Omega}) \) are variable coefficients with \( \theta_{i,j} = \theta_{j,i} \) and \( c \ge 0 \) on \( \overline{\Omega} \). Moreover, for some constant \( v > 0 \) there holds \( v \sum_{i=1}^d \xi_i^2 \le \sum_{i,j=1}^d \theta_{i,j}(x) \xi_i \xi_j \) for each \( x\in \overline{\Omega}, \, \xi \in \mathbb{R}^d \). Furthermore, \( \frac{\partial^\alpha}{\partial t^\alpha}, \, 0 < \alpha < 1 \), denotes the Caputo fractional derivative operator, and \( \frac{\partial u(x,t)}{\partial \textbf{n}} \) represents the outward normal derivative along the boundary \( \partial \Omega \). The source function \( F(x,t) \) is assumed to have separated variables, i.e., \( F(x,t) = p(t) f(x) \). The purpose of the paper is the simultaneous identification of the fractional order~\( \alpha \) and the time-dependent component \( p(t), t \ge 0 \), from the following additional data: \begin{align*} g(t) = \left\{\begin{array}{rl} u(0,t), \ t \ge 0, & \textup{if } d = 1, \\ \int_\Omega \mu(x) u(x,t) \, dx, \ t \ge 0, & \, \textup{if } d = 2, 3, \end{array}\right. \end{align*} where \( \mu \in C_0^\infty(\Omega) \) denotes a non-negative weight function. The authors show uniqueness of this simultaneous identification problem by using the Laplace transform, the two-parametric Mittag-Leffler function, and analytic continuation. Another topic of the paper is the stabilization of this inverse problem by using Tikhonov regularization on a finite observation interval: \[ J_{\delta}^\beta(\alpha,p) = \tfrac{1}{2} \Vert \mathbb{F}(\alpha,p) - g^{\delta} \Vert_{L^2(0,T)}^2 + \tfrac{\beta}{2} \Vert p \Vert_{L^2(0,T)}^2 \to \min, \] where \( \mathbb{F}(\alpha,p) := g \) denotes the forward operator, and \( (\alpha,p) \in D(\mathbb{F}) = [ \epsilon_0, 1-\epsilon_0] \times L^2(0,T) \) for some small positive constant \( \epsilon_0 \). In addition, \( g^{\delta} \in L^2(0,T) \) are given noisy data satisfying \( \Vert g^{\delta} - g \Vert_{L^2(0,T)} \le \delta \), and \( \beta > 0 \) is a regularization parameter. The authors consider a discrepancy principle as parameter choice strategy and show that \( \Vert \mathbb{F}(\alpha_\delta,p_\delta) - \mathbb{F}(\alpha_*,p_*) \Vert_{L^2(0,T)} \to 0 \) as \( \delta \to 0 \), where \( (\alpha_*,p_*) \in D(\mathbb{F}) \) denotes the exact solution of the given problem, and \( (\alpha_\delta,p_\delta) \in D(\mathbb{F}) \) are the approximations obtained by the considered discrepancy principle for noise level \( \delta > 0 \). For the implementation of the regularized optimization problem, an alternating minimization scheme based on \(\alpha\)- and \(p\)-optimization subproblems is considered, and the latter class of subproblems is treated in more detail. The paper concludes with some numerical illustrations.