Uniqueness for an inverse coefficient problem for a one-dimensional time-fractional diffusion equation with non-zero boundary conditions

Abstract

We consider initial boundary value problems for one-dimensional diffusion equation with time-fractional derivative of order α ∈ (0, 1) which are subject to non-zero Neumann boundary conditions. We prove the uniqueness for an inverse coefficient problem of determining a spatially varying potential and the order of the time-fractional derivative by Dirichlet data at one end point of the spatial interval. The imposed Neumann conditions are required to be within the correct Sobolev space of order α. Our proof is based on a representation formula of solution to an initial boundary value problem with non-zero boundary data. Moreover, we apply such a formula and prove the uniqueness in the determination of boundary value at another end point by Cauchy data at one end point.

Recovering the Potential and Order in One-Dimensional Time-Fractional Diffusion with Unknown Initial Condition and Source

“…Now we study the direct problem (1.1), especially the solution representation. One distinct feature of problem (1.1) is that it involves an inhomogeneous Neumann boundary condition, which has not been extensively studied in the literature ( [34,23] for relevant works). Following [34], we exploit the one-dimensional nature of problem (1.1), and derive a series representation of the solution u.…”

“…One distinct feature of problem (1.1) is that it involves an inhomogeneous Neumann boundary condition, which has not been extensively studied in the literature ( [34,23] for relevant works). Following [34], we exploit the one-dimensional nature of problem (1.1), and derive a series representation of the solution u. The derivation is based on the standard separation of variable technique (see, e.g., [35] and [14,Section 6.2]).…”

“…The recovery of the potential q in the classical diffusion equation from lateral Cauchy data has been extensively discussed, and several uniqueness results have been obtained [31,28,38]. The study on related inverse problems for time-fractional models is of more recent origin, starting from [6] (see [17] for an early tutorial) and there are a few works on recovering a spatially dependent potential from lateral Cauchy data [33,34,41,20]. Rundell and Yamamoto [33] showed that the lateral Cauchy data can uniquely determine the spectral data when u 0 ≡ f ≡ 0, and concluded the uniqueness of the potential by the classical Gel'fand-Levitan theory.…”

“…They also proposed a recovery procedure based on Newton's method and empirically studied the singular value spectrum of the linearized forward map, showing the severe ill-posed nature of the inverse problem. Later, they [34] significantly relaxed the regularity condition on the boundary excitation g(t) (in a suitable Sobolev space in time). Recently, Jing and Yamamoto [20] established the identifiability of multiple parameters (including order, spatially dependent potential, initial value and Robin coefficients in the boundary condition) simultaneously in the one-dimensional subdiffusion / diffusion-wave (i.e., α ∈ (0, 2)) equation with a zero boundary condition and source, excited by a nontrivial initial condition from the lateral Cauchy data at both end points (cf.…”

Recovering the potential and order in one-dimensional time-fractional diffusion with unknown initial condition and source *

“…The recovery of the space-dependent potential q in the classical diffusion equation from lateral Cauchy data has been extensively discussed, and several uniqueness results have been obtained [31,34,41]. The study on related inverse problems for time-fractional models is of more recent origin, starting from [7] (see [19] for an early tutorial) and there are a few works on recovering a spatially dependent potential from lateral Cauchy data [22,36,37,44]. Rundell and Yamamoto [36] showed that the lateral Cauchy data can uniquely determine the spectral data when u 0 ≡ f ≡ 0, and proved the uniqueness of the potential q by the classical Gel'fand-Levitan theory.…”