The uniqueness of inverse problems for a fractional equation with a single measurement

Abstract

This article is concerned with an inverse problem on simultaneously determining some unknown coefficients and/or an order of derivative in a multidimensional time-fractional evolution equation either in a Euclidean domain or on a Riemannian manifold. Based on a special choice of the Dirichlet boundary input, we prove the unique recovery of at most two out of four x-dependent coefficients (possibly with an extra unknown fractional order) by a single measurement of the partial Neumann boundary output. Especially, both a vector-valued velocity field of a convection term and a density can also be uniquely determined. The key ingredient turns out to be the time-analyticity of the decomposed solution, which enables the construction of Dirichlet-to-Neumann maps in the frequency domain and thus the application of inverse spectral results.

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

“…Third, the extension to the multi-dimensional case is very challenging, and requires more data for a unique determination, e.g. restricted Neumann-to-Dirichlet map [6] or one specially designed excitation [24]. Fourth and last, the design and analysis of relevant reconstruction algorithms can depart enormously from the more traditional (penalized) least-squares approach.…”

“…Remark 3.2. There have been several works on identifying multiple parameters from one single observation [22,24,46]. The recent work [22] is closest to the current one in some sense, which is concerned with the following model on Ω = (0, 1), with α ∈ (0, 2),…”

Well-Posedness for Weak and Strong Solutions of Non-Homogeneous Initial Boundary Value Problems for Fractional Diffusion Equations

“…We remark that the well-posdness for problem (1.4) with non-homogenous boundary conditions is meaningful also for other mathematical problems such as optimal control problems (see e.g., [26]) or inverse problems (see e.g., [3,7,11,12,15]).…”

An inverse problem for a multi‐term fractional differential equation with two‐parameter fractional derivatives in time and Bessel operator

“…An inverse problem of determining a space‐dependent source term for an equation involving only two fractional derivatives in time and Bessel operator was discussed in Agarwal et al 18 . The well‐posedness for ISP defined for multi‐term time fractional diffusion equation is proved in Li et al., 19 whereas uniqueness of solution of inverse problem with a single measurement is considered in Kian et al 20 . There are several techniques to handle IPs involving Carleman estimates 21 and unique continuation 22 .…”