Simultaneous uniqueness identification of the fractional order and diffusion coefficient in a time-fractional diffusion equation

This article is concerned with the uniqueness of simultaneously determining the fractional order of the derivative in time, diffusion coefficient, and Robin coefficient, in one-dimensional time-fractional diffusion equations with derivative order $\alpha \in (0,1)$ and non-zero boundary conditions. The measurement data, which is the solution to the initial–boundary value problem, is observed at a single boundary point over a finite time interval. Based on the expansion of eigenfunctions for the solution to the forward problem and the asymptotic properties of the Mittag-Leffler function, the uniqueness of the fractional order is established. Subsequently, the uniqueness of the eigenvalues and the absolute value of the eigenfunction evaluated at $x=0$ for the associated operator are demonstrated. Then, the uniqueness of identifying the diffusion coefficient and the Robin coefficient is proven via an inverse boundary spectral analysis for the eigenvalue problem of the spatial differential operator. The results show that the uniqueness of three parameters can be simultaneously determined using limited boundary observations at a single spatial endpoint over a finite time interval, without imposing any constraints on the eigenfunctions of the spatial differential operator.