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

TL;DR

This paper proves the uniqueness of determining a spatial potential and fractional order in a one-dimensional time-fractional diffusion equation with non-zero boundary conditions, using boundary data at a single point.

Contribution

It establishes the first uniqueness results for inverse problems involving time-fractional diffusion equations with non-zero boundary conditions.

Findings

Proved uniqueness for inverse coefficient and order determination.

Derived a representation formula for solutions with non-zero boundary data.

Extended results to boundary value determination at the opposite end.

Abstract

We consider initial boundary value problems for one-dimensional diffusion equation with time-fractional derivative of order $\alpha \in (0,1)$ 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 $\alpha$. 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.