Uniqueness for inverse problem of determining fractional orders for time-fractional advection-diffusion equations

TL;DR

This paper proves the uniqueness of determining the fractional order in a time-fractional advection-diffusion equation using data from a single spatial point over time, even when the domain and operator are unknown.

Contribution

It establishes the first uniqueness result for the inverse problem of identifying the fractional order with minimal data and unknown domain and operator conditions.

Findings

Uniqueness of fractional order determination from single-point data.

Applicable even when the domain and operator are unknown.

Uses eigenfunction expansions and Mittag-Leffler asymptotics.

Abstract

We consider initial boundary value problems of time-fractional advection-diffusion equations with the zero Dirichlet boundary value $\partial_t^{\alpha} u(x,t) = -Au(x,t)$, where $-A = \sum}{i,j=1}^d \partial_i(a_{ij}(x)\partial_j) + \sum{j=1}^d b_j(x)\partial_j + c(x)$. We establish the uniqueness for an inverse problem of determining an order $\alpha$ of fractional derivatives by data $u(x_0,t)$ for $0<t<T$ at one point $x_0$ in a spatial domain $\OOO$. The uniqueness holds even under assumption that $\OOO$ and $A$ are unknown, provided that the initial value does not change signs and is not identically zero. The proof is based on the eigenfunction expansions of finitely dimensional approximating solutions, a decay estimate and the asymptotic expansions of the Mittag-Leffler functions for large time.