Inverse problems for fractional equations with a minimal number of measurements

TL;DR

This paper demonstrates that for certain fractional differential equations, the unknown coefficients can be uniquely identified using minimal measurements, highlighting the nonlocality's role in inverse problem solutions.

Contribution

The paper establishes unique determination of coefficients in fractional equations with minimal measurements, revealing nonlocality's crucial role and introducing a new comparison principle.

Findings

Unique determination of coefficients with N+1 measurements for finite N

Infinite measurements needed for N=∞ case

Nonlocality enables results not possible in local equations

Abstract

In this paper, we study several inverse problems associated with a fractional differential equation of the following form: \[ (-\Delta)^s u(x)+\sum_{k=0}^N a^{(k)}(x) [u(x)]^k=0,\ \ 0<s<1,\ N\in\mathbb{N}\cup\{0\}\cup\{\infty\}, \] which is given in a bounded domain $\Omega\subset\mathbb{R}^n$, $n\geq 1$. For any finite $N$, we show that $a^{(k)}(x)$, $k=0,1,\ldots, N$, can be uniquely determined by $N+1$ different pairs of Cauchy data in $\Omega_e:=\mathbb{R}^n\backslash\overline{\Omega}$. If $N=\infty$, the uniqueness result is established by using infinitely many pairs of Cauchy data. The results are highly intriguing in that it generally does not hold true in the local case, namely $s=1$, even for the simplest case when $N=0$, a fortiori $N\geq 1$. The nonlocality plays a key role in establishing the uniqueness result. We also establish several other unique determination results by making use of a minimal number of measurements. Moreover, in the process we derive a novel comparison principle for nonlinear fractional differential equations as a significant byproduct.