An inverse problem for the heat equation

TL;DR

This paper investigates whether boundary flux measurements at the ends of a domain uniquely determine the spatially varying coefficient in a heat equation, addressing the inverse problem's uniqueness and informational content.

Contribution

The paper provides a definitive answer to the uniqueness of reconstructing the coefficient q(x) from boundary flux measurements in a heat equation inverse problem.

Findings

Flux measurement at x=0 determines q(x) uniquely.

Flux measurement at x=1 provides additional information about q(x).

The inverse problem's uniqueness depends on the boundary flux data used.

Abstract

Let $u_t = u_{xx} - q(x) u, 0 \leq x \leq 1$, $t>0$, $u(0, t) = 0, u(1, t) = a(t), u(x,0) = 0$, where $a(t)$ is a given function vanishing for $t>T$, $a(t) \not\equiv 0$, $\int^T_0 a(t) dt < \infty$. Suppose one measures the flux $u_x (0,t) := b_0 (t)$ for all $t>0$. Does this information determine $q(x)$ uniquely? Do the measurements of the flux $u_x (1,t) := b(t)$ give more information about $q(x)$ than $b_0 (t)$ does? The above questions are answered in this paper.