Applic. Analysis, 81, N4, (2002), 929-937

TL;DR

This paper proves the uniqueness of solutions for a heat equation inverse problem using derivative set completeness and Property C, highlighting conditions where uniqueness fails with additional flux data.

Contribution

It introduces a new proof of solution uniqueness for a heat equation inverse problem using derivative completeness and Property C, and discusses non-uniqueness with extra flux data.

Findings

Completeness of derivatives set is established.

Uniqueness of inverse heat problem solution is proved.

Non-uniqueness occurs with additional flux data at zero temperature point.

Abstract

Completeness of the set of products of the derivatives of the solutions to the equation $(av')'-{\l}v=0, v(0,\l)=0$ is proved. This property is used to prove the uniqueness of the solution to an inverse problem of finding conductivity in the heat equation $\dot{u}=(a(x)u')', u(x,0)=0, u(0,t)=0,\ u(1,t)=f(t)$ known for all $t>0$, from the heat flux $a(1)u'(1,t)=g(t)$. Uniqueness of the solution to this problem is proved. The proof is based on Property C. It is proved the inverse that the inverse problem with the extra data (the flux) measured at the point, where the temperature is kept at zero, (point $x=0$ in our case) does not have a unique solution, in general.