Inverse problems for parabolic equations 3

TL;DR

This paper addresses an inverse problem for a parabolic PDE, proposing a novel method to uniquely determine the unknown time-dependent coefficient a(t) from boundary and initial data.

Contribution

It introduces a new method for calculating the unknown coefficient a(t) in a parabolic equation, ensuring uniqueness and existence of the solution.

Findings

A new method for recovering a(t) is proposed.

The method guarantees uniqueness and existence of the solution.

The approach is applicable under continuous and bounded conditions for a(t).

Abstract

Let $u_t-a(t)u_{xx}=f(x, t)$ in $0\leq x \leq \pi,\,\,t\geq 0.$ Assume that $u(0,t)=u_1(t)$, $u(\pi,t)=u_2(t)$, $u(x,0)=h(x)$, and the extra data $u_x(0,t)=g(t)$ are known. The inverse problem is: {\it How does one determine the unknown $a(t)$?} The function $a(t)>a_0>0$ is assumed continuous and bounded. This question is answered and a method for recovery of $a(t)$ is proposed. There are several papers in which sufficient conditions are given for the uniqueness and existence of $a(t)$, but apparently there was no method proposed for calculating of $a$. The method given in this paper for proving the uniqueness and existence of the solution to inverse problem is new and it allows one to calculate the unknown coefficient $a(t)$.