Determine the source term of a two-dimensional heat equation

TL;DR

This paper addresses the inverse problem of identifying a heat source in a 2D heat equation using boundary and initial-final temperature data, employing Fourier transforms and regularization techniques.

Contribution

It introduces a novel approach combining Fourier transforms with Tikhonov regularization to uniquely determine the heat source from boundary and initial-final temperature data.

Findings

Unique determination of the heat source under given conditions

Development of a regularization method for the nonlinear ill-posed problem

Numerical validation demonstrating effectiveness of the approach

Abstract

Let $\Omega$ be a two-dimensional heat conduction body. We consider the problem of determining the heat source $F(x,t)=\varphi(t)f(x,y)$ with $\varphi$ be given inexactly and $f$ be unknown. The problem is nonlinear and ill-posed. By a specific form of Fourier transforms, we shall show that the heat source is determined uniquely by the minimum boundary condition and the temperature distribution in $\Omega$ at the initial time $t=0$ and at the final time $t=1$. Using the methods of Tikhonov's regularization and truncated integration, we construct the regularized solutions. Numerical part is given.