Determine the spacial term of a two-dimensional heat source

TL;DR

This paper addresses the inverse problem of identifying a spatial heat source in a 2D heat equation using boundary and initial data, proposing a regularization method with error estimates and numerical validation.

Contribution

It introduces a regularization approach for the ill-posed inverse problem of determining the spatial source term in a 2D heat equation, with theoretical and numerical analysis.

Findings

Unique determination of the source under certain conditions.

Development of a regularized solution using Fourier series.

Validation through numerical experiments with error estimates.

Abstract

We consider the problem of determining a pair of functions $(u,f)$ satisfying the heat equation $u_t -\Delta u =\varphi(t)f (x,y)$, where $(x,y)\in \Omega=(0,1)\times (0,1)$ and the function $\varphi$ is given. The problem is ill-posed. Under a slight condition on $\varphi$, we show that the solution is determined uniquely from some boundary data and the initial temperature. Using the interpolation method and the truncated Fourier series, we construct a regularized solution of the source term $f$ from non-smooth data. The error estimate and numerical experiments are given.