Regularization techniques for estimating the space-dependent source in an n-dimensional linear parabolic equation using space-dependent noisy data

Abstract

In this article, the mathematical study of the problem of identifying the space-dependent source term, in transport processes given by an n-dimensional linear parabolic equation, from space-dependent noisy measurements taken at an arbitrary fixed time is conducted. The problem is analytically solved using Fourier techniques, and it is shown that this solution is not stable. Three families of uniparametric regularization operators are proposed to address the instability of the solution. Each of them is designed to compensate for the factor that causes the instability of the inverse operator. Additionally, a selection rule for the regularization parameter is included, and an error bound for each estimation of Hölder type is obtained. Numerical examples of different characteristics are presented to demonstrate the benefits of the proposed strategies.