A collocation heat polynomials method for one-dimensional inverse Stefan problems

Abstract

The inverse one-phase Stefan problem in one dimension, aimed at identifying the unknown time-dependent heat flux $P\left(t\right)$ with a known moving boundary position $s\left(t\right)$, is investigated. A previous study (Kassabek et al., 2021) attempted to reconstruct the unknown heat flux $P\left(t\right)$ using the Variational Heat Polynomials Method (VHPM). In this paper, we develop the Collocation Heat Polynomials Method (CHPM) for the reconstruction of the time-dependent heat flux $P\left(t\right)$. This method constructs an approximate solution as a linear combination of heat polynomials, which satisfies the heat equation, with the coefficients determined using the collocation method. To address the resulting ill-posed problem, Tikhonov regularization is applied. As an application, we demonstrate the effectiveness of the method on benchmark problems. Numerical results show that the proposed method accurately reconstructs the time-dependent heat flux $P\left(t\right)$, even in the presence of significant noise. The results are also compared with those obtained in Kassabek et al. (2021) using the VHPM.