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

TL;DR

This paper introduces the Collocation Heat Polynomials Method (CHPM) for accurately reconstructing unknown time-dependent heat flux in one-dimensional inverse Stefan problems, demonstrating improved performance over previous methods even with noisy data.

Contribution

The paper develops a novel collocation-based approach using heat polynomials and Tikhonov regularization for inverse Stefan problems, enhancing reconstruction accuracy.

Findings

Accurately reconstructs heat flux P(t) with noisy data.

Outperforms previous VHPM method in benchmark tests.

Demonstrates effectiveness through numerical experiments.

Abstract

The inverse one-phase Stefan problem in one dimension, aimed at identifying the unknown time-dependent heat flux P(t) with a known moving boundary position s(t), is investigated. A previous study [16] attempted to reconstruct the unknown heat flux P(t) 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(t). 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(t), even in the presence of significant noise. The results are also compared with those obtained in [16] using the VHPM.