Inverse Stefan problems of determining the time-dependent source coefficient and heat flux function

TL;DR

This paper presents a mathematical approach to solving the inverse Stefan problem, focusing on determining time-dependent source coefficients and heat flux functions in heat transfer, with proofs of existence, uniqueness, and stability of solutions.

Contribution

It introduces a rigorous theoretical framework for the inverse Stefan problem, including existence, uniqueness, and stability analysis for determining source terms from temperature and heat flux data.

Findings

Proved existence and uniqueness of solutions.

Established continuous dependence on data.

Provided estimates for solution stability.

Abstract

This paper delves into the Inverse Stefan problem, specifically focusing on determining the time-dependent source coefficient in the parabolic heat equation governing heat transfer in a semi-infinite rod. The problem entails the intricate task of uncovering both temperature- and time-dependent coefficients of the source while accommodating Dirichlet and Neumann boundary conditions. Through a comprehensive mathematical model and rigorous theoretical analysis, our study aims to provide a robust methodology for accurately determining the source coefficient from observed temperature and heat flux data in problems with different cases of the source functions. Importantly, we establish the existence and uniqueness, and estimate the continuous dependence of a weak solution upon the given data for some inverse problems, offering a foundational understanding of its solvability.