Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Targyn A. Nauryz, Khumoyun Jabbarkhanov

TL;DR

This work tackles inverse Stefan problems in a heat-conduction setting on a moving boundary, aiming to recover time-dependent source coefficients and, in some formulations, heat flux from boundary and interior data. A spectral theory approach is employed, combining a fixed-domain transformation $\xi=x/s(t)$ with eigenfunction expansions to derive weak solutions for four problem variants under Dirichlet and Neumann conditions, and for two source-term forms: $G(x,t,u)=R(t)f(x,t)$ and $G(x,t,u)=P(t)u(x,t)+f(x,t)$. The authors establish existence, uniqueness, and continuous dependence of the weak solutions on the input data, and provide explicit representations for the unknown time-dependent coefficients via boundary conditions and spectral coefficients. These results contribute a rigorous solvability framework for inverse Stefan-type problems, offering mathematical foundations and representations that can support practical identification of heat sources and boundary fluxes in problems involving phase change and moving interfaces.

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.

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

TL;DR

This work tackles inverse Stefan problems in a heat-conduction setting on a moving boundary, aiming to recover time-dependent source coefficients and, in some formulations, heat flux from boundary and interior data. A spectral theory approach is employed, combining a fixed-domain transformation with eigenfunction expansions to derive weak solutions for four problem variants under Dirichlet and Neumann conditions, and for two source-term forms: and . The authors establish existence, uniqueness, and continuous dependence of the weak solutions on the input data, and provide explicit representations for the unknown time-dependent coefficients via boundary conditions and spectral coefficients. These results contribute a rigorous solvability framework for inverse Stefan-type problems, offering mathematical foundations and representations that can support practical identification of heat sources and boundary fluxes in problems involving phase change and moving interfaces.

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.
Paper Structure (6 sections, 18 theorems, 185 equations)

This paper contains 6 sections, 18 theorems, 185 equations.

Table of Contents

  1. Introduction
  2. Inverse Stefan problem 1.
  3. Inverse Stefan problem 2.
  4. Inverse Stefan problem 3.
  5. Inverse Stefan problem 4.
  6. Conclusion

Key Result

Lemma 1

If $\widetilde{\psi}\in C^4(\Omega_0)$ which satisfy the conditions $\widetilde{\psi}(0)=\widetilde{\psi}(1)=\widetilde{\psi}'(0)=\widetilde{\psi}'(1)=\widetilde{\psi}"(0)=\widetilde{\psi}"(1)=0$, then for some positive constant $c=\sqrt{2}\left[\sum_{n=1}^{\infty}\frac{1}{\lambda_n}\right]^{1/2}$, where $\widetilde{\psi}_n=\int_0^{1}\widetilde{\psi}(\xi)\phi_n(\xi)d\xi$.

Theorems & Definitions (33)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Theorem 1
  • proof
  • ...and 23 more