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A class of moving boundary problems with an exponential source term

Julieta Bollati, Ernesto A. Borrego Rodriguez, Adriana C. Briozzo, Colin Rogers

TL;DR

The paper addresses moving boundary problems for a nonlinear evolution equation with an exponential source term by establishing a systematic link to Stefan-type problems using reciprocal and Cole–Hopf transformations. This approach maps Stefan problems to nonlinear transport problems with exponential sources and yields parametric explicit similarity solutions in selected cases, including Dirichlet and Robin–Neumann boundary regimes. The authors derive a chain of equivalent formulations (P1) through (P4), and demonstrate convergence of Robin–Neumann to Dirichlet solutions as heat-transfer coefficients grow. The results provide a rigorous analytical framework for understanding the dynamics of phase-change–type systems with exponential sources and offer explicit solutions that can serve as benchmarks for more complex/heterogeneous settings.

Abstract

This work investigates a class of moving boundary problems related to a nonlinear evolution equation featuring an exponential source term. We establish a connection to Stefan-type problems, for different boundary conditions at the fixed face, through the application of a reciprocal transformation alongside the Cole-Hopf transformation. For specific cases, we derive explicit similarity solutions in parametric form. This innovative approach enhances our understanding of the underlying dynamics and offers valuable insights into the behavior of these systems.

A class of moving boundary problems with an exponential source term

TL;DR

The paper addresses moving boundary problems for a nonlinear evolution equation with an exponential source term by establishing a systematic link to Stefan-type problems using reciprocal and Cole–Hopf transformations. This approach maps Stefan problems to nonlinear transport problems with exponential sources and yields parametric explicit similarity solutions in selected cases, including Dirichlet and Robin–Neumann boundary regimes. The authors derive a chain of equivalent formulations (P1) through (P4), and demonstrate convergence of Robin–Neumann to Dirichlet solutions as heat-transfer coefficients grow. The results provide a rigorous analytical framework for understanding the dynamics of phase-change–type systems with exponential sources and offer explicit solutions that can serve as benchmarks for more complex/heterogeneous settings.

Abstract

This work investigates a class of moving boundary problems related to a nonlinear evolution equation featuring an exponential source term. We establish a connection to Stefan-type problems, for different boundary conditions at the fixed face, through the application of a reciprocal transformation alongside the Cole-Hopf transformation. For specific cases, we derive explicit similarity solutions in parametric form. This innovative approach enhances our understanding of the underlying dynamics and offers valuable insights into the behavior of these systems.
Paper Structure (8 sections, 12 theorems, 131 equations)

This paper contains 8 sections, 12 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Connection between Stefan type problems with a class of moving boundary problems with an exponential source term
  3. Stefan type problem with Dirichlet condition
  4. Stefan type problem with Robin-Neumann conditions
  5. Explicit solutions for particular cases
  6. Particular case with a Dirichlet condition
  7. Particular case with a Robin-Neumann condition
  8. Conclusions

Key Result

Lemma 2.1

If $w$ satisfies P1-1 in a bounded $D_T=\lbrace(z,t): 0< z< s(t), \quad 0<t<T\rbrace$ and $w$ is continuous in $D_T\cup B_T$ where $B_T=\lbrace(z,t):z=0,\quad 0\leq t<T\rbrace\cup \lbrace(z,t):z=s(t),\quad 0\leq t<T\rbrace$ then, we have

Theorems & Definitions (30)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Remark 2.1
  • Remark 2.2
  • Theorem 2.4
  • proof
  • ...and 20 more