Heat equations with Dirichlet boundary conditions: Exponential relaxation and normalized dynamics

 

Mathematik  |  Informatik

 

Hongjia Meng, 2009 | Altdorf, UR

 

In this paper, we first prove that solutions of the inhomogeneous heat equation on an open domain with Dirichlet boundary conditions converge exponentially in L^2 to the Poisson steady state. We then implement a Python program that evaluates the analytical solution for heat diffusion in a copper wire with internal Joule heating, and the resulting plots illustrate the theoretical prediction.
Secondly, we consider a control problem where the source term is such that the L^2-norm of the solution is preserved. We establish global well-posedness in finite-dimensional subspaces and show, using Galerkin methods, that the solutions convergence to a weak solution of the full problem. Furthermore, we show that the time-dependent scalar function is monotonically decreasing.

Introduction

In this paper, we explore two problems concerning heat equations with Dirichlet boundary conditions. The first is about how solutions of general inhomogeneous heat equations converge toward the corresponding steady-state (Poisson) solutions, while the second focuses on the properties of a heat equation with global constraint on the L^2 norm of the solution.

Methods

The analysis relies on the variational formulation in the Sobolev space H_0^1(Omega). Using Green’s formulas, we justify integration by parts and express the equations in weak form. This is applied together with standard functional analysis tools such as the Lax–Milgram theorem to prove properties including existence and uniqueness (as explored in works of H. Brezis). We apply standard inequalities to bound norms, as well as spectral decomposition to study the convergence rate and Galerkin methods to approximate a weak solution.

Results

In this paper, we examine two aspects of heat flow dynamics with Dirichlet boundary data.
The first part demonstrates that the solution of the inhomogeneous heat equation converges exponentially in L^2 toward the steady-state solution of the corresponding Poisson equation. By expressing the problem in terms of the Laplacian’s eigenfunctions, the partial differential equation reduces to a set of ordinary differential equations, each showing exponential decay. This analytic result is accompanied through a Python plot of the temperature in a copper wire with fixed endpoints, where the temperature curve visibly approaches the steady parabola exponentially.
The second part focuses on the heat flow normalized by a feedback function. We first deduce that the feedback function decreases monotonically and approaches the first eigenvalue. Afterwards, we establish that the solution of this heat equation remains well-defined for all time. Lastly, we apply Galerkin methods to do a finite dimensional approximation and ultimately approximate a weak solution.

Discussion

The convergence of solutions to the inhomogeneous heat equation toward a steady state is a classical and well-established result (see, e.g., the works of Y. Pinchover and J. Rubinstein, and H. Brezis).
Within this broader setting, by introducing a time-dependent scalar function whose role is to enforce normalization along the flow, we extend the analysis to a non-linear, constrained heat equation where the constraint normalizes the norm of the solution to 1. We establish global well-posedness in finite-dimensional subspaces and analyze the behavior of the scalar function over time. To the best of our knowledge, these results are original.

Conclusions

This work studies classical convergence of the inhomogeneous heat equation and a non-linear variant with an L^2 norm constraint. While the exponential approach to steady-state is classical, the introduction of a feedback term motivates more ideas. Future directions include generalizing the normalization to other norms, exploring alternative feedback functions or source terms and applying the framework to more complex PDEs. Furthermore, the geometrical structure of the domain could be studied, as specific structures could yield constraints on the equation.

 

 

Würdigung durch die Expertin

Dr. Laura Kobel-Keller

Die vorliegende Arbeit zur mathematischen Analyse ausgewählter Fragestellungen zur klassischen Wärmeleitungsgleichung zeichnet sich durch hohe fachliche Präzision & Rigorosität, eigenständige tiefgreifende Herangehensweise und klare Präsentation der gefundenen Resultate sowie der komplexen entsprechenden mathematischen Beweise aus. Der physikalische Hintergrund der untersuchten Fragen wird ebenfalls erschlossen.

Prädikat:

Gold

Sonderpreis «Regeneron International Science and Engineering Fair (ISEF)» gestiftet von der Gebauer Stiftung

 

 

 

Kantonale Mittelschule Uri, Altdorf
Lehrerin: Nuria Canta