Heat flow in polygons with reflecting edges

TL;DR

This paper derives an asymptotic formula for the heat content in polygonal domains with reflecting edges, considering initial heat distribution within a subpolygon and Neumann boundary conditions.

Contribution

It provides a new asymptotic analysis of heat flow in polygons with reflecting edges, extending previous results to more general polygonal geometries.

Findings

Derived an asymptotic formula for heat content as time approaches zero.

Extended heat flow analysis to polygons with reflecting edges.

Provided mathematical insights into heat distribution in polygonal domains.

Abstract

We investigate the heat flow in an open, bounded set $D$ in $\mathbb{R}^2$ with polygonal boundary $\partial D$. We suppose that $D$ contains an open, bounded set $\widetilde{D}$ with polygonal boundary $\partial \widetilde{D}$. The initial condition is the indicator function of $\widetilde{D}$ and we impose a Neumann boundary condition on the edges of $\partial D$. We obtain an asymptotic formula for the heat content of $\widetilde{D}$ in $D$ as time $t\downarrow 0$.