Heat equation on the arithmetic von Koch snowflake
type: Item
instance of: scholarly article
title: Heat equation on the arithmetic von Koch snowflake
zbMATH Open document ID: 0963.35072
publication date: 2001-05-06T00:00:00Z
review text: The author studies the Dirichlet problem for the heat equation \[ u_t'=\Delta u,\;x\in D,t>0,\quad u(0,x)=0,\;x\in D,\quad u(t,x)=1,\;x\in\partial D\tag{1} \] in a bounded domain \(D\subset\mathbb{R}^n\) with a boundary \(\partial D=K_s,\) where \(K_s\) is the \(s\)-adic arithmetic von Koch snowflake. Using the probabilistic representation of the solution to (1) he derives an approximate functional equation for the total amount \(E_{K_s}(t)= \int_{K_s}^{}u(t,x)dx\) of heat in \(K_s\) at time \(t\). This functional equation takes after some transformations the form of an inhomogeneous renewal equation which is further analyzed in the paper. Some asymptotic formulas for \(E_{K_s}(t)\) are derived.
Mathematics Subject Classification ID: 35K05
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 28A80
zbMATH DE Number: 1547920
zbMATH Keywords: probabilistic representation of the solution
zbMATH Keywords: inhomogeneous renewal equation
author: M. van den Berg
reviewed by: Yana Belopolskaya
MaRDI profile type: Publication
full work available at URL: s004400000068
OpenAlex ID: W4205719418
DOI: 10.1007/S004400000068
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 1547920