On wavelet fundamental solutions to the heat equation -- heatlets
type: Item
instance of: scholarly article
title: On wavelet fundamental solutions to the heat equation -- heatlets
zbMATH Open document ID: 0953.35051
author: Jianhong Shen
author: Gilbert Strang
published in: Journal of Differential Equations
publication date: 2000-12-06T00:00:00Z
review text: The application of wavelet theory in differential equations has been mostly focused on numerical computation. The present paper treats an application of wavelet theory to the theory of differential equations. Consider the initial value problem for the heat equation \[ u_t={\sigma^2\over 2}u_{xx}, \quad -\infty<x <\infty,\;t>0 \] with initial data \(u(x,0)= \varphi(x)\). When the initial function \(\varphi(x)\) is a wavelet function, then the solution of the problem is called a heatlet. The paper gives a fundamental property of heatlet, and using this property the authors give a heatlet decomposition theorem of the solution to the initial value problem for the heat equation (heat evolution). As an example the authors investigate the Haar heatlet derived from the Haar multiresolution. The authors treat also the case of multidimensional heat equation, and construct multidimensional heatlets, give the result about the self-similarity and other fundamental properties. Moreover they give the \(n-D\) heatlet decomposition to the heat evolution of dimension \(n\).
Mathematics Subject Classification ID: 35K05
Mathematics Subject Classification ID: 42C40
zbMATH DE Number: 1437333
zbMATH Keywords: completeness
zbMATH Keywords: scale similarity
zbMATH Keywords: translation similarity
zbMATH Keywords: Haar multiresolution
reviewed by: M. Nagase
MaRDI profile type: Publication
full work available at URL: jdeq.1999.3707
OpenAlex ID: W1983635033
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DOI: 10.1006/JDEQ.1999.3707
description: scientific article; zbMATH DE number 1437333