Averaging of ordinary differential equations with slowly varying averages
type: Item
instance of: scholarly article
title: Averaging of ordinary differential equations with slowly varying averages
zbMATH Open document ID: 1207.34055
published in: Discrete and Continuous Dynamical Systems. Series B
publication date: 2010-09-22T00:00:00Z
review text: The averaging method asserts that a good approximation to solutions of a time varying ordinary differential equation with small amplitude \[ \frac{dx}{dt}=\varepsilon f(x,t) \] are solutions of the averaged equation \[ \frac{dx}{dt}=\varepsilon f^0(x), \] where \[ f^0(x)=\lim_{t\to\infty}\frac{1}{T}\int_0^Tf(x,t)\,dt. \] The error between solutions with the same initial conditions is maintained small on a long time interval. In the paper, a similar result is established allowing the averaged equation to vary in time thus allowing slowly varying averages of the original equation. Both the modeling aspect and a discussion including a comparison of the model with a model from the literature and possible modifications of other available forms of the equation are presented. Examples that illustrate various possibilities of slowly varying averages and the application of the method established in the paper are given.
Mathematics Subject Classification ID: 34C29
zbMATH DE Number: 5790136
zbMATH Keywords: averaging
zbMATH Keywords: time-varying differential equations
zbMATH Keywords: varying averages
reviewed by: Eugene Ershov
MaRDI profile type: Publication
full work available at URL: dcdsb.2010.14.353
OpenAlex ID: W2075106117
DOI: 10.3934/DCDSB.2010.14.353
author: Zvi Artstein
description: scientific article; zbMATH DE number 5790136