Precise asymptotics of boundary layers for damped simple pendulum equations
type: Item
instance of: scholarly article
title: Precise asymptotics of boundary layers for damped simple pendulum equations
zbMATH Open document ID: 1214.34018
author: Tetsutaro Shibata
published in: Results in Mathematics
publication date: 2010-10-14T00:00:00Z
review text: The author considers the simple pendulum equation with friction \[ -u''(t)- |u'(t)|+ g(u(t))\equiv\lambda \sin u(t),\quad t\in (-T,T), \] \[ u(t)> 0,\quad t\in (-T, T),\qquad u(\pm T)= 0, \] where \(T> 0\) and \(\lambda> 0\) is a parameter; \(g\in C^m(\mathbb{R})\) \((m\geq 1)\) and \(g(u)> 0\) for \(u> 0\); \(g(0)= g'(0)= 0\); \(g(u)/u\) is strictly increasing for \(0\leq u\leq n\). The author establishes the asymptotic formula for the boundary layers of the solution \(u_\lambda\) for the equation above, and shows that its boundary slope is steeper than that of the solution without damping term \(|u'(t)|\).
Mathematics Subject Classification ID: 34B15
zbMATH DE Number: 5800218
zbMATH Keywords: slope of boundary layers
zbMATH Keywords: damped simple pendulum
MaRDI profile type: Publication
full work available at URL: s00025-009-0009-5
OpenAlex ID: W2012861015
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DOI: 10.1007/S00025-009-0009-5
description: scientific article; zbMATH DE number 5800218