Symmetry analysis of a canonical MEMS model
type: Item
instance of: scholarly article
title: Symmetry analysis of a canonical MEMS model
zbMATH Open document ID: 1175.34046
author: J. Regan Beckham
author: John A. Pelesko
published in: Methods and Applications of Analysis
publication date: 2009-06-04T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.maa/1239396533
review text: In the study of electrostatically actuated MEMS devices the semi-linear elliptic equation \[ \Delta u=\frac{\lambda}{u^2}\;\;\text{in}\;\;\Omega,\quad u=1\;\;\text{on}\;\;\partial\Omega \] arises. Such equation models the deflection of an elastic membrane under the influence of an applied electric field. \(u\) is the position of the membrane. In this article the authors study the one-dimensional version of this generalized equation, i.e. \[ \frac{d^2u}{dx^2}=\frac{\lambda f(x)}{u^2}\quad \text{with } \frac{du}{dx}(0)=0\;\;\text{ and }u(1/2)=1.\tag{1} \] The function \(f(x)\) is assumed to be symmetric about the origin. The authors perform a symmetry analysis of (1) to determine properties of \(f(x)\) that allow invariance under a one-parameter Lie group of transformations. Then the analysis of the problem is made for two particular functions which arise from the found symmetries. It is shown how symmetry is used to convert boundary value problem to initial value problem and to construct bifurcation diagrams.
reviewed by: Oleg V. Makeev
Mathematics Subject Classification ID: 34C14
Mathematics Subject Classification ID: 34C60
Mathematics Subject Classification ID: 74K15
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34B15
zbMATH DE Number: 5562039
zbMATH Keywords: MEMS
zbMATH Keywords: microelectomechanical system
zbMATH Keywords: Lie symmetry
zbMATH Keywords: semi-linear elliptic equation
MaRDI profile type: Publication
DOI: 10.4310/MAA.2008.V15.N3.A5
description: scientific article; zbMATH DE number 5562039