Topics in the asymptotic analysis of linear and nonlinear eigenvalue problems

In Applied Mathematics, linear and nonlinear eigenvalue problems arise frequently when characterizing the equilibria of various physical systems. In this thesis, two specific problems are studied, the first of which has its roots in micro engineering and concerns Micro-Electro Mechanical Systems (ME...

Full description

Bibliographic Details
Main Author: Lindsay, Alan Euan
Language:English
Published: University of British Columbia 2010
Online Access:http://hdl.handle.net/2429/26271
id ndltd-UBC-oai-circle.library.ubc.ca-2429-26271
record_format oai_dc
spelling ndltd-UBC-oai-circle.library.ubc.ca-2429-262712018-01-05T17:24:24Z Topics in the asymptotic analysis of linear and nonlinear eigenvalue problems Lindsay, Alan Euan In Applied Mathematics, linear and nonlinear eigenvalue problems arise frequently when characterizing the equilibria of various physical systems. In this thesis, two specific problems are studied, the first of which has its roots in micro engineering and concerns Micro-Electro Mechanical Systems (MEMS). A MEMS device consists of an elastic beam deflecting in the presence of an electric field. Modelling such devices leads to nonlinear eigenvalue problems of second and fourth order whose solution properties are investigated by a variety of asymptotic and numerical techniques. The second problem studied in this thesis considers the optimal strategy for distributing a fixed quantity of resources in a bounded two dimensional domain so as to minimize the probability of extinction of some species evolving in the domain. Mathematically, this involves the study of an indefinite weight eigenvalue problem on an arbitrary two dimensional domain with homogeneous Neumann boundary conditions, and the optimization of the principal eigenvalue of this problem. Under the assumption that resources are placed on small patches whose area relative to that of the entire domain is small, the underlying eigenvalue problem is solved explicitly using the method of matched asymptotic expansions and several important qualitative results are established. Science, Faculty of Mathematics, Department of Graduate 2010-07-09T18:16:35Z 2010-07-09T18:16:35Z 2010 2010-11 Text Thesis/Dissertation http://hdl.handle.net/2429/26271 eng Attribution-NonCommercial-NoDerivatives 4.0 International http://creativecommons.org/licenses/by-nc-nd/4.0/ University of British Columbia
collection NDLTD
language English
sources NDLTD
description In Applied Mathematics, linear and nonlinear eigenvalue problems arise frequently when characterizing the equilibria of various physical systems. In this thesis, two specific problems are studied, the first of which has its roots in micro engineering and concerns Micro-Electro Mechanical Systems (MEMS). A MEMS device consists of an elastic beam deflecting in the presence of an electric field. Modelling such devices leads to nonlinear eigenvalue problems of second and fourth order whose solution properties are investigated by a variety of asymptotic and numerical techniques. The second problem studied in this thesis considers the optimal strategy for distributing a fixed quantity of resources in a bounded two dimensional domain so as to minimize the probability of extinction of some species evolving in the domain. Mathematically, this involves the study of an indefinite weight eigenvalue problem on an arbitrary two dimensional domain with homogeneous Neumann boundary conditions, and the optimization of the principal eigenvalue of this problem. Under the assumption that resources are placed on small patches whose area relative to that of the entire domain is small, the underlying eigenvalue problem is solved explicitly using the method of matched asymptotic expansions and several important qualitative results are established. === Science, Faculty of === Mathematics, Department of === Graduate
author Lindsay, Alan Euan
spellingShingle Lindsay, Alan Euan
Topics in the asymptotic analysis of linear and nonlinear eigenvalue problems
author_facet Lindsay, Alan Euan
author_sort Lindsay, Alan Euan
title Topics in the asymptotic analysis of linear and nonlinear eigenvalue problems
title_short Topics in the asymptotic analysis of linear and nonlinear eigenvalue problems
title_full Topics in the asymptotic analysis of linear and nonlinear eigenvalue problems
title_fullStr Topics in the asymptotic analysis of linear and nonlinear eigenvalue problems
title_full_unstemmed Topics in the asymptotic analysis of linear and nonlinear eigenvalue problems
title_sort topics in the asymptotic analysis of linear and nonlinear eigenvalue problems
publisher University of British Columbia
publishDate 2010
url http://hdl.handle.net/2429/26271
work_keys_str_mv AT lindsayalaneuan topicsintheasymptoticanalysisoflinearandnonlineareigenvalueproblems
_version_ 1718582520501501952