City University of Hong Kong
DSpace
 

CityU Institutional Repository >
3_CityU Electronic Theses and Dissertations >
ETD - Dept. of Mathematics  >
MA - Doctor of Philosophy  >

Please use this identifier to cite or link to this item: http://hdl.handle.net/2031/5049

Title: On two nonlinear singularly perturbed boundary value problems
Other Titles: Guan yu liang ge fei xian xing qi yi she dong bian zhi tiao jian wen ti de yan jiu
關於兩個非線性奇異攝動邊值條件問題的研究
Authors: Zhao, Yi (趙毅)
Department: Dept. of Mathematics
Degree: Doctor of Philosophy
Issue Date: 2007
Publisher: City University of Hong Kong
Subjects: Boundary value problems
Singular perturbations (Mathematics)
Notes: CityU Call Number: QA379.Z43 2007
Includes bibliographical references (leaves [86]-90)
Thesis (Ph.D.)--City University of Hong Kong, 2007
v, 90 leaves : ill. ; 30 cm.
Type: Thesis
Abstract: Boundary layer theory is a collection of perturbation methods which are frequently used to derive asymptotic solutions to differential equations with a small parameter ". However, it is often very difficult to prove rigorously that the approximations obtained are truly representations of exact solutions. Sometimes, boundary layer theory even leads to spurious solutions. In this thesis, based on shooting method, we study two nonlinear singularly perturbed boundary value problems rigorously, which the method of matched asymptotics cannot be applied directly. First, we are concerned with the positive solutions of the boundary-value problem 8>< >: "u00 ¡ ¾(u) = ¡°; u(0) = u(1) = 0; where ° is a positive constant. The nonlinear term ¾(u) behaves like a cubic, and it vanishes only at u = 0, where ¾0(0) > 0 and ¾00(0) < 0. This problem arises in a study of phase transitions in a slender circular cylinder composed of an incompressible phase-transforming material. Here, we determine the number of solutions to the problem for any given °, derive asymptotic formulas for these solutions, and show that the error terms associated with these formulas are exponentially small, except for one critical value of °. Then, we investigate the Carrier’s problem 8>< >: "2y00 = 1 ¡ y2 ¡ 2(1 ¡ x2)y; y(¡1) = y(1) = 0; where " is again a small positive parameter. Let N" denote the maximum number of spikes that a solution to Carrier’s problem can have. We show that N" is asymptotically equal to [K="], where K = 0:4725 ¢ ¢ ¢ , and the square brackets represent the greatest integer less than or equal to the quantity inside. If n(") stands for the number of solutions to this problem, then it is also shown that 4N" ¡ 3 · n(") · 4N". Our approach is based on the shooting method and on the construction of an envelope function for the minimum values of the solutions as " approaches zero.
Online Catalog Link: http://lib.cityu.edu.hk/record=b2218032
Appears in Collections:MA - Doctor of Philosophy

Files in This Item:

File Description SizeFormat
fulltext.html157 BHTMLView/Open
abstract.html157 BHTMLView/Open

Items in CityU IR are protected by copyright, with all rights reserved, unless otherwise indicated.

 

Valid XHTML 1.0!
DSpace Software © 2013 CityU Library - Send feedback to Library Systems
Privacy Policy · Copyright · Disclaimer