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Please use this identifier to cite or link to this item: http://hdl.handle.net/2031/5052

Title: On a singularly perturbed two-point boundary value problem with nested boundary layers
Other Titles: Ju you kan tao bian jie ceng de liang dian bian zhi qi yi she dong wen ti
具有嵌套邊界層的兩點邊值奇異攝動問題
Authors: Liang, Xiangdong (梁向東)
Department: Dept. of Mathematics
Degree: Master of Philosophy
Issue Date: 2007
Publisher: City University of Hong Kong
Subjects: Boundary layer
Boundary value problems -- Numerical solutions
Singular perturbations (Mathematics)
Notes: 3, 48 leaves : ill. ; 30 cm.
CityU Call Number: QA379.L53 2007
Includes bibliographical references (leaves [47]-48)
Thesis (M.Phil.)--City University of Hong Kong, 2007
Type: Thesis
Abstract: In this thesis, we consider the di®erential equation ²3xy00 + x2y0 ¡ (x3 + ²)y = 0; subject to the boundary conditions y(0) = 1; y(1) = pe; where 0 < ² ¿ 1 is a small positive parameter. Searching through the literature, this singularly perturbed boundary-value prob- lem was given by Carl M. Bender and Steven A. Orszag in the classic book Advanced mathematical methods for scientists and engineers. Bender and Orszag construct a simple asymptotic solution by means of matching techniques for nested boundary layers, which means that there is a boundary layer lying inside another one. Indeed, they show that two boundary layers occur near x = 0 with the thickness ² and ²2 respectively, and that a uniformly valid asymptotic solution is given by yunif(x) = 2px ² K1 µ 2px ² ¶ + e¡ ² x + e x2 2 ¡ 1; where K1(z) is the modi¯ed Bessel function of order one. Although this solution appears to behave like the true solution, a natural question to ask now is \In what sense does yunif(x) approximate the true solution y(x)?" From Bender and Orszag's arguments, one can readily see that while the matching tech- niques are very useful in obtaining uniform asymptotic solutions to practical problems, the approximations are often not established on ¯rm mathematical grounds. The aim of this thesis is to reinvestigate the above problem and to derive a uniformly valid asymptotic solution to this problem by using a mathematically rigorous argument. Instead of using the method of matched asymptotics, we refer to the Liouville- Green approximation. First, we ¯nd an approximate equation via Liouville trans- formation and convert it into an integral equation. Then, we construct a uniform approximate solution and provide a mathematically rigorous proof by using Picard's method of successive approximation. Finally, we show the leading term for the inner layer is exponentially small, which has never been mentioned in the existing literature.
Online Catalog Link: http://lib.cityu.edu.hk/record=b2218007
Appears in Collections:MA - Master of Philosophy

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