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Title:  Asymptotic solutions of a fourth order differential equation 
Other Titles:  Yi ge si jie wei fen fang cheng de jian jin jie 一個四階微分方程的漸近解 
Authors:  Zhang, Haiyu (張海愉) 
Department:  Dept. of Mathematics 
Degree:  Master of Philosophy 
Issue Date:  2006 
Publisher:  City University of Hong Kong 
Subjects:  Differential equations, Partial  Asymptotic theory 
Notes:  CityU Call Number: QA377.Z438 2006 Includes bibliographical references (leaves [76]78) Thesis (M.Phil.)City University of Hong Kong, 2006 ii, 78 leaves : ill. ; 30 cm. 
Type:  Thesis 
Abstract:  In this thesis, we derive uniform asymptotic expansions of solutions to the fourth order differential equation y(4) + λ2(zy11 + y) = 0 with large parameter λ and real variable z. The solutions of this differential equation can be expressed by Laplace integrals of the form y(z) = ∫Lτ−2 exp(τz −τ−1 +λ−2τ/3)dτ. The uniform asymptotic expansions are derived by the wellknown method first suggested by C. Chester, B. Friedman and F. Ursell in 1957 for the integral ∫c g(t) exp[λf(t, z)]dt, where g(t) and f(t, z) are analytic functions of t, z is a bounded real parameter, and f(t, z) have two saddle points t±(z) which coalesce as z tends to some real number z0. This method begins with a cubic transformation that converts the integral into a canonical form, and then applies an integrationbyparts procedure repeatedly. We prove the cubic transformation given by Chester, Friedman and Ursell is indeed onetoone and analytic in a domain containing the paths of integration, and obtain the uniform asymptotic expansion for large values of λ in terms of Airy functions and their derivatives. There are two advantages of this approach: (i) the coefficients in the expansion are defined recursively, and (ii) the remainder is given explicitly. Moreover, by using a recent method of Olde Daalhuis and Temme, we can extend the validity of the uniform asymptotic expansions to include all real values of z. 
Online Catalog Link:  http://lib.cityu.edu.hk/record=b2147190 
Appears in Collections:  MA  Master of Philosophy

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