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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
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 well-known 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 integration-by-parts procedure repeatedly. We prove the cubic transformation given by Chester, Friedman and Ursell is indeed one-to-one 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.
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