Asymptotic solutions of a fourth order differential equation

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.

Date of Award	14 Jul 2006
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Sue Cheun Roderick WONG (Supervisor)