Synopsis for C6.3a: Perturbation Methods

Level: M-level Method of Assessment: Written examination.
Weight: Half-unit (OSS paper code 2A68)

Recommended Prerequisites

Part A Differential Equations and Core Analysis (Complex Analysis). B5, B6 and B8 are helpful but not officially required.

Overview

Perturbation methods underlie numerous applications of physical applied mathematics: including boundary layers in viscous flow, celestial mechanics, optics, shock waves, reaction-diffusion equations, and nonlinear oscillations. The aims of the course are to give a clear and systematic account of modern perturbation theory and to show how it can be applied to differential equations.

Synopsis

Asymptotic expansions. Asymptotic evaluation of integrals (including Laplace's method, method of stationary phase, method of steepest descent). Regular and singular perturbation theory. Multiple-scale perturbation theory. WKB theory and semiclassics. Boundary layers and related topics. Applications to nonlinear oscillators. Applications to partial differential equations and nonlinear waves.

1. E.J. Hinch, Perturbation Methods (Cambridge University Press, 1991), Chs. 1–3, 5–7.
2. C.M. Bender and S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (Springer, 1999), Chs. 6, 7, 9–11.
3. J. Kevorkian and J.D. Cole, Perturbation Methods in Applied Mathematics (Springer-Verlag, 1981), Chs. 1, 2.1–2.5, 3.1, 3.2, 3.6, 4.1, 5.2.