C5.5 Perturbation Methods - Archived material for the year 2016-2017

2016-2017
Lecturer(s): 
Prof. James Oliver
General Prerequisites: 

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

Course Term: 
Michaelmas
Course Lecture Information: 

16 lectures

Course Weight: 
1.00 unit(s)
Course Level: 
M

Assessment type:

Course 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.

Course Synopsis: 

Introduction to regular and singular perturbation theory: approximate roots of algebraic and transcendental equations. Asymptotic expansions and their properties. Asymptotic approximation of integrals, including Laplace's method, the method of stationary phase and the method of steepest descent. Matched asymptotic expansions and boundary layer theory. Multiple-scale perturbation theory. WKB theory and semiclassics.

Reading List: 
  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.