C5.5 Perturbation Methods - Material for the year 2019-2020

2019-2020
Lecturer(s): 
Prof. Eamonn Gaffney
General Prerequisites: 

Knowledge of core complex analysis and of core differential equations will be assumed, respectively at the level of the complex analysis in the Part A (Second Year) course Metric Spaces and Complex Analysis and the phase plane section in Part A Differential Equations I. The final section on approximation techniques in Part A Differential Equations II is highly recommended reading if it has not already been covered

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.