## Perturbation methods

### Aims and objectives

Perturbation methods underlie almost all applications of physical
applied mathematics: for example, in boundary layer theory of 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 how it can be applied to differential
equations.
#### Prerequisites

a1 differential equations;
a2 complex analysis;
(b5 and b6 are relevant but not essential).
### Synopsis

Asymptotic expansions. Regular perturbation methods:
Poincaré-Lindstedt method, method of multiple scales, method of
averaging. Application to partial differential equations. Singular
perturbation theory: boundary layers, transition layers; application
to partial differential equations; WKB method. Asymptotic evaluation
of integrals: Laplace's method, method of steepest descent; Stokes
phenomenon, exponential asymptotics.
### Outline of lectures

#### A. Preamble

- Asymptotic expansions.
- Algebraic equations.
#### B. Integrals

- Laplace's method.
- The method of stationary phase.
- The method of steepest descents I.
- The method of steepest descents II.
- Stokes phenomenon.
#### C. Boundary value problems

- Boundary layer theory: transition layers.
- Method of strained coordinates.
- Boundary layer theory for pdes: subcharacteristics.
- Helmholtz's equation, WKB method.
#### D. Evolution equations

- Regular perturbation methods. Poincaré-Lindstedt method.
- The method of multiple scales.
- Method of averaging, Kuzmak's method.
#### E. Exotic options

- Large activation energy asymptotics.
- Exponential asymptotics.

### Reading

- E.J. Hinch 1991 Perturbation methods. C.U.P., Cambridge. Chs. 1-3,
5-7.
- C.M. Bender and S.A. Orszag 1978 Advanced mathematical methods
for scientists and engineers. McGraw-Hill, New York. Chs. 6, 7, 9-11.
- J. Kevorkian and J.D. Cole 1981 Perturbation methods in applied
mathematics. Springer-Verlag, Berlin. Chs. 1, 2.1-2.5, 3.1, 3.2, 3.6,
4.1, 5.2.

### Course materials

These can be downloaded as postscript files; currently available are