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.


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


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

  1. Asymptotic expansions.
  2. Algebraic equations.

    B. Integrals

  3. Laplace's method.
  4. The method of stationary phase.
  5. The method of steepest descents I.
  6. The method of steepest descents II.
  7. Stokes phenomenon.

    C. Boundary value problems

  8. Boundary layer theory: transition layers.
  9. Method of strained coordinates.
  10. Boundary layer theory for pdes: subcharacteristics.
  11. Helmholtz's equation, WKB method.

    D. Evolution equations

  12. Regular perturbation methods. Poincaré-Lindstedt method.
  13. The method of multiple scales.
  14. Method of averaging, Kuzmak's method.

    E. Exotic options

  15. Large activation energy asymptotics.
  16. Exponential asymptotics.


Course materials

These can be downloaded as postscript files; currently available are