Perturbation theory and calculus of variations

Module code: 840G1
Level 6
15 credits in spring teaching
Teaching method: Lecture
Assessment modes: Coursework, Unseen examination

The aim of this module is to introduce you to a variety of techniques primarily involving ordinary differential equations, that have applications in various branches of applied mathematics. No particular application is emphasised.

Topics covered include –

Dimensional analysis and scaling:

  • physical quantities and their measurement
  • dimensions
  • change of units
  • physical laws
  • Buckingham Pi Theorem
  • scaling.

Regular perturbation methods:

  • direct method applied to algebraic equations and initial value problems (IVP)
  • Poincar method for periodic solutions
  • validity of approximations.

Singular perturbation methods:

  • finding approximate solutions to algebraic solutions
  • finding approximate solutions to boundary value problems (BVP) including boundary layers and matching.

Calculus of Variations:

  • necessary conditions for a function to be an extremal of a fixed or free end point problem involving a functional of integral form
  • isoperimetric problems.

Module learning outcomes

  • Systematic understanding of the concept of dimensions of physical quantities and how to express problems involving them in a dimensionless form using appropriate scaling.
  • Ability to apply perturbation methods and be able to handle problems that generate secular terms.
  • Ability to tackle singular perturbation problems using scaling to obtain the inner solution valid in the boundary layer.
  • Systematic understanding of the calculus of variations and its use in solving simple extremal problems.