Perturbation theory and calculus of variations (840G1)

15 credits, Level 6

Spring teaching

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.


100%: Lecture


20%: Coursework (Portfolio, Problem set)
80%: Examination (Computer-based examination)

Contact hours and workload

This module is approximately 150 hours of work. This breaks down into about 33 hours of contact time and about 117 hours of independent study. The University may make minor variations to the contact hours for operational reasons, including timetabling requirements.

We regularly review our modules to incorporate student feedback, staff expertise, as well as the latest research and teaching methodology. We’re planning to run these modules in the academic year 2022/23. However, there may be changes to these modules in response to COVID-19, staff availability, student demand or updates to our curriculum. We’ll make sure to let our applicants know of material changes to modules at the earliest opportunity.