Perturbation theory and calculus of variations
Module code: 840G1
15 credits in spring teaching
Teaching method: Lecture
Assessment modes: Unseen examination, Coursework
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
- change of units
- physical laws
- Buckingham Pi Theorem
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.