Mathematics

Perturbation theory and calculus of variations

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

The aim of this module is to introduce you to a variety of techniques, involving primarily ordinary differential equations, that have applications in several branches of mathematics applied to the natural and social sciences. Once a practical problem is modelled and the relevant ordinary differential equations are constructed, techniques such as dimensional analysis and scaling are used to check that the model is correct and to reduce the equations to their simplest form.

Since many of these equations do not have explicit/closed-form solutions, the module will introduce regular and singular perturbation methods which give rise to approximate solutions.

The final part of the module introduces concepts from the calculus of variations which studies functions whose input is another function rather than a numerical value, known as functionals. The goal is to then find the extremals of these functionals, that is functions that minimise or maximise the output of the functional, with many real-world problems falling under this setting. For example, the isoperimetric problem which aims to determine a plane figure of the largest possible area whose boundary has a specified length is a special application of the calculus of variations.

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.