Analytical Methods is a graduate course run through the London Taught Course Centre. It will be given in 5 2-hour lectures, on Mondays from 10th Nov to 12th Dec 2025, from 15:15-17:15 at De Morgan House.

The major texts are Perturbation methods by Hinch; and PDEs with complex variables and transform methods by Weinberger.

Lecture notes

Exercise Sheets [ Week 1 | Week 2 | Week 3 | Week 4 | Week 5 ]

Worked Solutions [ Week 1 | Week 2 | Week 3 | Week 4 | Week 5 ]

Course structure

This course is a mixture of perturbation methods and PDEs with the emphasis being on the practical ability to solve problems rather than the theoretical underpinnings. I intersperse the two major subject areas each week, and try to provide examples which combine them. The topics covered each week are:

1. Introductory material. Dimensional analysis. First-order PDEs. Characteristics. Regular perturbation expansions.
   Link: Named dimensionless groups in heat and mass transfer.
2. Scaling. Similarity solutions. Homogeneous and inhomogeneous wave equation. Characteristics in second order systems. Stretching coordinates.
3. Scale and stretch with nonlinear DEs. Classification of PDEs. Separable solutions.
4. Laplace's equation: complex characteristics, analytic functions, Cauchy-Riemann equations. Conformal maps. Matched asymptotic expansions. Intermediate variables; finding the boundary layer.
5. Further boundary layer examples. Matching and logarithms. Steepest descents.

Lecturer contact details: Phone 020 7679 1302; Email helen.wilson@ucl.ac.uk.