Synopsis for B5a: Techniques of Applied Mathematics

Level: H-level Method of Assessment: Written examination.
Weight: Half-unit (OSS paper code 2A44)

Recommended Prerequisites:

Calculus of Variations and Fluid Mechanics from Part A are desirable but not essential. The introductory Michaelmas Term course B568a is a prerequisite for both parts of the course, and the material in that course will be assumed to be known.

Overview

This course develops mathematical techniques which are useful in solving `real-world' problems involving differential equations, and is a development of ideas which arise in the second year differential equations course. The course aims to show in a practical way how equations `work', what kinds of solution behaviours can occur, and some techniques which are useful in their solution.

Learning Outcomes

Students will know how differential equations can be used to model real-world phenomena and be able to describe the behaviour of the types of solutions that can occur. They will be familiar with the use of delta functions and the Fredholm Alternative and will be able to solve Sturm–Liouville systems. They will develop the theory of ODEs with regular singular points, including special functions.

Synopsis

Introduction to distributions; the delta function. Green's functions revisited. [3 lectures]

Fredholm alternative. [1 lecture]

Sturm–Liouville systems, adjoints, eigenfunction expansions. Integral equations and eigenfunctions. [6 lectures]

Singular points of differential equations; special functions. [4 lectures]

1. A. C. Fowler, Techniques of Applied Mathematics, Mathematical Institute Notes (2005).
2. J. P. Keener, Principles of Applied Mathematics: Transformation and Approximation (revised edition, Perseus Books, Cambridge, Mass., 2000).
3. E. J. Hinch, Perturbation Methods (Cambridge University Press, Cambridge, 1991).
4. J. R. Ockendon, S. D. Howison, A. A. Lacey and A. B. Movchan, Applied Partial Differential Equations (revised edition, Oxford University Press, Oxford, 2003).
5. R. Haberman, Mathematical Models (SIAM, Philadelphia, 1998).
6. S. D. Howison, Practical Applied Mathematics: Modelling, Analysis, Approximation (Cambridge University Press, Cambridge, 2005).