Synopsis for Mathematical Methods

• Calculus of Variations [2 lectures]: This is covered in many text books, for example Keener (Chapter 5) or Hildebrand (Chapter 2).
• Optimal Control [1 lecture]: Many real problems are control problems and this lecture will show how the ideas of the calculus of variations can be used to solve a special class of control problems. This section is covered in Hocking (Chapters 1–4).
• Nonlinear Analysis [3 lectures]: This section overlaps with the optional course on Nonlinear Systems so that those of you who want to take the subjct further can take that course next term. I will start with a very quick revision of phase plane methods which I will assume everyone knows (see Jordan and Smith (Chapters 1 and 2)) followed by two main topics:
  1. Bifurcation theory (Drazin (Chapters 1 and 2));
  2. Poincaré Lindstedt method (Howison (Chapter 22)).

The aim of this course is to show you some techniques that are not covered in the Mathematical Methods course. It will be a fast run through of the basic ideas — hopefully giving you enough information to find out more for yourselves if you need to use any of these ideas. Some of you may have already covered some of the material, for example you may have done a course on the calculus of variations already. Nevertheless you should still try the examples and you may find the lectures useful: the emphasis here will be on using the methods to solve problems rather than on proving theorems.

1. J. P. Keener, Principles of Applied Mathematics: Transformation and Approximation (revised edition, Perseus Books, Cambridge, Mass., 2000).
2. F. B. Hildebrand, Methods of Applied Mathematics (2nd edition, Dover Publications, 1992).
3. L.M. Hocking, Optimal Control: An Introduction to the Theory with Applications (Oxford University Press, 1991).
4. D. W. Jordan and P. Smith, Nonlinear Ordinary Differential Equations, An Introduction to Dynamical Systems (4th Edition, Oxford University Press, 2007).
5. P. G. Drazin, Nonlinear Systems (Cambridge University Press, Cambridge, 1992).
6. S. D. Howison, Practical Applied Mathematics: Modelling, Analysis, Approximation (Cambridge University Press, 2005).