2007 Handbooks

This unit develops techniques and skills that are fundamental in the study and application of mathematics at an advanced level. In any successful application, two contrasting but complementary skills must be developed: the ability to formulate a given 'real-world' problem in appropriate mathematical terms, and sufficient knowledge to obtain useful information and testable predictions from that model, by analytical and numerical means.

The unit shows how differential equations arise as mathematical models of such real phenomena in science, engineering and the social sciences, and introduces some tools -- Fourier series and numerical methods -- for the study and eventual solution of these equations. Fourier series and transforms are particularly useful in those situations where the system response (and indeed many functions) can be seen as a complex sum of simpler vibrations or oscillations. When analytical methods fail, or provide only limited information about the model, numerical techniques are essential to quantify its behaviour precisely; some simple methods are introduced and the conditions under which reliable and accurate solutions may be obtained are described.