For most equations that arise in modelling applications it is unlikely that exact solutions or convergent series approximations can be found. However, asymptotic expansions can give good approximations, especially when there is some small or large parameter in the problem. They are usually divergent if summed to infinity but a few terms can often give excellent and well defined approximations. This unit introduces the basic ideas and shows how they can be applied to algebraic and differential equations, and to the evaluation of integrals. Usually coordinate values, or some parameter, are either large or small which leads to an expansion of a solution in a small parameter.
Most emphasis is placed on the latter, singular perturbations.
Practical problems are used as illustrations. These techniques are especially useful when numerical solutions are hard, or impossible, to obtain. The primary content of the course is taught using lectures, with reference to texts and the use of example sheets to reinforce the material presented. At the end of the unit, the students should be able to take a wide range of mathematical problems and modify the equations in order to find perturbation solutions for at least part of the parameter and coordinate range of interest. Assessment Method One three-hour examination in April comprising five questions, the best four answers to count.