Emphasis is placed on methods of developing asymptotic solutions. For most equations that arise in modelling applications it is unlikely that exact solutions can be found. Even convergent series approximations are often not available, or they are of limited use if they converge very slowly. They are typically 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.
Most emphasis is placed on the latter, singular perturbations.
Practical problems are used as illustrations. These techniques are especially useful when accurate numerical solutions are hard, or impossible, to obtain. This unit is a sequel to Level 3 Methods , and develops further techniques useful throughout applied mathematics. It is a prerequisite for Waves and Instabilities . The primary content of the course is taught using lectures, with reference to texts and the use of problem sheets to reinforce the material presented.