For most model equations it is unlikely that exact solutions or convergent series approximations can be found. However, asymptotic expansions can give excellent approximations when there is some small, or large, parameter in the problem. The basic ideas are introduced and applied to integrals of practical importance that may be difficult to evaluate because of rapidly varying integrands. Applications to o.d.e.s are covered, with WKB and balance methods. Boundary layers, matched asymptotic expansions and multiple scales are introduced for both o.d.e.s and p.d.e.s.
Emphasis is put on practical problems that are difficult to solve numerically.
This course provides the theoretical and numerical methods for analysing the nonlinear dynamical behaviour of a wide class of important physical systems. The material reaches the research frontiers of this rapidly developing and very popular subject area. Topics discussed include centre manifold reduction, numerical methods, global and symbolic dynamics, resonance and tori, control and synchronisation, C-bifurcations in piecewise systems. The introductory part of this course is on the use of computers, packages, such as Maple, Matlab, etc. The numerical part covers the numerical solution of partial differential equations using finite-difference schemes for parabolic, elliptic and hyperbolic equations.