Time: Mondays and Wednesdays 1:15-2:30 pm, Physics 235
Many mathematical problems do not admit explicit solutions, so it is very useful to have methods for approximating their solution. Asymptotic analysis and perturbation methods provide powerful techniques for obtaining approximate solutions to complicated problems. Often these problems involve a parameter that is very large or very small, and one wants to exploit this structure to obtain a good approximation. These techniques are very useful in many different fields of research. For example, they have been applied to mathematical problems in fluid dynamics, quantum mechanics, classical mechanics, wave propagation, imaging, population biology, ordinary and partial differential equations, number theory, combinatorics, probability and stochastic processes, and many other fields.
This course will focus on asymptotics of integrals and asymptotics of ordinary differential equations; there also will be some discussion of asymptotic problems for algebraic equations and for partial differential equations. In particular, the course covers: regular and singular perturbation, asymptotic expansions, Laplace's method, the method of stationary phase, the method of steepest-descent, WKB theory, asymptotics of boundary value problems, boundary layers, multiple-scale analysis, and matched asymptotic expansions, and an introduction to homogenization theory for PDEs.
The course is a graduate level course intended for students in the sciences, engineering, statistics, and mathematics. Students outside the mathematics department are encouraged to register.
The prerequisites include ordinary differential equations and complex analysis at the undergraduate level.