Saverio Spagnolie, Fall 2020:
This is a first-year course for all incoming PhD and Master students interested in pursuing research in applied mathematics. The course introduces advanced mathematical methods with a particular focus on asymptotic analysis of algebraic, differential, and integral equations, including the methods of stationary phase, steepest descent, multiple scales, matched asymptotics, and WKB approximations. Other topics include divergent series, Fourier transform, complex integration, dimensional analysis, phase-plane analysis, Floquet theory, Green's functions, self-adjoint operators, Sturm-Liouville problems, and calculus of variations.
Gheorge Craciun, Fall 2016:
Previous description: The course introduces methods to solve mathematical problems that arise in areas of application such as physics, engineering, chemistry, biology, and statistics. Roughly speaking, we can divide these problems into two categories: (i) equilibrium (statics problems), and (ii) departures from equilibrium (dynamics problems). The first part of the course will be devoted to the study of equilibrium: linear algebra provides a unifying framework for discrete equilibrium problems from several application areas. This algebraic structure is also the basis for numerical solution of both discrete and continuous equilibrium systems. In the continuous case, equilibrium mechanics leads to boundary value problems for differential equations: in one dimension, one finds ordinary differential equations, e.g., Sturm-Liouville equations; for higher dimensional systems, one finds partial differential equations, e.g., Laplace's equation, Poisson's equation and the equations for Stokes flow. After review of some basic techniques for solving differential equations, asymptotic methods for the global analysis of ordinary differential equations will be introduced (boundary layer theory and WKB theory). The calculus of variations will also enable us to understand the different formulations of mechanics (by Newton, Lagrange and Hamilton).