Math 703: Methods of Applied Mathematics 1

The course introduces methods to solve mathematical problems that arise in areas of application such as physics, engineering and statistics.  Roughly speaking, we can divide these problems into two categories: (i) equilibrium, i.e., statics problems and (ii) departures from equilibrium, i.e., 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.  Dynamics problems become initial value problems for ordinary and partial differential equations.  Thus we will study some well-known techniques for solving differential equations, e.g., Separation of Variables and Green's functions.  Asymptotic methods for the global analysis of ordinary differential equations will be introduced, e.g., boundary layer theory and WKB theory. The Calculus of Variations will enable us to understand the different formulations of mechanics (by Newton, Lagrange and Hamilton).  We will finish with an introduction to Fluid Dynamics.