Brief description:

This is a graduate topics course on asymptotic methods and their applications in differential equations.

Mathematical modeling often involves differential equations (ODE's and PDE's) where exact solution formulas are either too complicated to yield useful information or simply not available. Asymptotic methods concern those regimes of solutions where certain physical parameters are large or small, and the hope is to simplify solutions order by order. Such physical parameters are usually associated with length scales of the modeling problems, for example, width of a phase transition layer, wave length of an oscillating medium, time scale of evolution etc. The asymptotic methods are complementary to direct numerical simulations.

We shall first cover:

•  Laplace method and stationary phase method, with applications to linear dispersive waves;
•  Regular/singular perturbation methods, matching asymptotics;
•  Geometric optics and weakly nonlinear asymptotics to linear/nonlinear wave equations;
•  Homogenization methods.

• Front and free boundary reduction of advection-reaction-diffusion models, and homogenization;
• Amplitude equations and modulation analysis of nonlinear Schrödinger equations;
• Fluid limit of kinetic equations.

Prerequisite:  The prerequisite is basic knowledge of real analysis, ODE and PDE.

Textbook:   None.

Consent of Instructor Required:  No.

Prof. Jack Xin

RLM 10.164

471-3149

Email: jxin@math.utexas.edu

Homepage: http://www.math.utexas.edu/users/jxin/