Description: Differential equations model most natural phenomena we know. Yet their solutions can be notoriously difficult to understand. In place of "exact"solutions, a rich array of asymptotic methods have been developed to qualitatively understand the solutions. These are based on either the intrinsic variables or an external parameter in the problem being large or small. The field is vast and ranges from the fundamental theory of asymptotic expansions and perturbation methods, developed by Poincare for the study of the solar system, to modern advanced techniques which deal with cases where conventional asymptotics fails. This course will include asymptotics of boundary layers, WKB and multiscale methods (depending on the background of the class) and modern techniques developed to model dendritic growth (such as snowflakes), fluid flow (existence of solitary waves), and the onset of chaos. The only background needed is the basic theory of differential equations and complex analysis.

Prof. Nalini Joshi