In many cases, the understanding and interpretation of exact solutions requires numerical computation and/or further approximation. The study of asymptotic and perturbation methods provides a systematic framework for the approximate analysis of PDE solutions. Here, the presentation will begin from the far-field approximation of integral solutions, and continue onto developments in multiple-scale, averaging and boundary-layer methods.
Lectures will be based upon a case-study approach of PDE examples. Computational illustration will be an important tool for the lectures and assigned work. The rudiments of numerical computing will be developed through the use and modification of downloadable Matlab scripts.
The purpose of this meeting will be to discuss the interests of the class members, explain the computing environments, and answer general questions about the course. Registered students who miss this meeting should arrange an alternative meeting with the instructor ASAP.
The image above shows the streamlines and wind
vectors for a surface dipole from a model of atmospheric flow. The
dipole is obtained as a solution to a three-dimensional Laplace PDE
with mixed boundary conditions. Its computation involves a numerical
implementation of an integral Hankel transform -- an example
of applied special function theory.