Overview

Description

CALENDAR DESCRIPTION:

Analysis and computation of classical problems from applied mathematics such as eigenfunction expansions, integral transforms, and stability and bifurcation analyses. Methods include perturbation, boundary layer and multiple-scale analyses, averaging and homogenization, integral asymptotics and complex variable methods as applied to differential equations.

COURSE DETAILS:

Exact & Approximate Methods for Understanding DEs

One aim of this course is to provide an introduction to exact methods for the solution of ordinary and partial differential equations (ODEs & PDEs).  Fourier series methods for solving linear DEs are extended to integral solution methods that include the Fourier and Laplace transforms.  Investigation of this solution perspective establishes the close connection between complex variable theory and DEs.  A different generalization of the Fourier idea leads to the development of Sturm-Liouville eigenfunctions, function (Hilbert) spaces and special function theory.

But many ODEs and PDEs encountered in applications are not amenable to exact solution, particularly those involving nonlinearity.  Another aim is to introduce a variety of approximation methods that extend our analytical toolbox beyond exact theory.  Nonlinear ODEs systems provide many example contexts for the development of these powerful tools.  The results can also be useful in benchmarking numerically-computed solutions, and even decoding exact solutions whose formula complexity defies interpretation.  

Perturbation theory analyzes problems that are "nearby" to those with known exact properties.  This perspective also gives mathematical insight into the consequences of approximations that neglect complicating effects in the reduction of model equations. Finally, yet other types of asymptotic methods address singular situations where small changes to DE problems have a large impact on their solution.

Lectures will be based upon a case-study approach of ODE & PDE examples that draw from the interests of course participants. Computational graphics will be an important tool for the lectures and assigned work. Visualization and numerical computing will involve the use and modification of Python & Matlab scripts.

Calendar course prerequisites:  Undergraduate introduction to ODEs and linear PDEs.  Other useful background includes real & complex analysis, elementary numerical analysis &/or scientific computing.  (SFU undergraduates with Math 418 credit are welcome to consider joining.)

Grading

Materials

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