What is the “best” way to approximate a function? Taylor series can provide a great local representation but may diverge outside a disappointingly small radius of convergence, as with the witch of Agnesi. In contrast, Fourier series and other spectral methods have been developed to provide a globally convergent approximation, and are often exceptionally efficient at doing so.
The goal of this course is to study spectral methods for problems arising in ordinary and partial differential equations, such as solving boundary value problems and finding eigenvalues/eigenfunctions. The course will follow the book “Chebyshev and Fourier Spectral Methods” by John P. Boyd, with supplemental material from “Spectral Methods in MATLAB” by Lloyd Trefethen and other sources. Programming in the course will be in MATLAB.
This course is a seminar and, unlike a traditional lecture course, almost all of the talking in our meetings is done by the students. The main work of the seminar meeting will be presentations on selected topics, discussion of the readings and presentations, discussion of student generated questions, and presentation/discussion of homework problems. For details of how a similar class has run in the past, click here.
Prerequisite:Ideally students will have taken (or are concurrently taking) MA 776 (Partial Differential Equations), and will have some basic computer programming experience. However this is not strictly necessary. If you have any questions about whether this course is right for you, feel free to send me an email!