Advanced Methods for Ordinary Differential Equations
ARC G070, MWF 1:30-2:30pm
Prereqs: Differential Equations
Instructor: Bernard Deconinck
Office Hours: M 10-11, T10-12
Course DescriptionOverview of perturbation methods and techniques from asymptotics, especially as applied to approximating solutions of differential equations.
TextbookThe textbook for this course is Carl Bender and Steven Orszag's Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory (Springer, 1978, reprinted 1999). This a classic text with a delightful writing style (and Sherlock Holmes quotes!). As all math books, it is unreasonably priced ($99 at the publishes, $79.20 at one of the better-known on-line vendors), but it should be a great resource for you far beyond this course. Since there's only one edition of this book, you have no reason not to look for used copies.
Other useful books from which on occasion material will be used are:
- Peter Miller, Applied Asymptotic Analysis, AMS, 2006
- Frank Olver, Asymptotics and Special Functions, AKP Classics, 1997
- William Paulsen, Asymptotic Analysis and Perturbation Theory, CRC Press, 2013
Course Canvas Page
I will use Canvas to post homework sets, link to the class message board, etc. You will need a UW account and be enrolled in the course to access this page.
The following topics will be covered, time permitting, in some order to be decided.
- Approximate solutions of linear differential equations (classification of singular points, local behavior, the method of Fuchs and Frobenius, asymptotic series)
- Asymptotic expansion of integrals (Watson's lemma, Laplace's method, stationary phase, steepest descent)
- Perturbation series (regular perturbation theory, singular perturbation theory, matching)
- Boundary layer theory (inner, outer and intermediate limits)
- WKB theory (turning point problems, tunneling)
- Multiple scale analysis (resonance and secular behavior)
Homework sets are assigned weekly. Homework is due at the beginning of class on its due date. Late homework is not accepted. Every homework set you hand in should have a header containing your name, student number, due date, course, and the homework number as a title. Your homework should be neat and readable. Your homework score may reflect the presentation of your homework set. Your course grade will be calculated by weighing your homework, midterm, and final exam scores in the proportions 55%, 15%, and 30%, respectively.