## Advanced Methods for Ordinary Differential Equations

SLN 10254
ARC G070, MWF 1:30-2:30pm
Prereqs: Differential Equations

### Instructor: Bernard Deconinck

Lewis 313
bernard@amath.washington.edu
Tel: 206-543-6069
Office Hours: M 10-11, T10-12

#### Course Description

Overview of perturbation methods and techniques from asymptotics, especially as applied to approximating solutions of differential equations.

#### Textbook

The 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.

#### Syllabus

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)