## Department of Mathematics Syllabus

This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.

**Approved:**2015-10-01, Joseph Biello and John Hunter

**Suggested Textbook:**(actual textbook varies by instructor; check your instructor)

**Suggested Schedule:**

Selected material from Chapters 1-4. Basic sections are listed below.

### Asymptotic approximations (Chapter 1.1--1.5 and Appendix C)

Non-dimensionalization and scaling (2 lectures)

Regular versus singular perturbations (2 lectures)

- Introductory examples
- Algebraic equations
- Dominant balance and distinguished limits

Asymptotic expansions (2 lectures)

- Big "oh" and little "oh" notation
- Gauge functions
- Asymptotic versus convergent series

Asymptotic expansion of integrals (3 lectures)

- Integration by parts
- Laplace's method
- Stationary phase

### Method of matched asymptotic expansions

(Chapter 2.1--2.3)

Initial layers (3 lectures)

- Inner and outer expansions and matching
- Fast/slow systems

Two-point boundary value problems (4 lectures)

- Boundary layers
- Matched asymptotic and composite solutions
- Examples

### Method of multiple scales (Chapter 3.1--3.4)

Failure of regular perturbation theory (1 lecture)

- Secular terms
- Solvability conditions

Poincare-Lindstedt method for periodic solutions (3 lectures)

- Stretched variables
- Conservative systems
- Limit cycles

Multiple scale expansions (5 lectures)

- Forced/damped nonlinear oscillators
- Examples

### WKB method (Chapter 4.1--4.2)

WKB solution and applications (2 lectures)

Total 27 Lectures

**Assessment:**