Syllabus Detail

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.

MAT 207C: Methods of Applied Mathematics

Approved: 2015-10-01, Joseph Biello and John Hunter
Suggested Textbook: (actual textbook varies by instructor; check your instructor)
Introduction to Perturbation Methods, Mark H. Holmes, 2nd Ed., Texts in Applied Mathematics 20, Springer, 2013. (An e-book is available from the UCD Library)
Graduate standing or consent of 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

The recommendation is one in-class midterm exam and a 2 hour, timed final exam during final exam week.