Introduction to Perturbation Methods

Volume: 10 weeks × (2 lecture+1 seminar periods)/Week

Textbook: M.H. Holmes, Introduction to Perturbation Methods, Texts in Applied Mathematics 20, 1995, Springer Verlag (Reprinted by Beijing World Publishing Corporation, 1999; Cover Price 48.00 RMB)

Prerequisites: Calculus 156/186. Some knowledge of ordinary differential equations is required, so students who attended the course “Differential Equations” in the fall term 2007 will feel comfortable. However, the basic knowledge taught in Calculus 156 (Chapter 9 of Stewart's Calculus) and 186 should be sufficient for dedicated students. Elementary concepts will be refreshed as necessary.

The number of participants is limited to 50. Please register for the course with Ms. Miao of the Academic Services office by the 24th of January. Participants will be admitted on a “first come - first served” basis. The participants will be divided into groups of at most five students each, and each group will hold a presentation on an application (as given in the textbook's exercises) in the 45-minute seminar period.

Contents: The lecture is intended as an introduction to methods of asymptotic analysis and perturbation theory, with a strong emphasis on applications in the sciences. The lecture will cover basic methods, such as asymptotic analysis, matched asymptotic expansions, multiple scales and WKB approximations and their applications to solving ordinary differential equations in physics, engineering and chemistry. In particular, approximate solutions to nonlinear differential equations and physical problems such as forced oscillations will be discussed. In order to keep the prerequisites required of the participants to a minimum, applications to partial differential equations and difference equations will not be discussed, even though they are treated in the textbook.

Overview: This overview will be kept up to date to reflect the real progress made in the lecture.

Week	Lecture Subject	Chapters
1	Introduction, Landau symbols, and asymptotic approximations	1.1-1.4
2	Asymptotic expansions and solutions to algebraic equations	1.4-1.5
3	Solutions to transcendental equations and ordinary differential equations; Uniformity	1.5-1.7
4	Matched Expansions - Introduction	2.1-2.2
5	Matched Expansions - Boundary Layers	2.3-2.4
6	Matched Expansions - Interior Layers	2.4-2.5
7	Multiple Scales - Introduction	3.1-3.2
8	Multiple Scales - The pendulum	3.3

Materials:

Lecture Slides (for beamer): Full Lecture, Chapter 1, Chapter 2, Chapter 3.
Lecture Slides (A4 format for printing): Full Lecture, Chapter 1, Chapter 2, Chapter 3.
Mathematica Notebooks (for beamer): Chapter 1, Chapter 2, Chapter 3.

Presentations by students: