Advanced Analytic Methods in Science and Engineering

Syllabus

Course Meeting Times

Lectures: 2 sessions / week, 1.5 hours / session

Course Outline

Each of the first five chapters in the textbook were covered briefly in one lecture. The discussions did not cover all the topics in the chapter. Rather, they aimed to communicate the basics and make it easier for the students to read the materials in these chapters on their own. While these materials were not tested in the exams, they are fundamental and it is important for the students to learn them well. It was not a requirement for students to hand in the homeworks in the lectures, but students had to do as many as they could on their own.

The following topics were covered in detail:

1. Series solutions of linear homogeneous ODE. Discussed how to make expansions around irregular as well as regular singular points.

2. The WKB Method. Covered not only the lowest-order WKB approximation but also the higher-order WKB approximations.

3. Asymptotic expansion of integrals. Covered the Laplace method, the method of stationary phase, and the saddle point method.

4. Boundary layer theory. Demonstrated how to solve the standard boundary layer problems. Discussed some of the boundary layer problems either considered unsolvable or solved incorrectly in the literature.

5. Small non-linear oscillations. This includes the two-scale method, the counter term method in renormalized perturbation and the method of renormalization group.

Prerequisites

Complex Variables with Applications (18.04), or Advanced Calculus for Engineers (18.075) or Functions of a Complex Variable (18.112)

Textbook

Cheng, Hung. Advanced Analytic Methods in Continuum Mathematics: Fundamentals for Science and Engineering. Boston, MA: Luban Press, 2004. ISBN: 0975862502.

Reference Books

Bender, C., and S. Orszag. Advanced Mathematical Methods for Scientists and Engineers. New York, NY: McGraw-Hill, 1978. Reprinted by Springer-Verlag, New York, 1998. ISBN: 0387989315.

Mei, C. C. Mathematical Analysis in Engineering. Cambridge, England: Cambridge University Press, 1995. ISBN: 0521460530.

Exam

Two open-book exams. 40% each of total grade. No final.

Homeworks

Homeworks assigned will be collected on the following Monday. Late homeworks will not be accepted. You may discuss the homework problems with others, but you must write up the solutions on your own. They constitute 20% of total grade.

ACTIVITIES	PERCENTAGES
Homeworks	20%
Exams	80%