Course Overview

Covers asymptotic evaluation of integrals (stationary phase and steepest descent), perturbation methods (regular and singular methods, and inner and outer expansions), multiple scale methods, and applications to differential and integral equations.

Approximate, asymptotic methods are indispensable tools for the applied mathematician, in league with computational and exact solution methods. Very often in applications, there are physical, nondimensional parameters p ∈ Rn that are small, yet essentially nonzero. Perturbative methods allow one to extend knowledge of the (often well-understood) p = 0 case to the small |p| case and beyond. These tools are not only beneficial for applications in the physical, biological, and social sciences themselves but also for numerical methods where many computational difficulties arise when parameters are small, e.g., the numerical solution of stiff ODEs or PDEs. Also, many precise results in analysis, number theory, and probability/statistics are formulated in terms of asymptotics.

Course Information

Primary Text: C. M. Bender, S. A. Orszag Advanced Mathematical Methods for Scientists and Engi- neers, Springer, NY 1999.

Additional Text: E. J. Hinch Perturbation Methods, Cambridge University Press, Cambridge, UK 1991; P. D. Miller Applied Asymptotic Analysis, American Mathematical Society, Providence, RI 2006.

Lecture Times and Location

Instructor Room Number Time
Keith Julien ECCR 257, APPM Newton Lab MWF 11 to 11.50AM

Office Hours

Instructor/TA Room Number Office Hours
Keith Julien  ECOT 321 W 1 to 3PM

Homeworks

Seven approximately bi-weekly problem sets will be assigned. You are encouraged to discuss homework with other students or with me. You must write up your own work legibly and clearly. It is self evident that you must comprehend the material and be able to solve the problems on your own. Homework will be graded on a 0-100 scale based on the amount attempted and detailed grading of selected problems. Your time and my time are valuable. Therefore, if I cannot understand your results (e.g. incomprehensible proofs, poorly written calculations, etc.), then you will get the problem wrong. Homework is due by 5pm on the due date. Late homework will not be accepted.

Projects

Individual projects will be a substantial component of the course. Projects should based on exsiting or published works where asymptotic or perturbation methods are emphasized in a significant way.

Grading: 60% for seven problem sets, 20% for in-class 25 minute project presentation, 20% for project report. The standard grading scale, subject to possible downshifting, will be used, e.g. 90- 100% A, 80-89% B, etc.

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