Asymptotic Analysis - Fall 2025
Instructor
Zhao Yang
Email: yangzhao@amss.ac.cn
819 Siyuan Hall
Office hours: M 1-2p, W 1-2p, or by appointment
Course
info
Time and Place: M 10:00-11:40, TBD.
Textbook: Introduction to Perturbation Methods, Second Edition, by Mark H. Holmes.
Notes: Lecture notes will be posted after classes. However, they cannot be used as a substitute of your textbook.
Aim and Scope:
This course introduces the fundamental ideas and techniques of
asymptotic analysis and perturbation methods, which play a
central role in applied mathematics, physics, and engineering. Many
real-world problems involve small (or large) parameters for which exact
solutions are either unavailable or impractical. Asymptotic methods provide
systematic approximations that capture the essential features of such
problems and yield insight into their behavior.
The course emphasizes both methodology and applications:
- Developing asymptotic expansions and understanding their validity.
- Distinguishing between regular and singular perturbations.
- Applying key tools such as boundary layer theory, matched asymptotic
expansions, the method of multiple scales, and WKB analysis.
- Exploring applications to ordinary differential equations,
with selected extensions to partial differential equations
and physical models.
By the end of the course, students will be able to recognize perturbation
structures in mathematical models, construct appropriate asymptotic
approximations, and interpret the results in the context of physical and
engineering problems.
Prerequisite: This course will require undergraduate background in Mathematical Analysis, Linear Algebra, Ordinary Differential Equations. However, the more a student brings to the course, the
more the student will get out of it, so graduate courses in these areas can only help. Interested students who are not sure if they have sufficient background are encouraged to email me and discuss their readiness.
Grading Policy: homework 60% (six assignments 10% each); Final 40%.
Homework: Mathematics (and problem solving in general) is a collaborative
discipline. You are strongly encouraged to work in groups and to discuss homework problems with your classmates. However, you
must write-up the solution on
your own and it
must be
in your own words. Anything else is plagiarism and will be treated as
such.
Your solution needs to be complete and correct (of
course!), but to recieve full credit your write-up should also meet the
following criteria.
-
All of the important logical steps in the proof should be present and fully explained.
- All assumptions should be clearly identified.
- Your write-up should be clear and concise. (I.e., if a sentence or
paragraph does not advance the reader's understanding of the solution,
it does not belong in the write-up.)
- Use full and complete English sentences. Symbols should be used
only as a tool to distill a complex mathematical relationship into
a readable format.
I
highly suggest that you write up your solution sets using the typsetting program LaTeX which is designed for the production of technical and scientific documentation.
Overleaf is an online LaTeX editor that allows real-time collaboration and online compiling of projects to PDF format. This online editor also provides a number of beautiful templates for completing math homework assignments.
Materials
I will upload lecture notes and homework assignments here as the course progresses.