Math 744    Asymptotic Analysis   Fall 2010
 

DR. N. KEVLAHAN

Office: HH 324, Tel: x23412
Email: kevlahan@mcmaster.ca
Web: www.math.mcmaster.ca/kevla
Office hours: by appointment.

Lectures: Tuesdays and Thursdays, 12:30-14:00 in HH 312.

Purpose of the course

We will cover techniques from advanced applied mathematics which are especially useful in the physical sciences and engineering. Highlights include: asymptotic expansions, singular perturbation theory, multiple scales analysis, boundary layer theory, and WKB theory.

The mathematical methods will be illustrated with results from physics and engineering. The emphasis will be on the use of these methods to solve problems arising in the physical sciences. In particular, we will point out how to use these analytical methods to complement numerical methods when tackling difficult problems in physics and engineering.

Text

The primary text for the course is Advanced Mathematical Methods for Scientists and Engineers by C.M. Bender & S.A. Orszag (Springer) and the secondary text is Perturbation Methods by E.J. Hinch (Cambridge University Press).

Outline

1. Introduction

• What are asymptotics and perturbations (convergence and asymptoticness)?
• Why asymptotic?
• How are they used?
• Why use analytical methods when powerful computers are available?

2. Asymptotic expansion of integrals

• Integration by parts.
• Laplace's Method and Watson's Lemma.
• Method of Stationary Phase.
• Method of Steepest Descents.
• Non-local contributions (splitting the range of integration).

3. Perturbation series solution

• Regular perturbation problems in partial differential equations.
• Singular perturbations.
• Asymptotic matching.

4. Boundary layer theory

• Outer approximation.
• Inner approximation (i.e. boundary layer solution).
• How thick is the boundary layer (i.e. distinguished limits)?
• Logarithms.
• Where is the boundary layer (internal boundary layers)?
• Composite approximations.

5. Multiple scales methods

• van der Pol oscillator.
• Multiple scales analysis.
• WKB approximation.
• Inhomogeneous WKB.
• Floquet theory.

Course notes

Weeks 1-3
Week 4
Week 5
Weeks 6-7
Week 8
Week 9
Week 10
Week 11
Week 12

Maple worksheets

Boundary layers
Multiple scales
WKB1

Evaluation

There will be four assignments, and a final exam.

Assignments

Four assignments will be given:

Assignment given	Assignment due	Assignment	Solutions
23 September	7 October	Assignment 1	Assignment 1 solutions
14 October	28 October	Assignment 2	Assignment 2 solutions
4 November	18 November	Assignment 3	Assignment 3 solutions
18 November	2 December	Assignment 4	Assignment 4 solutions

Final exam

There will be a take-home final exam covering the entire course:

Exam given	Exam due	Exam	Solutions
15 December (9:30 am)	17 December (9:30am)	Exam

Final mark

The final mark will be calculated as follows:
 

Homework	60%
Final exam	40%

I reserve the right to change the weight of any portion of this marking scheme. If changes are made, your grade will be calculated using the original weightings and the new weightings, and you will be given the higher of the two grades. At the end of the course the grades may be adjusted but this can only increase your grade and will be done uniformly. I will use the grade equivalence chart in the university calendar to convert between letter grades, grade points and percentages.

Official notices

Excused Absences
Exemptions from the assignments or tests for valid reasons are possible, but must be requested through the
office of the Associate Dean of the Faculty that you are registered with. In the event of an exemption, no
make up test or assignment will be administered, but your course grade will be re-weighted by increasing
the weight of the final examination to compensate for the missed test or assignment.

Academic Integrity
You are expected to exhibit honesty and use ethical behaviour in all aspects of the learning process. Academic
credentials you earn are rooted in principles of honesty and academic integrity.

Academic dishonesty is to knowingly act or fail to act in a way that results or could result in unearned
academic credit or advantage. This behaviour can result in serious consequences, e.g. the grade of zero
on an assignment, loss of credit with a notation on the transcript (notation reads: Grade of F assigned for
academic dishonesty), and/or suspension or expulsion from the university.

It is your responsibility to understand what constitutes academic dishonesty. For information on the
various types of academic dishonesty please refer to the Academic Integrity Policy, located at:

www.mcmaster.ca/academicintegrity

The following illustrates only three forms of academic dishonesty:

1. Plagiarism, e.g. the submission of work that is not one’s own or for which other credit has been obtained.
2. Improper collaboration in group work.
3. Copying or using unauthorized aids in tests and examinations.

Attention is drawn to the "Statement on Academic Ethics'' and "Senate Resolutions on Academic Dishonesty'' as found in the Senate Policy Statements distributed at Registration and available in the Senate Office. Any student who infringes on one of these resolutions will be treated according to the published policy. In particular, it is expected that the assignments shall be done and submitted as individual work.  Students may discuss general problems or approaches, but the final solution must be a result of the student's own effort.

The Faculty of Science is concerned with ensuring an environment that is free of all adverse discrimination. If there is a problem that cannot be resolved by discussion among the persons concerned, individuals are reminded that they should contact their Department Chair, the Sexual Harassment Office or the Human Rights Consultant, as soon as possible.

2010-12-02