Fall 2019 - APMA 900 G100
Asymptotic Analysis of Differential Equations (4)
Class Number: 4137
Delivery Method: In Person
Analysis and computation of classical problems from applied mathematics such as eigenfunction expansions, integral transforms, and stability and bifurcation analyses. Methods include perturbation, boundary layer and multiple-scale analyses, averaging and homogenization, integral asymptotics and complex variable methods as applied to differential equations.
One aim of this course is to provide an overview of ordinary and partial differential equations (ODEs & PDEs) that are solved by exact methods. Fourier series methods for solving linear DEs are extended to integral solution methods that include the Fourier and Laplace transforms. Investigation of this solution perspective establishes the close connection between complex variable theory and DEs. Other generalizations lead to the development of Sturm-Liouville eigenfunctions, function (Hilbert) spaces and special function theory.
But many ODEs and PDEs encountered in applications are not amenable to exact solution. The other aim of this course is to introduce a variety of asymptotic methods that extend our analytical toolbox beyond exact theory. These approximate approaches can also be useful in benchmarking numerically-computed solutions, and even decoding exact solutions whose formula complexity defies interpretation. Perturbation theory analyzes problems that are “nearby” to those with known exact properties. This perspective also gives mathematical insight into the consequences of approximations which neglect complicating effects in the development of model
equations. Yet other types of asymptotic methods address singular situations where small changes to DE problems have a large impact on the solution. Examples include techniques known as: multiple-scale, averaging, WKB (geometrical optics) and boundary-layer methods.
Lectures will be based upon a case-study approach of ODE & PDE examples. Computational illustration will be an important tool for the lectures and assigned work. Computer visualization and numerical computing will involve the use and modification of Matlab scripts.
- Bi-weekly assignments, equally weighted 60%
- Exam 1 15%
- Exam 2 25%
Calendar course prerequisites: Undergraduate introduction to ODEs and linear PDEs. Other useful background includes real & complex analysis, elementary numerical analysis &/or scientific computing. (SFU undergraduates with Math 418 credit are encouraged to consider joining.)
Introduction to perturbation methods [electronic resource] / Mark H. Holmes.
NOTE: This is available for unlimited user access via the SFU library.
Applied asymptotic analysis / Peter D. Miller
Hard copy available at WAC Bennett Library (QA 431 M477 2006)
Foundations of applied mathematics / Michael D. Greenberg.
Hard copy available at WAC Bennett Library (TA 330 G73 2013)
Applied Partial Differential Equations / J. Ockendon
Hard copy available at WAC Bennett Library (QA 377 A675 1999)
Graduate Studies Notes:
Important dates and deadlines for graduate students are found here: http://www.sfu.ca/dean-gradstudies/current/important_dates/guidelines.html. The deadline to drop a course with a 100% refund is the end of week 2. The deadline to drop with no notation on your transcript is the end of week 3.
SFU’s Academic Integrity web site http://www.sfu.ca/students/academicintegrity.html is filled with information on what is meant by academic dishonesty, where you can find resources to help with your studies and the consequences of cheating. Check out the site for more information and videos that help explain the issues in plain English.
Each student is responsible for his or her conduct as it affects the University community. Academic dishonesty, in whatever form, is ultimately destructive of the values of the University. Furthermore, it is unfair and discouraging to the majority of students who pursue their studies honestly. Scholarly integrity is required of all members of the University. http://www.sfu.ca/policies/gazette/student/s10-01.html
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