Fall 2023 - APMA 900 G100

Asymptotic Analysis of Differential Equations (4)

Class Number: 2201

Delivery Method: In Person


  • Course Times + Location:

    Sep 6 – Oct 6, 2023: Tue, Thu, 2:30–4:20 p.m.

    Oct 11 – Dec 5, 2023: Tue, Thu, 2:30–4:20 p.m.



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.


Exact & Approximate Methods for Understanding DEs

One aim of this course is to provide an introduction to exact methods for the solution of ordinary and partial differential equations (ODEs & PDEs).  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.  A different generalization of the Fourier idea leads 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, particularly those involving nonlinearity.  Another aim is to introduce a variety of approximation methods that extend our analytical toolbox beyond exact theory.  Nonlinear ODEs systems provide many example contexts for the development of these powerful tools.  The results 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 that neglect complicating effects in the reduction of model equations. Finally, yet other types of asymptotic methods address singular situations where small changes to DE problems have a large impact on their solution.

Lectures will be based upon a case-study approach of ODE & PDE examples that draw from the interests of course participants. Computational graphics will be an important tool for the lectures and assigned work. Visualization and numerical computing will involve the use and modification of Python & Matlab scripts.

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 welcome to consider joining.)


  • Bi-weekly assignments, equally weighted 60%
  • Exam 1 15%
  • Exam 2 25%



Holmes, Mark H
New York, NY: Springer New York, 2009
online access from SFU library.


Applied asymptotic analysis / Peter D. Miller

Hard copy not currently available at WAC Bennett Library (QA 431 M477 2006)

Due to the remediation of the SFU library 5th floor, this book is currently
not accessible.  See the instructor if you are interested in this reference work.

Foundations of applied mathematics / Michael D. Greenberg.

Hard copy currently available at WAC Bennett Library (TA 330 G73 2013)

Applied Partial Differential Equations / J. Ockendon

Hard copy not currently available at WAC Bennett Library (QA 377 A675 1999)

Due to the remediation of the SFU library 5th floor, this book is currently
not accessible.  See the instructor if you are interested in this reference work.

Holmes, Mark H
Germany: Springer
Introduction to Perturbation Methods, 2013, Vol.20
online access from SFU library.


Your personalized Course Material list, including digital and physical textbooks, are available through the SFU Bookstore website by simply entering your Computing ID at: shop.sfu.ca/course-materials/my-personalized-course-materials.

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.

Registrar Notes:


SFU’s Academic Integrity website 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


Students with a faith background who may need accommodations during the semester are encouraged to assess their needs as soon as possible and review the Multifaith religious accommodations website. The page outlines ways they begin working toward an accommodation and ensure solutions can be reached in a timely fashion.