Methods of Asymptotic Analysis- Math 550/605

Office hours: By appointment (email usually works best)

PDE's: Review Exercises

Project Deadlines:
March 21: choose project (inform instructor of your choice)
April 27: hand-in project

Suggestions for Projects
You may also choose your own project - subject to instructor approval!

Localized Perturbation References
  • References (PS)

    WKB methods
    Naeh, T.; Klosek, M. M.; Matkowsky, B. J.; Schuss, Z. A direct approach to the exit problem. SIAM J. Appl. Math. 50 (1990), no. 2, 595--627.
    Large deviation theory for stochastic difference equations, R. Kuske and J.B.Keller, EJAM 8 (1997),567-580. (PDF)
  • Additional WKB references (PS)

    Turing Patterns

    Long wavelength instabilities of square patterns, R. Hoyle, Physica D, 67(1993), p. 198-223. Domain Boundaries in Convection patterns, B. Malomed, A. Nepomnyaschy, M. I. Tribelsky, Phys. Rev. A, 42,1990, 7244-. Nonpotential effects in dynamics of fronts between convection patterns, A. Hari and A. A. Nepomnyaschy, Phys. Rev. E, 61, 4835-.
  • Additional Pattern references (PS)

    Free boundaries
    American options with dividends near expiry , J.D. Evans, R. Kuske, and Joseph B. Keller, Mathematical Finance, 2002.

    Project Guidelines
    No two students may choose the same paper.
    Read the selected paper, and fill in ALL mathematical details as completely as possible
    Provide a summary of the paper in words, describing methods used and main results
    Rules: Each project is to be independent student work
    You must understand everything you have written down.
    If there are some numerical results in the paper, discuss how they relate to the analysis.
    Score on the final project will be based on completeness and clarity of presentation
    The difficulty level of the paper will also be taken into account.

    Course Review Notes

    Review of PDE's
  • PS version
  • PDF version

    Introduction to regular perturbation
    Asymptotic evaluation of integrals
    Boundary layer theory for ODE's
  • PS version
  • PDF version
    (550) Multi-scale analysis for ODE's
  • PS version
  • PDF version
    (605) WKB analysis (ODE's)
  • PS version
  • PDF version
    (605) WKB analysis (PDE's) (Thanks to M. Ward)
  • PDF version
    (550) Quasi-steady approximations:
  • PS version
  • PDF version
    (605)Bifurcation analysis for nonlinear boundary value problems
  • PS version
  • PDF version
    (605)Real Ginzburg-Landau equation
  • PDF version
    (605)Stability and 2D Ginzburg-Landau equation
  • PDF version
    (605) Free boundary problems
  • PS version
  • PDF version
    (605) Free boundary problems II
  • PDF version