Math 550 2003-2004
Methods of Asymptotic Analysis
(Part of core sequence in IAM curriculum, recommended prereq for
Math 551 and Math 605E)
Math 550: Prof. Rachel Kuske: rachel@math.ubc.ca
This is a course in modern applied methods of asymptotic analysis,
which play important roles in mathematical modeling, applied analysis, and approximations.
The material provides valuable skills and resources complementary to
scientific computations, mathematical modeling in
applications, analysis of pde's and dynamical systems.
The general concepts and methods are developed in the context
of a wide variety of applications. Connections with other
areas of mathematics, such as computations, analysis and modeling,
are also covered.
The course is designed so that the two semesters ( 550 followed by 551 and/or 605E)
are complementary.
Some topics are covered in both semesters (without repetition of material),
including different applications of multiple scales, boundary layers, bifurcations, etc.
Those who take both semesters will gain in-depth experience with these
topics, as well as gaining an expertise with a broad range of techniques.
Course Outline
Math 550: Fall 2003
1. Introduction to asymptotic expansions
2. Dimensional analysis
i) Small and large parameters/non-dimensional numbers
ii) Regular expansions
3. Asymptotic approximation of integral representations of solutions
i) Green's functions and Laplace-type integrals
ii) Fourier integrals and stationary phase
4. Boundary layer I
i) Singular Perturbation and Scales
ii) Boundary value problems for ODE's
iii) Convection-diffusion equations: PDE's
iv) Parabolic approximations
v) Flow past obstacles
5. Methods of Multiple Scales
i) quasi-static approximations
ii) ODE models of oscillators
chemical/biological oscillators,Van der Pol and circuits,Poincare-Lindstedt
iii) Envelope equations: nonlinear waves and patterns:
NLS, KdV, GL, dispersion relations
6. Introduction to WKB approximations
i) Eikonal and transport equations
ii) Applications in quantum mechanics/waves
7. Bifurcations
i) Systems of ODE's and boundary value problems
ii) Turing instabilities
iii) Pattern formation, Ginzburg-Landau equations
iv) PDE's and Boundary value problems:
elasticity, combustion...
8. Similarity Methods
i) Diffusion and nonlinear diffusion
ii) Free boundary problems: Stefan problem, option pricing
Math 551: Spring 2003
( We recommend taking both semesters to cover all topics. In
some exceptional cases, such as a student who has a conflict with Math 550,
it may be possible to take Math 551 with minimal background work, such as reading
sections of the notes of Math 550.)
8. WKB Methods
i) Connection formulae
ii) Eigenvalue asymptotics
iii) Maslow theory
iv) Diffraction theory
9. Homogenization methods
i) Review of method of multiple scales
ii) Micro- vs. Macro-scale
iii) Bubbly fluids,
iv) Wave propagation on rough boundaries
10. Oscillators and dynamical systems
i) Floquet theory
ii) Melnikov's method
iii) stick-slip problems
11. Boundary layer II
i) Holes in the domain
ii) Low/High Reynolds number flow
iii) Exponentially ill-conditioned problems
12. Complex Variable methods
i) Hele-Shaw flow and free boundaries
ii) singularities in the complex plane
iii) Stokes phenomenon, selection processes
iv) Steepest Descent
13. Geometric Problems
i) Interface motion in Multi-Dim.
ii) Motion of curves by curvature
iii) Geometric laws from PDE's
iv) Dynamics of filaments
twisting of DNA coils, super-conductivity, etc.