MATH 4250-5250: Asymptotic analysis
When/where: Tues Thurs 11:35-12:55, Chase building 319 (subject to change)
Instructor:
Theodore Kolokolnikov, Chase building 304,
tkolokol@mathstat.dal.ca, http://www.mathstat.dal.ca/~tkolokol;
Phone: 494-6295.
Office hours: drop by anytime I'm in my office. In addition, I will try to
be there every week TR 10:30-11:15; You can also email me to set up an appointment.
Lecture notes:
- Roots of polynomials and transcendental equations;
iteration method, singular perturbations and ill-conditioning.
- Eigenvalue perturbation
problems; bifurcation from steady state;
domain perturbations.
- Integration, Part I: integration by parts,
divergent asymptotic series, Watson's lemma, Laplace/Mellin transforms,
local analysis.
- Integration, Part II: Laplace's method, Sterling's
formula, steepest descent, Airy ODE, asymptotic expansions of
special functions. Singular integrals, splitting the integration range,
Euler's constant.
- Boundary layers, matching, composite solutions,
interior boundary layers; exponentially ill-conditioned problem.
- Method of multiple scales, nonlinear Hopf analysis, singular Hopf
bifurcation.
- Delay differential equations, multiple scales,
period doubling and chaos.
- WKB theory, turning points, wave propagation,
in thin membrates, delayed bifurcations.
Maple worksheets:
Homework sets
Outline:
Mathematical models of many physical systems
have a naturally occurring small parameter
which may be exploited using asymptotic analysis techniques.
In this
course, we will study a variety of physical systems which illustrate
many of the common approaches used in asymptotic analysis. Topics
covered may include
- Asymptotic expansions, (non)convergence, algebraic equations with
small parameters, eigenvalue problems.
- Asymptotic evaluation of integrals: Laplace's method, mehtod of stationary phase
- Boundary layers, principle of dominant balance,
matched asymptotics with applications to physical problems.
- Boundary layers in PDE's
- Method of multiple scales, WKB theory,
- Exponentially ill conditioning, stability of fronts, reaction-diffusion systems
- Delay differential equations
References:
The textbook is
M.H. Holmes, Introduction to Perturbation Methods. However much of material
will come from various other sources. Lecture notes will be posted on this
webpage.
Evaluation:
The evaluation will consist of bi-weekly homework sets and a take-home
final. The homework sets will be posted on this website.
Graduate students will also be expected to make a presentation. This
presentation is optional for undergraduates; details will
be provided later.
The grading scheme is 50% HW, 30% final, 20% presentation
presentation or 60% HW, 40% final (if not doing a talk).