MATH 4250-5250: Asymptotic analysis
When/where: Tues Thurs 11:35-12:55, Chase building 319 (subject to change)
Theodore Kolokolnikov, Chase building 304,
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.
Mathematical models of many physical systems
have a naturally occurring small parameter
which may be exploited using asymptotic analysis techniques.
course, we will study a variety of physical systems which illustrate
many of the common approaches used in asymptotic analysis. Topics
covered may include
- Roots of polynomials and transcendental equations;
iteration method, singular perturbations and ill-conditioning.
- Eigenvalue perturbation
problems; bifurcation from steady state;
- Integration, Part I: integration by parts,
divergent asymptotic series, Watson's lemma, Laplace/Mellin transforms,
- Integration, Part II: Laplace's method, Sterling's
formula, steepest descent, Airy ODE, asymptotic expansions of
special functions. Singular integrals, splitting the integration range,
- Boundary layers, matching, composite solutions,
interior boundary layers; exponentially ill-conditioned problem.
- Method of multiple scales, nonlinear Hopf analysis, singular Hopf
- Delay differential equations, multiple scales,
period doubling and chaos.
- WKB theory, turning points, wave propagation,
in thin membrates, delayed bifurcations.
- 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
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
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).