** 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.

- Using maple to compare numeric and asymptotic solutions to x exp(-x)=epsilon
- An example of a boundary layer problem: Demonstrates how to use maple's dsolve/numeric to numerically solve ODE boundary value problems.
- Method of multiple scales: Van der Pol oscillator with periodic oscillations.

- Homework 1 (due 18 sep) Solutions
- Homework 2 (due 30 sep) Solutions
- Homework 3 (due 14 oct) Solutions
- Homework 4 (due 30 oct) Solutions
- Homework 5 (due 14 nov) Solutions
- Homework 6 (due 24 nov) Solutions See also: an example of simulating amplitude equations.
- Homework 7 (due at the latest 6 Dec 5pm) Solutions

- 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).