Syllabus Math 558

Math 558. Methods of Applied Mathematics Instructor Syllabus

1 lecture = 80 minutes ( = 1.5 hours of class time)

Topic 1: Nondimensionalization, scaling.
(6 Lectures)

  • Lecture 1.1. Dimensional analysis, examples
  • Lecture 1.2. Buckingham Pi-Theorem.
  • Lecture 1.3. Examples of Theorem, e.g. measuring size of nuclear blast from data, Pythagoras' theorem, drag on a sphere
  • Lecture 1.4. Rescaling variables in ODE, PDE. Examples, e.g. diffusion equation, reaction-diffusion equation, height of projectile (lead to asymtotic questions for ODE, PDE)
  • Lecture 1.5. Random Walk on 1D, diffusion equation approximation; Biased random walk, Fokker-Planck equation
  • Lecture 1.6. Liouville and F-P equations, counting process to Poisson distribution, generator for chemical reaction system.

Topic 2: Asymptotics for ODE, PDE, SDE, Markov chains
(14 lectures)

  • Lecture 2.1. Asymptotics for algebraic equations, definition of singular vs. regular, Gronwall's Theorem
  • Lecture 2.2. Applications of Gronwall, matrix norm, singular values, asymptotics of eigenvalues/eigenvectors
  • Lecture 2.3. SIR models, generating mean-field models, breakdown of mean-field (epidemic size calculation)
  • Lecture 2.4. Example of why naive asymptotics are bad (celestial mechanics)
  • Lecture 2.5. Method of Multiple Scales, resonances
  • Lecture 2.6. Averaging for fast-slow stochastic systems 1.
  • Lecture 2.7. Averaging for fast-slow stochastic systems 2.
  • Lecture 2.8. Boundary layers, WKB
  • Lecture 2.9. Reaction-diffusion, small diffusion regime 1.
  • Lecture 2.10. Reaction-diffusion, small diffusion regime 2.
  • Lecture 2.11. Reaction-diffusion, large diffusion regime.
  • Lectures 2.12--2.14. Scattering/inverse scattering for NLS, Lax pairs, connection to KdV, solitions 1.

Topic 3: Complex systems
(6 Lectures) [some instructor choice in this Topic]

  • Lecture 3.1. Phase oscillators, linear coupling, Gershgorin's theorem
  • Lecture 3.2. Nonlinearly-coupled phase oscillators, Kuramoto model
  • Lecture 3.3. Coupled integrate-and-fire neurons, conditions for stability
  • Lecture 3.4. Integrate-and-fire neurons in the presence of noise, connection to Erdős-Rényi
  • Lecture 3.5. Condition for synchrony in complex systems 1.
  • Lecture 3.6. Condition for synchrony in complex systems 2.

+3 Lectures of leeway and instructor choice

Recommended references:

G. I. Barenblatt
Scaling, self-similarity, and intermediate asymptotics.
Cambridge University Press, Cambridge, 1996

M. H. Holmes
Introduction to the foundations of applied mathematics.
Springer, New York, 2009

C. W. Gardiner
Handbook of stochastic methods for physics, chemistry and the natural sciences.
Springer-Verlag, Berlin, third edition, 2004

J. P. Keener
Waves in excitable media.
SIAM J. Appl. Math, 39(3):528--548, 1980

Approved by GAC, May 2010.