B19
Course Coordinator: 
Eliot Fried
Methods of Mathematical Modeling II
Description: 

In this course, students will learn to apply regular and singular perturbation methods to ordinary and partial differential equations. They will also be exposed to boundary-layer theory, long-wave asymptotic methods for partial differential equations, methods for analyzing weakly nonlinear oscillators and systems with multiple time scales, the method of moments, the Turing instability, pattern formation, and Taylor dispersion. 

Aim: 
This is the second of a two-part series of lectures designed to provide students the ability to formulate and extract insight from basic mathematical models.
Detailed Syllabus: 

Perturbation methods

  1. Asymptotic expansions
  2. Regular expansions
  3. Singular perturbation problems
  4. Applications of perturbation methods

Boundary-layer theory

  1. Inner and outer asymptotic solutions
  2. Distinguished limits
  3. Matching
  4. Applications of boundary-layer theory

Long-wave asymptotic solutions to partial differential equations

  1. Separation of variables
  2. Dirichlet problem for a slender rectangle
  3. Application to wires

Weakly nonlinear oscillators

  1. Linear oscillators
  2. Poincare–Lindstedt expansions
  3. Method of multiple time scales

Fast/slow dynamical systems

  1. Strongly nonlinear oscillators
  2. Chemical reactions
  3. Enzyme kinetics

Reduced models for problems involving partial differential equations

  1. Method of moments
  2. Turing instability and pattern formation
  3. Taylor dispersion and enhanced diffusion
Course Type: 
Elective
Credits: 
2
Assessment: 
Assignments 50%; Final exam 50%
Text Book: 
Witelski & Mark Bowen, Methods of Mathematical Modelling — Continuous Systems and Differential Equations. Springer, 2015. ISBN 978-3-319-23041-2.
Reference Book: 
Supplemental notes will be supplied as appropriate
Prior Knowledge: 

B18 Methods of Mathematical Modeling I