B18
Course Coordinator: 
Eliot Fried
Methods of Mathematical Modeling I
Description: 

In this course, students will learn to formulate mathematical models leading to rate equations, transport equations, and variational principles. They will also learn techniques for extracting qualitative and quantitative information from those models. In particular, they will study phase line analysis, phase plane analysis, the method of characteristics, dimensional analysis, and methods for constructing similarity solutions.

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

Rate equations

  1. Particle motion
  2. Chemical reaction kinetics
  3. Ecological and biological models
  4. One-dimensional phase-line analysis
  5. Two-dimensional phase-plane analysis

Transport equations

  1. Reynold’s transport theorem
  2. Deriving conservation laws
  3.  Linear advection equation
  4.  Systems of linear advection equations
  5.  Method of characterisitcs
  6.  Quasilinear equations and shocks

Variational principles

  1. Functionals
  2. Necessary and sufficient conditions for extrema
  3. Essential and natural boundary conditions
  4. Application to classical mechanics
  5. Treatment of constraints

Dimensional scaling analysis

  1. Dimensional quantities
  2. Dimensional homogeneity
  3. Nondimensionalization
  4. Applications
  5. Buckingham Pi theorem

Self-similar solutions of partial differential equations

  1. Scale-invariant symmetries
  2. Similarity variables and solutions
  3. Application to the heat equation
  4. Application to a nonlinear diffusion equation
Course Type: 
Elective
Credits: 
2
Assessment: 
Assignments 50%; Final exam 50%
Text Book: 
Thomas Witelski & Mark Bowen, Methods of Mathematical Modelling — Continuous Systems and Differential Equations. Springer, 2015. ISBN 978-3-319-23041-2.
Reference Book: 
Other readings will be supplied
Prior Knowledge: 

Prior knowledge of elementary calculus