Topics in differential equations
Fall 2018

• Introduction: persistence of vegetation patterns in near-desert conditions
  • Related FlexPDE script
• Notes on Turing bifurcation
• Multiple scales example: Rayleygh oscillator
• Using complex variables for multiple scale computation for u''+u=eps u^2
• Delay differential equations, multiple scales, period doubling and chaos
• Using multiple scales to get a Complex Ginzburg Landau PDE from a PDE with a delay See also:
  • - Matlab code to numerically an ODE with delay: simpledde.m
  MEMS model
• Subsolution/supersoluton/max principle See also:
  • Maple worksheet for MEMS bifurcation diagram
• Mathematical model of crime
• Galante-Wisen-Bhaya-Levy model of bacterial aggregation
  • Accompaning movies
  • Related paper
  • Matlab code
• Traffic model: method of characteristics, shock formation, Burger's pde... See also FlexPDE script, burger.pde
• Heteroclinic orbits, slow travelling waves
• Travelling waves in thin domains
• Mean first passage time for small traps
• Metastability: exponentially slow motion of interfaces in 1D Allen-Cahn model
  • See the FlexPDE script front1d-two.pde
• Spike patterns in GM system 1: Stability of K spikes, large and small eigenvalues
• Dynamics of a single interior spike in the GM system
• (for reference only) Wei's proof of stability of NLEP problem
• FlexPDE script: two boundary spikes in GM model
• FlexPDE script: Dynamics of a single interior spike.
• Swarm aggregation models
  • Matlab code: swarm.m

Term project:
For term project, you will be required to read a mutually agreed upon research paper, give an in-class presentation on it, and submit a report. More information will be given during the class.

Homework sets

• Homework 1, due Wed, Sep 19 | Solutions
• Homework 2, due Wed, Oct 10 | Solutions
• Homework 3, due Mon, Oct 22 | Solutions
• Homework 4, due Mon, Nov 5 | Solutions
• Homework 5, due Mon, Nov 19 | Solutions |
• Homework 6, due Mon, Dec 3 |

• My notes on various topics
• See also notes from last time class was offered (2016)
• PPlane software