Notes on pattern formation in PDEs

Lecture notes.

Persistence of vegetation patterns Part 1: Turing instability
- Associated FlexPDE script
Persistence of vegetation patterns Part 2: delayed Turing bifurcation due to noise and slow parameter drift
- see related paper here
Patch model of organic farming
Notes on MEMS devices
Turing instability
Derivation of a mathematical model of crime
Mean first passage time for small traps
Perturbation of eigenvalues; domain perturbations
Metastability: exponentially slow motion of interfaces in 1D Allen-Cahn model
- See the FlexPDE script front1d-two.pde
Travelling waves in thin domains
Coarsening phenomenon for mesa patterns
Stability of mesa patterns in the Brusselator | supplement: Examples of patterns in Brusselator
Swarm aggregation models
- Matlab code: swarm.m
Spike patterns in GM system 1: introduction
Dynamics of a single interior spike in the GM system
Spike patterns in GM system 2: Wei's proof of stability of NLEP problem
Spike patterns in GM system 3: Stability of K spikes, large and small eigenvalues
Shallow wave theory

More notes on various topics (Warning: some may be rough sketches only)

Random matrix theory and Wigner's semi-circle law
Multiple scales:
Some [rather rough] notes on traffic models: derivation, method of characteristics, shock formation, Burger's pde... See also FlexPDE script, burger.pde
Rapidly convergent series for the regular part of modified Green's function on a rectangle: resummation.
Fourier series: convergence, discrete fourier series, FFT, and Gibbs phenomenon.
The motion of a hanging chain | Maple worksheet with animation

Supplemental questions/material (Note that some of homework questions are included in the notes above).
Note1: For *.pde files, you will need to download FlexPDE program ( they have a free student version, which is sufficient for these simulations).
Note2: to run Maple worksheets, you need to also download the following file: tools.mpl. Then replace the line read `/theodore/tools.mpl` inside the worksheets by whatever directory you put it in.

Maple worksheet to compute the steady state and the eigenvalue of the interface by using numerical boundary problem solver.
Questions on construction of mesa patterns
Questions on NLEP problem
Questions on NLEP problem part 2
Computing a fold point for the ODE u''-u+u^2*(1+ax^2)=0 (Maple worksheet)
FlexPDE simulation of spike instabilities in the GM model:
Simulation of a simple pulse-splitting model (FlexPDE script file)
Simulation of a pulse-splitting behaviour in GS model model (FlexPDE script file)