Overview

Via this web page you may create a PDF book on rational mathematical modelling of dynamics tailored to the interests you specify. Currently this is an early draft version, an alpha version.

Among the sections that follow, choose the interesting ones

At the end click Submit to process, typeset and produce a PDF book tailored just for you. The processing automatically detects all prerequisite sections and includes them as well. Only select what interests you.

But first, what computer algebra package do you prefer?
Reduce (recommended)
Maple (as yet only available in Chapter 1)
Mathematica (as yet only available in Chapter 1, thanks to Rick Eller)

Asymptotic methods solve algebraic and differential equations
Perturbed algebraic equations solved iteratively
Power series solve ordinary differential equations
The normal form of oscillations give their amplitude and frequency

Basic fluid dynamics
Flow description
Conservation of mass
Conservation of momentum
The state space

Centre manifolds emerge
Couette flow
The metastability of quasi-stationary probability distributions
The centre manifold
Construct centre manifolds iteratively
Taylor vortices form in a pitchfork bifurcation
Irregular centre manifolds encompass novel applications

High fidelity numerical models use centre manifolds
Introduction to some numerical methods
Introduce holistic discretisation on just two elements
Holistic discretisation in one-D
Model physical boundary conditions
Dispersion along pipes invokes cylindrical elements

Normal forms usefully illustrate
Normal forms simplify evolution
Separate the fast and slow dynamics
Appropriate initial conditions empower accurate forecasts
Resonance couples slow modes with fast oscillations
N/A Normal forms for homogenisation

Hopf bifurcation has oscillation within the centre manifold
Linear stability of double diffusion
Oscillations on the centre manifold
Modulation of oscillations
Nonlinear evolution of double diffusion
N/A Chaos appears in triple convection

Avoid memory in modelling non-autonomous or stochastic systems
Averaging is often a good first modelling approximation
Normal forms separate slow from fast in non-autonomous dynamics
Basic stochastic calculus
Strong and weak models of stochastic dynamics

Large-scale spatial variations
Poiseuille pipe flow
Dispersion in pipes
Thin fluid films evolve slowly over space and time
Resolve inertia in thicker faster fluid films

Slow manifolds guide the mean dynamics
Incompressible and other approximations
Sub-centre slow manifolds are useful but do not emerge
N/A Water waves

Patterns form and evolve
One-dimensional introduction
N/A Two-dimensional spatio-temporal patterns
N/A Embedding slow dynamics

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Print the book 2up

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