Week 1
What is a mathematical model?
The modelling cycle; units and dimensions; a first ODE from a rate law.
Week 2
Dimensional analysis and nondimensionalisation
Characteristic scales; nondimensionalising an ODE; solution collapse.
Week 3
The Buckingham Π theorem
Dimension matrices; finding Π-groups systematically; physical similarity.
Coming soon
Week 4
First-order ODEs: rate laws and separable equations
Deriving ODEs from rate laws; separable solutions; exponential growth and decay.
Coming soon
Week 5
Logistic growth: equilibria and stability
The logistic equation; finding and classifying equilibria; phase-line analysis.
Coming soon
Week 6
Linear first-order ODEs and the integrating factor
Forcing and relaxation; the integrating factor method; fitting parameters to data.
Coming soon
Week 7
Harvesting, thresholds, and bifurcation
Harvesting models; tipping points; bifurcation diagrams; catastrophe.
Coming soon
Week 8
Two-species systems and phase planes
Coupled ODEs; nullclines; phase portraits; the Lotka–Volterra predator–prey model.
Coming soon
Week 9
Second-order ODEs and simple harmonic motion
Constant-coefficient ODEs; spring–mass–damper; overdamping, critical damping, oscillation.
Coming soon
Week 10
Numerical ODE solving: Euler, RK2, and error
Implementing Euler and RK2 from scratch; global error; stability; comparison with
solve_ivp.
Coming soon
Week 11
Self-similarity and scaling capstone
Scaling collapse revisited; dimensional analysis in action; project and report clinic.
Coming soon