Equations & Models

ρ(∂u/∂t + u·∇u) = −∇p + μ∇²u + f

The fundamental equations of viscous fluid motion.

ẋ=σ(y−x), ẏ=x(ρ−z)−y, ż=xy−βz

A three-variable model that birthed chaos theory.

dS/dt=−βSI, dI/dt=βSI−γI, dR/dt=γI

The compartmental model of infectious-disease spread.

dx/dt=αx−βxy, dy/dt=δxy−γy

Predator–prey population oscillations.

iħ ∂ψ/∂t = Ĥψ

The governing equation of quantum states.

∂u/∂t = α∇²u

Diffusion of heat through a medium.

∂²u/∂t² = c²∇²u

Propagation of waves at speed c.

∇·E=ρ/ε₀, ∇·B=0, ∇×E=−∂B/∂t, ∇×B=μ₀J+μ₀ε₀∂E/∂t

The unified laws of electromagnetism.

∂u/∂t=Dᵤ∇²u−uv²+F(1−u), ∂v/∂t=D_v∇²v+uv²−(F+k)v

A reaction–diffusion system that forms Turing patterns.

F = ma

Force equals mass times acceleration.

F = G m₁m₂ / r²

The gravitational attraction between masses.

ẍ − μ(1−x²)ẋ + x = 0

A self-sustaining oscillator with a limit cycle.

∂V/∂t + ½σ²S²∂²V/∂S² + rS∂V/∂S − rV = 0

The PDE for pricing options.

∇²φ = −ρ/ε

Relates a potential to its source.

∂c/∂t = D∇²c

How concentration spreads by diffusion.

p + ½ρv² + ρgh = const

Energy conservation along a streamline.

xₙ₊₁ = r xₙ(1 − xₙ)

A simple map that routes to chaos.

mẍ + cẋ + kx = 0

Oscillation with energy loss.

∂ρ/∂t + ∇·(ρu) = 0

Conservation of mass in a flow.

v = Vmax[S] / (Km + [S])

The rate law for enzyme-catalyzed reactions.