Diffusion Equation – Green's Function

Concentration heatmap over time →

Diffusivity D0.50 Source width σ0.05 Time step Δt0.001

The heat/diffusion equation:

∂u/∂t = D ∇²u

Green's function (fundamental solution) for a delta-function initial condition:

G(x,t) = 1/√(4πDt) · exp(−x²/4Dt)

This is a Gaussian whose width σ(t) = √(2Dt) grows as the square-root of time — the hallmark of diffusive spreading.

⟨x²⟩ = 2Dt

General solutions are convolutions: u(x,t) = ∫G(x−y,t)u₀(y)dy. Click "Add Source" to superpose multiple point sources.