Diffusion Equation — Fundamental Solution

The heat/diffusion equation ∂u/∂t = D ∇²u has a fundamental solution G(x,t) = (4πDt)^(-1/2) exp(-x²/4Dt) — a Gaussian that spreads as σ = √(2Dt). The width grows as t^(1/2), the peak falls as t^(-1/2), and the total integral is conserved (probability). Multiple initial conditions are shown simultaneously.

G(x,t) = (4πDt)^{−½} · exp(−x²/4Dt)
Diffusivity D0.50
t (log scale)1.00
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t:
σ = √(2Dt):
Peak:
∫G dx: