Heat Equation — Green's Function & Mode Decomposition

u_t = κ u_xx: Gaussian spreading from δ-function IC; Fourier mode decay

Time t: 0.000 Peak: -- σ²(t): -- σ²∝t: slope=1
About: The 1D heat equation u_t = κ u_xx has Green's function G(x,t) = (4πκt)^(−1/2) exp(−x²/4κt) — a Gaussian broadening with variance σ² = 2κt (slope 1 on log-log). The purple curve shows the current temperature field; faint snapshots at earlier times. For a square pulse IC, Fourier modes u_k ∝ exp(−κk²t) decay at rate κk²: high-frequency modes vanish fastest. The bottom panel shows log σ² vs log t with slope 1 reference.