Heat Equation

Significance

Temperature smooths itself out: peaks flatten, dips fill, at a rate set by the local curvature. Fourier invented his transform to solve exactly this, and the same equation under other names governs diffusion, option prices (Black–Scholes) and image blurring. Kelvin used it to date the Earth at twenty to forty million years and was wrong — though not only for the reason usually given. Radioactivity was indeed undiscovered, but his own former assistant John Perry showed in 1895 that a convecting interior alone stretches the answer by the factor needed. Conduction was the wrong model, not merely the wrong number.