heat equation
Hold one end of a metal spoon in hot tea and the warmth creeps up to your fingers; smell perfume slowly fill a room; watch a drop of dye disperse through still water. The heat equation is the single PDE that captures all of these — the law of how a quantity that is conserved but freely spreading, like thermal energy or concentration, redistributes itself over time. It is the parabolic model equation and the prototype of all diffusion.
In one space dimension it reads u_t = k u_xx, where u(x, t) is the temperature, the subscript t is the time derivative and xx is the second space derivative, and k is the thermal diffusivity (always positive). The physics is captured in a sentence: the rate at which the temperature rises at a point equals a constant times the curvature of the temperature profile there. Where the profile dips below its neighbors (a local minimum, positive curvature) heat flows in and the point warms; where it bulges above (a local maximum, negative curvature) heat flows out and it cools. So bumps decay and dents fill in, and the whole profile relaxes toward a flat or steady state. In higher dimensions the u_xx is replaced by the Laplacian nabla^2 u, giving u_t = k nabla^2 u.
Derived by Joseph Fourier — who invented Fourier series precisely to solve it — the heat equation is the workhorse of thermal engineering, but its reach is far wider: chemical diffusion (Fick's law), the smoothing of images in computer vision, the spread of populations and diseases in simple models, and the Black-Scholes equation of finance, which becomes the heat equation after a change of variables. Its hallmark behaviors — instant smoothing, an arrow of time, a maximum principle — are all consequences of that curvature-driven mechanism.
A rod of length L with both ends held at 0 and initial temperature f(x) has the separated solution u(x, t) = sum of b_n sin(n pi x / L) e^{-k (n pi / L)^2 t}. Every Fourier mode decays exponentially, and the higher the mode (larger n) the faster it dies — which is exactly why diffusion smooths.
High-frequency wiggles (large n) decay fastest, so even a jagged start becomes smooth almost instantly.
The heat equation cannot be run backward in time in any stable way: small errors in late data are amplified without bound when you try to recover the past, which is the precise reason deblurring and unmixing are so hard.