The Heat Equation

The heat equation is simply the rule for how temperature evens itself out over time: cold spots warm up and hot spots cool down. The same equation describes anything that spreads by diffusion, so it's just as good for ink in still water or a drug moving through tissue as for a hot rod. The equation $u_t = k\,u_{xx}$ has three small pieces: $u_t$ (how fast the temperature at a point is changing), $u_{xx}$ (how much colder or hotter that point is than its neighbors' average), and $k$ (how quickly the material lets heat spread). So the rule is "the more out of line a point is with its neighbors, the faster it catches up." It looks like a lot of symbols, but that one sentence is the whole equation.

Two physical facts go in. First, heat flows downhill, from hot to cold, and the steeper the drop, the faster it flows. That's Fourier's law $\phi = -K_0u_x$ above; the minus sign is there because heat moves opposite to the direction the temperature increases. Second, energy is conserved: the heat in a piece of rod only changes by what crosses its ends.

Derivation. Take a thin slice of rod from $x$ to $x + \Delta x$. Its heat energy changes because of the flux in at the left minus the flux out at the right:

With Fourier's law $\phi = -K_0u_x$, dividing by $c\rho\,\Delta x$ gives $u_t = k\,u_{xx}$ with $k = \frac{K_0}{c\rho}$. Notice where the second derivative came from: conservation turned the flux into its derivative, and the flux was already a derivative of $u$.

Why curvature? If $u_{xx} > 0$ at a point, the profile curves upward there, so the neighbors on either side are warmer on average. Heat flows in and the point warms up. If $u_{xx} < 0$ the point is a local hot spot and it cools. Either way, bumps get flattened, which is why heat always smooths things out.

Boundary conditions:

• Fixed temperature (Dirichlet): $u(0, t) = T_0$, an end held in an ice bath.
• Insulated (Neumann): $u_x(0, t) = 0$, no heat flows through the end.
• Newton cooling (Robin): heat leaves at a rate proportional to the difference with the surroundings.

Equilibrium. When nothing changes in time, $u_t = 0$, so $u_{xx} = 0$. The only functions with zero second derivative are straight lines, so a steady temperature profile is a straight line between the end temperatures. That's the line the rod settled onto in the first picture of this unit.

In 2D and 3D: $u_t = k\nabla^2u$, with $\nabla^2u = u_{xx} + u_{yy} + u_{zz}$.

The same equation describes heat in engine blocks and CPUs, the diffusion of chemicals and drugs, the spread of a pollutant in still water, and the smoothing of images.