Heat equation
The heat equation is a partial differential equation that describes the distribution of heat over time in solid (static) materials. It was first formulated by Joseph Fourier in the early 1800s, along with a method of solving using Fourier series.
The heat equation is described in more detail in undergraduate physics texts and at many web sites. See Wikipedia Heat-Equation entry and Wikipedia's entry on Heat in general. Also see Wolfram's Heat Conduction Equation page.
In spite of all the math, understand that the heat equation really describes something very intuitive and simple. Namely that things tend to flatten out and get smooth. Sharp edges and points wear away. Quickly at first, when they are still sharp, and then more slowly as they get duller and smoother.
The heat equation describes a lot of other things besides heat. It describes the diffusion of liquids and gases, the spread of disease, the erosion of mountains, the melting of ice, and the rocks in your rock tumbler. It also describes random walks and the bell-curves of normal probability distributions. In the financial world it is used to establish option prices.
The images on this page show how a spikey heat landscape smooths out over time as it approaches thermal equilibrium. In the first image at the top (click to enlarge), a lot of sharp random heat has been added to the sheet. The next image down shows the same sheet after about 25 iterations of the heat solver have taken off the roughest edges.
The next image, to the left (click to enlarge) shows the same sheet after about 25 more solve iterations. The smoothing is more pronounced; the slopes and high-order derivatives are greatly reduced.
The final image is the smoothest yet. But there is still a long way to go, even though this sheet has gone through more than 100 solve iterations since the previous image. The heat equation works quickly with sharp edges, but gets slower and slower as the sheet approaches static equilibrium.
So even though the heat equation is a partial differential equation awash in a sea of complicated math, at its heart it is very simple. It is just describing edges wearing down and peaks flattening over time, as illustrated in these images. You still need the math to solve the equation and draw these pictures, but it turns out even that isn't very complicated. We'll talk about that next.