Heat equation

Besides being useful in engineering, it was very important historically from a "development of mathematics point of view", e.g. it was the initial motivation for the Fourier series.

Some interesting properties:

TODO confirm: for a fixed boundary condition that does not depend on time, the solutions always approaches one specific equilibrium function.
This is in contrast notably with the wave equation, which can oscillate forever.
TODO: for a given point, can the temperature go down and then up, or is it always monotonic with time?
information propagates instantly to infinitely far. Again in contrast to the wave equation, where information propagates at wave speed.

Sample numerical solutions:

 Ancestors