= Heat equation
{wiki}
Besides being useful in engineering, it was very important historically from a "development of mathematics point of view", e.g. <history of the Fourier series>[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:
* with <FreeFem>:
* <heat-dirichlet.1d.freefem>
* <heat-dirichlet-2d-freefem>