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:
- The best articles by Ciro Santilli
- Complete basis
- Computational physics
- Fourier transform
- History of the Fourier series
- Important partial differential equation
- Laplace's equation
- Robin boundary condition
- Separation of variables
- Solving partial differential equations with the Fourier series
- Wave equation