The heat equation is a fundamental partial differential equation that describes how the distribution of heat (or temperature) in a given region evolves over time. It is a mathematical model used in various fields such as physics, engineering, and finance to study heat conduction, diffusion, and other related processes.
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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: