Heat equation

The heat equation is a partial differential equation of the form

$$\frac{\partial u}{\partial t}=k\Delta u$$

where $$\Delta$$ denotes the Laplacian. When there is only one spatial dimension being considered, the heat equation takes the form

$$\frac{\partial u}{\partial t}=k\frac{\partial^{2}u}{\partial x^{2}}$$

The heat equation is so called because it was studied by Joseph Fourier as a way of describing the way heat spreads over time. The heat equation can also be used to describe other processes such as diffusion.

On an infinite spatial domain, the heat equation is usually solved using the Fourier transform. On a finite spatial domain, separation of variables is often used along with the Fourier Series.