heat equation

The heat equation in 1-dimension (for example, along a metal wire) is a partial differential equation of the following form:

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

also written as

$$u_t=c^2\cdot u_{xx}$$

Where $u:{ℝ}^2\to ℝ$ is the function giving the temperature at time $t$ and position $x$ and $c$ is a real valued constant. This can be easily extended to 2 or 3 dimensions as

$$u_t=c^2\cdot (u_{xx}+u_{yy})$$

and

$$u_t=c^2\cdot (u_{xx}+u_{yy}+u_{zz})$$

Note that in the steady state, that is when $u_t=0$, we are left with the Laplacian of $u$:

$$\mathrm{\Delta }u=0$$