# 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 $u:\mathbb{R}^{2}\to\mathbb{R}$ 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 $u$:

 $\Delta u=0$
Title heat equation HeatEquation 2013-03-22 12:45:36 2013-03-22 12:45:36 drini (3) drini (3) 5 drini (3) Definition msc 35Q99 DifferentialEquation Laplacian