# Definition:Heat Equation

## Equation

The heat equation is a second order PDE of the form:

$a^2 \paren {\dfrac {\partial^2 w} {\partial x^2} + \dfrac {\partial^2 w} {\partial y^2} + \dfrac {\partial^2 w} {\partial z^2} } = \dfrac {\partial w} {\partial t}$

## Also presented as

The heat equation can also be presented as:

$\nabla^2 u = c^2\dfrac {\partial u} {\partial t}$

where $\nabla^2$ denotes the Laplacian of $u$.

## Also known as

This equation is also known as the diffusion equation.

Some sources refer to this equation as Fourier's equation, for Joseph Fourier, but while the attribution is clear, this usage is rare.

## Also see

• Results about the heat equation can be found here.