Definition:Heat Equation

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The heat equation is a second order partial differential equation which models the flow of heat in a body.

For a $3$-dimensional body, the equation for the temperature $w$ at time $t$ and position $\tuple {x, y, z}$, it is 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

The heat 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.