# Definition:Heat Equation

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## Equation

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**.

## Sources

- 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next): Chapter $\text{VI}$: On the Seashore - 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $1$: The Nature of Differential Equations: $\S 1$: Introduction - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**heat equation** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**heat equation** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**heat equation**