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

• Solution to Heat Equation

• 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): diffusion equation
• 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