PlanetMath: heat equation

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heat equation

(Definition)

The heat equation in 1-dimension (for example, along a metal wire) is a partial differential equation of the following form:

$\displaystyle \frac{\partial u}{\partial t} = c^2 \cdot \frac{\partial^2 u}{\partial x^2}$

also written as

$\displaystyle u_{t} = c^2 \cdot u_{xx}$

Where $u:\mathbb{R}^2\to\mathbb{R}$ is the function giving the temperature at time and position and is a real valued constant. This can be easily extended to 2 or 3 dimensions as

$\displaystyle u_{t} = c^2 \cdot ( u_{xx} + u_{yy} )$

and

$\displaystyle u_{t} = c^2 \cdot ( u_{xx} + u_{yy} + u_{zz} )$

Note that in the steady state, that is when $u_{t} = 0$ , we are left with the Laplacian of :

$\displaystyle \Delta u = 0$

"heat equation" is owned by drini. [ owner history (2) ]

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See Also: differential equation, Laplacian

Attachments:: example of solving the heat equation (Example) by pahio
time-dependent example of heat equation (Example) by pahio

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Cross-references: Laplacian, dimensions, real, function, partial differential equation
There are 6 references to this entry.

This is version 2 of heat equation, born on 2002-06-07, modified 2002-06-07.
Object id is 3067, canonical name is HeatEquation.
Accessed 6438 times total.

Classification:

AMS MSC:

35Q99 (Partial differential equations :: Equations of mathematical physics and other areas of application :: Miscellaneous)

Pending Errata and Addenda

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