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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:

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

also written as

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

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

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

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

$$\Delta u = 0 $$




"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
derivation of heat equation (Derivation) by pahio
using Laplace transform to solve heat equation (Example) by pahio
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Cross-references: Laplacian, dimensions, real, function, partial differential equation
There are 8 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 8198 times total.

Classification:
AMS MSC35Q99 (Partial differential equations :: Equations of mathematical physics and other areas of application :: Miscellaneous)

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hi!! by harry on 2005-12-10 23:20:56
hi how to find the transient heat equation solution using FEM
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  • Re: hi!! by rspuzio on 2005-12-11 14:15:56

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