Deriving heat diffusion equation with Gauss' Divergence Theorem?

**GeoH2102** · Nov 25th 2009, 12:05 PM

Hello!

I've been given a question on an assignment and I'm not quite sure where to start. It's supposed to be one of the harder questions, and I'm really unsure what I'm doing with it!

We're supposed to derive the Heat Diffusion equation ( $u_{t} = ku_{xx}$ ) using Gauss' Divergence Theorem, which is (as you probably know if you're reading this!):

$\iiint_{V} (\nabla \cdot \textbf{F}) \enspace dV = \iint_{S} \textbf{F} \cdot \textbf{n} \enspace dS$

Any help would be much appreciated!

-Geo

**qmech** · Nov 25th 2009, 12:38 PM

Can you use continuity of thermal energy:

$<br /> \frac{\partial \rho}{\partial t} + {\nabla} \cdot j = 0<br />$

and the assumption that

$<br /> j = -k {\nabla} \rho<br />$

to get your result?

Your result is in 1-D, so you can simplify $\nabla$ to partial derivatives.

**GeoH2102** · Nov 25th 2009, 12:45 PM

I'm not entirely sure really. At my uni we're given a Group Project module and we're all a bit stuck with it. We have to give presentations each week - the first week we had to prove the Divergence Theorem which was easy enough, and this week we have to derive the heat equation as above. Literally no other information has been given, you can see the assignment sheet here: http://www.mas.ncl.ac.uk/~nzal/MAS30...jectsA1_A4.pdf - I'm doing the fourth project.

**GeoH2102** · Nov 25th 2009, 04:06 PM

Right, I've found an eBook here: Introduction to partial differential ... - Google Books - it seems to answer my questions. However, I'm not quite clear on how it gets from (2.4) to (2.5) on page 158.

Any help, as ever, is much appreciated!

EDIT: Never mind, got it

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