What Linear Transformations Satisfy the 2D Heat Equation?

[stripes]

Homework Statement

For the heat equation in two space variables find all the linear transformations of the form (x,y) = a(x',y') for real number a such that

$\frac{\partial u}{\partial t} = \frac{k}{\sigma}\Delta u \Leftrightarrow \frac{\partial v}{\partial t} = \Delta'v$

where u(x,y,t) = v(x',y',t) and Δ' is the Laplacian with respect to the primed coordinates.

Homework Equations

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The Attempt at a Solution

I hate to say it so early in the semester, but I have not a clue where to start. This is a fourth year course with second year prerequisites, and I've satisfied all the prerequisites, but why do I find that I don't even know where to begin? I guess what has me stumped is the fact that the question is very open ended. Also, I'm not sure where the sigma constant has come from.

So after staring at it for a while, I figure I should write out the Laplacian for a function in 2 space, and then start manipulating it? Since t is not a spatial variable, how do I account for that? But the Tex stuff written about is just information that we know to be true, how do I use it to find linear transformations? I'm only familiar with linear transformations within the topics of linear algebra and I don't think my mathematical maturity allows me to extend what I learned in that course to other courses. In other words, I may have taken courses that have covered these topics before, but I can't really put it all together. Any help on where to start would be appreciated. I have two more heat equation questions to do so I'm a bit worried. Thanks very much.

 

[voko]

What the problem tells you is that a coefficient at the Laplacian can be reduced to unity by a linear transformation of the spatial variables. You need to find this transformation.

Start by writing down the general form of the linear transformation (it should have four arbitrary constants). Then apply the chain rule to the unprimed Laplacian and compare that with the primed Laplacian.