Scaling the Heat Equation to Standard Form

[Somefantastik]

I don't understand where to even start with this problem. This book has ZERO examples. I would appreciate some help.

Show that by a suitable scaling of the space coordinates, the heat equation

u_{t}=\kappa\left(u_{xx}+u_{yy}+u_{zz}\right)

can be reduced to the standard form

v_{t} = \Delta v where u becomes v after scaling. \Delta is the Laplacian operator

 

[Mute]

What you want to do is scale the spatial variables such that (using vector notation) \mathbf{r} \rightarrow \alpha \mathbf{r}. Basically, using the problem's notation, you define the function v such that

u(x,y,z,t) = v(\alpha x, \alpha y, \alpha z,t)

To proceed from there, plug that into your equation for u and use the chain rule to figure out what \alpha should be in terms of \kappa to get the pure laplacian.