Scaling a PDE

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

[tex]u_{t}=\kappa\left(u_{xx}+u_{yy}+u_{zz}\right)[/tex]

can be reduced to the standard form

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

Mute

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

[tex]u(x,y,z,t) = v(\alpha x, \alpha y, \alpha z,t)[/tex]

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