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Scaling a PDE

  1. May 29, 2008 #1
    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
     
    Last edited: May 29, 2008
  2. jcsd
  3. May 29, 2008 #2

    Mute

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    Homework Helper

    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.
     
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