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Scaling Problem in Diffusion Equation

  1. Feb 21, 2013 #1
    Hello everyone,

    I have a question that is bothering me a bit. I would be happy if you could give an idea or tell me a specific point to look at.

    Lets say that we have an arbitrary function that obeys diffusion equation:

    f = f(η), here η is the scaling parameter for pde which equals to x/√t
    (Diffusion eq. becomes [itex]df/d\eta=-2\eta d^{2}f/d\eta^{2}[/itex] after scaling)

    As I understood, diffusion eq. becomes an ode after scaling and it contains all the possible solutions for x and t pairs in η. Can we say that f(η) is invariant with respect to x and t? or in other words say ∂f/∂t=0 or not? What kind of physical information can we get just by lloking at f(η)?

    Thanks in advance!
     
  2. jcsd
  3. Feb 22, 2013 #2

    HallsofIvy

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    No, [itex]\partial f/\partial x[/itex] and [itex]\partial f/\partial x[/itex] are NOT 0 because [itex]\eta[/itex] is itself a function of x and t.

    [tex]\frac{\partial f}{\partial x}= \frac{df}{d\eta}\frac{\partial \eta}{\partial x}= \frac{1}{\sqrt{t}}\frac{df}{d\eta}[/tex]
    [tex]\frac{\partial f}{\partial t}= \frac{df}{d\eta}\frac{\partial \eta}{\partial t}= -\frac{x}{\sqrt{t^3}}\frac{df}{d\eta}[/tex]
     
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