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Change of variables for PDE

  1. Apr 8, 2006 #1
    hi,
    i am having difficulty trying to find a change of variables to solve this partial differential equation
    [tex]\frac{\partial f}{\partial t} = t^\gamma \frac{\partial ^2 f}{\partial x^2}[/tex]
    not sure how to pluck out a change of variables by looking at the equation as its definately not obvious to the untrained eye (me).
    have tried
    [tex]f(x,t) = p(\eta) [/tex] (not sure if i am even on the right track doing this)
    where
    [tex]\eta = \frac{x}{t^{\gamma}} [/tex]
    but that just gets me in a giant mess with
    [tex]-\gamma\frac{x}{t}\frac{\partial p}{\partial \eta} = \frac{\partial ^2 p}{\partial \eta ^2} [/tex]
    i may have done [tex]\frac{\partial ^2 f}{\partial x^2}= \frac{\partial \eta}{\partial x}\frac{\partial ^2 p}{\partial \eta ^2}\frac{\partial \eta}{\partial x} [/tex] wrongly.

    any suggestions?
    thanks for the time
     
  2. jcsd
  3. Apr 9, 2006 #2
    I don't know about a substitution, but if you're just after a solution you can set f(x,t) = X(x)T(t) and the equation becomes separable.

    -Dan
     
  4. Apr 10, 2006 #3

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    There is no need to get "x" involved in your change of variable. You want to go from
    [tex]\frac{\partial f}{\partial t} = t^\gamma \frac{\partial ^2 f}{\partial x^2}[/tex]
    or, similarly,
    [tex]t^{-\gamma}\frac{\partial f}{\partial t} = \frac{\partial ^2 f}{\partial x^2}[/tex]
    to something like
    [tex]\frac{\partial f}{\partial \eta} = \frac{\partial ^2 f}{\partial x^2}[/tex]
    which means you want
    [tex]\frac{\partial f}{\partial \eta}= \frac{\partial f}{\partial t}\frac{dt}{d \eta}= t^{-\gamma}\frac{\partial f}{\partial t}[/tex]
    So you must have
    [tex]\frac{dt}{d\eta}= t^{-\gamma}[/tex]
    or
    [tex]\frac{d\eta}{dt}= t^{\gamma}[/tex]
     
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