Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Change of variables for PDE

  1. Apr 8, 2006 #1
    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)
    [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.

  4. Apr 10, 2006 #3


    User Avatar
    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]
    [tex]\frac{d\eta}{dt}= t^{\gamma}[/tex]
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook