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Change of variables in PDEs

  1. Feb 16, 2007 #1
    I need guidance regarding PDE.
    If u have a nonlinear PDE as
    where U is function of (s,t) and a,b are constants.
    by introducing new variable x=s-t we will get
    Ut means partial derivative w.r.t time
    Us means partial derivative w.r.t s.

    How can we get the second equation from the first one?
    Last edited: Feb 16, 2007
  2. jcsd
  3. Feb 16, 2007 #2


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    Science Advisor

    By using the chain rule.

    [tex]U_s= U_t \frac{\partial t}{\partial s}+ U_x\frac{\partial x}{\partial s}[/tex]
    Note: if you are going to use x= s- t to replace s only, you will need to think of s as a function of the other variable, t.
    If x= s- t, then s= s+ t so both partial derivatives are 1:
    [tex]U_s= U_t+ U_x[/itex]
    [tex]U_ss= (U_t+ U_x)_s= (U_t+ U_x)_t + (U_t+ U_x)_x= U_tt+ 2U_tx+ U

    [tex]U_sss= U_ttt+ 3Uttx+ 3Utxx+ Uxxx[/tex]
    Sustitute those into you equation.
  4. Feb 16, 2007 #3
    Subsituting these , will not give me the desired equation.
  5. Feb 16, 2007 #4
    I have solve it, same concept of chain rule but with different approach.
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