Results 1 to 2 of 2

Math Help - DE Hyperbolic change of scale

  1. #1
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5

    DE Hyperbolic change of scale

    x=\mu\xi \ \ \ y=\nu\eta

    I don't understand what the book is getting at and how this is done.

    The final step is a "change of scale," where mu and nu are chosen so that in the transformed equation the coefficients of \omega_{\xi\xi}, \ \omega_{\eta\eta}, \ \mbox{and} \ \omega are equal in absolute value. We have

    \displaystyle\frac{\partial^2}{\partial x^2}=\frac{1}{\mu^2}\frac{\partial^2}{\partial\xi^  2}, \ \frac{\partial^2}{\partial y^2}=\frac{1}{\nu^2}\frac{\partial}{\partial\eta^2  },

    and equation \omega_{xx}-4\omega_{yy}-\frac{5}{4}\omega=0.

    The codition

    \displaystyle\frac{1}{\mu^2}=\frac{4}{\nu^2}=\frac  {5}{4}

    will be satisfied if \mu=\frac{2}{\sqrt{5}}, \ \nu=\frac{4}{\sqrt{5}}.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Jester's Avatar
    Joined
    Dec 2008
    From
    Conway AR
    Posts
    2,383
    Thanks
    51
    What the book is getting at is to transform your PDE to

    \omega_{\xi \xi}-\omega_{\eta \eta}-\omega=0\;\;\;(*)

    Under the change of variables we obtain

    \dfrac{1}{\mu^2}\omega_{\xi \xi}-4\dfrac{1}{\nu^2}\omega_{\eta \eta}-\dfrac{5}{4}\omega=0.

    To hit the target (*) choose

    \dfrac{1}{\mu^2} = \dfrac{5}{4},\;\;\;4\dfrac{1}{\nu^2} = \dfrac{5}{4}.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 0
    Last Post: July 19th 2011, 12:45 PM
  2. scale?
    Posted in the Algebra Forum
    Replies: 13
    Last Post: April 17th 2010, 05:30 AM
  3. Log-scale
    Posted in the Math Topics Forum
    Replies: 5
    Last Post: December 22nd 2009, 11:14 AM
  4. Richter scale
    Posted in the Algebra Forum
    Replies: 1
    Last Post: October 4th 2009, 12:37 PM
  5. Replies: 0
    Last Post: November 27th 2008, 04:35 AM

/mathhelpforum @mathhelpforum