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Thread: DE Hyperbolic change of scale

  1. #1
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    DE Hyperbolic change of scale

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

    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 $\displaystyle \omega_{\xi\xi}, \ \omega_{\eta\eta}, \ \mbox{and} \ \omega$ are equal in absolute value. We have

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

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

    The codition

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

    will be satisfied if $\displaystyle \mu=\frac{2}{\sqrt{5}}, \ \nu=\frac{4}{\sqrt{5}}.$
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  2. #2
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    What the book is getting at is to transform your PDE to

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

    Under the change of variables we obtain

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

    To hit the target (*) choose

    $\displaystyle \dfrac{1}{\mu^2} = \dfrac{5}{4},\;\;\;4\dfrac{1}{\nu^2} = \dfrac{5}{4}.$
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