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Homework Help: From PDE to ODE ? + research

  1. Mar 13, 2010 #1
    From PDE to ODE ?!! + research

    1. The problem statement, all variables and given/known data

    In the attached research, What are the steps that we work to transform the equation (1) to (8)

    2. Relevant equations

    (1) and (8)

    3. The attempt at a solution

    I know that they used similarity transformations but I do not know how to do it by myself (how to substitute correctly) ...

    can you help me to I understand the steps ... then I will do the others In the same way.

    Thanks
     

    Attached Files:

  2. jcsd
  3. Mar 13, 2010 #2

    phyzguy

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    Re: From PDE to ODE ?!! + research

    Do you know how to apply the chain rule for partial differentiation? Try going to Wikipedia and searching on "chain rule", and reading the section on several variables. Then I think if you apply this to the research paper you attached you will see how they arrived at the equations. Basically, they chose a new set of variables so that after applying the chain rule and collecting terms, many terms cancelled and the original set of PDEs was converted into a set of ODEs.
     
  4. Mar 13, 2010 #3
    Re: From PDE to ODE ?!! + research

    Thank you for this hint...I will try it :) and read:

    http://en.wikipedia.org/wiki/Chain_rule

    https://www.physicsforums.com/library.php?do=view_item&itemid=353
     
  5. Mar 13, 2010 #4
    Re: From PDE to ODE ?!! + research

    Q: How do I choose the appropriate similarity transformations to a particular system?!​
     
  6. Mar 13, 2010 #5
    Re: From PDE to ODE ?!! + research

    please ... help me ... just the first term in eq(1)

    I try but failed :(
     
  7. Mar 13, 2010 #6

    phyzguy

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    Re: From PDE to ODE ?!! + research

    Show us your calculations on the first term in Equation 1. Learn how to use TeX so you can type it in.
     
  8. Mar 13, 2010 #7
    Re: From PDE to ODE ?!! + research

    I will try :(
     
  9. Mar 13, 2010 #8
    Re: From PDE to ODE ?!! + research

    is it right that u is in 3 variables : x and Fi Dash and M ?
     
  10. Mar 13, 2010 #9
    Re: From PDE to ODE ?!! + research

    please ... Is this true?!! I took the second term in eq(1)



    [tex]\partial w[/tex] / [tex]\partial z[/tex]

    = ( [tex]\partial w[/tex] / [tex]\partial \varphi[/tex] ) . ( [tex]\partial \varphi[/tex] / [tex]\partial \eta[/tex] ) . ( [tex]\partial \eta[/tex] / [tex]\partial z[/tex] )

    = (- [tex]\sqrt{b\nu}[/tex] ) . ( [tex]\grave{\varphi}[/tex] ) .( [tex]\sqrt{b/\nu}[/tex] )

    = -b [tex]\grave{\varphi}[/tex]​
     
  11. Mar 13, 2010 #10

    phyzguy

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    Re: From PDE to ODE ?!! + research

    I think this is correct. Now if you look at the first term in equation 1, you see:

    [tex]\frac{\partial{u}}{\partial{x}} = b \phi'(\eta)[/tex]

    so the two terms cancel and equation 1 is automatically satisfied. Keep going!
     
  12. Mar 13, 2010 #11
    Re: From PDE to ODE ?!! + research

    Thanks ...I appreciate it

    but how do I get the equation (8) ?!!

    and

     
    Last edited: Mar 13, 2010
  13. Mar 13, 2010 #12

    phyzguy

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    Re: From PDE to ODE ?!! + research

    Yes, you need to consider u to be a function of x, [tex]\phi'[/tex], and M. You need to evaluate all of the partial derviatives, like you did for

    [tex]\frac{\partial{w}}{\partial{z}}[/tex]

    and plug these into equations 2-5, and you should come out with equations 8-12.
     
  14. Mar 13, 2010 #13
    Re: From PDE to ODE ?!! + research

    ok, but eq (1) say:

    [tex]\frac{\partial{u}}{\partial{x}}[/tex] + ( [tex]\partial w[/tex] / [tex]\partial z[/tex]) =0

    so, I have evaluate all of the partial derviatives but How can I have eq (8) :( I do not understand !!!!!!!
     
  15. Mar 13, 2010 #14

    phyzguy

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    Re: From PDE to ODE ?!! + research

    Equation 8 comes from equation 2. Since u has a term proportional to [tex]\phi'[/tex], when you evaluate
    [tex]\frac{\partial^2{u}}{\partial{z^2}}[/tex] , you will get a term in [tex]\phi'''[/tex] . Do you see?
     
  16. Mar 14, 2010 #15
    Re: From PDE to ODE ?!! + research

    ok ... I will try :)

    Thank you
     
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