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Thread: method of characteristic

  1. #1
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    method of characteristic !!!

    Hi everyone,
    i have a problem for solving a partial differential equation with the method of characteristic. I already spent lot of time and I have not yet found the solution


    In fact, when I try to solve du/dt after I already obtained before dx/dt and dy/dt, I have an expression an expression which depends on t and u. Impossible to solve...
    If someone can help me pleasseeeee!!!!

    Thankss
    Last edited by metouka; Nov 21st 2012 at 01:46 PM.
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  2. #2
    Newbie
    Joined
    Nov 2012
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    france
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    Re: method of characteristic

    hi,
    I advanced in my problem but I'm again in trouble
    I transformed my expression with polar coordinates $\displaystyle v(\frac{dv}{dr})=r$
    so $\displaystyle v=r+c_{3}$ (c is my constant) $\displaystyle u=sqrt(x^2+y^2)+c_{3}$

    Now I have to find the value of my constant by caracteristics method
    so I continue my reasoning:

    $\displaystyle \frac{dr}{dt}=v=r+c_{3}$ such as $\displaystyle r=e^(t)c_{1}-c_{3}$
    $\displaystyle \frac{d(\theta)}{dt}=0 $ such as $\displaystyle \theta=c_{2}$
    $\displaystyle \frac{dv}{dt}=r $ we find before that $\displaystyle v=r+c_{3}$

    moreover with my initial condition $\displaystyle v(rcos(\theta),1)=\sqrt{(r^2.cos^2(\theta)+1)}$
    so i guess $\displaystyle r=(\frac{s}{cos(\theta)})$
    $\displaystyle \theta=1$
    $\displaystyle v=sqrt{(s^2+1)}$

    for t=0, $\displaystyle c_{1}-c_{3}=\frac{s}{cos(\theta)}$
    $\displaystyle c_{2}=1$
    $\displaystyle c1=sqrt{(s^2+1)}$ donc que$\displaystyle c_{3}=sqrt{(s^2+1)}-\frac{s}{cos(\theta)}$

    so $\displaystyle u=e^(t)sqrt{(s^2+1)}-(sqrt{(s^2+1)}-\frac{s}{cos(\theta)})=sqrt{(s^2+1)}(e^(t)-1) + (\frac{s}{cos(\theta)}$

    in short I feel that's wrong
    someone can help me please??
    thanks
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