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Math Help - shock PDE help

  1. #1
    Newbie
    Joined
    Aug 2011
    Posts
    16

    shock PDE help

    Hi, I need a bit of help understanding what this question wants.

    (\frac{1}{2^5}u^4)_x + u_t =0

    1. How do I find the Rankine-Hugoniot shock speed condition for this?
    EDIT: Got this:

    s=\frac{F(u_L)-F(u_R)}{(u_L-u_R)}

    2. How do I find the rarefaction fan for this such that u(x,t)=g(x/t) is a solution.

    Thank You for any help.
    Last edited by ellenu485; August 11th 2011 at 11:57 PM.
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  2. #2
    Senior Member
    Joined
    Mar 2010
    Posts
    280

    Re: shock PDE help

    May be this helps.

    Inserting

    g(z)=g(x/t)

    to the equation we get

    \frac{1}{8} \; g^3 \; g_z \; z_x + g_z \; z_t =0


    g_z \; (\frac{1}{8} \; g^3 \; \frac{1}{t} \; -  \; \frac{x}{t^2}\; ) \; = \; 0

    g^3 \; = \; 8 \ z


    g \; = \; 2 \ z^{1/3} \; .
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