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Shock solution by conservation law

  1. Apr 20, 2010 #1
    1. The problem statement, all variables and given/known data

    I need to find the shock solution for the initial value problem

    [tex]u_t-u^2 u_x =0[/tex]

    with

    [tex]u(x,0)=g(x)=\begin{cases}-\frac{1}{2}\quad x\leq 0 \\ 1\quad 0<x<1 \\ \frac{1}{2} \quad x\geq 0\end{cases} [/tex]

    2. Relevant equations



    3. The attempt at a solution

    Using the conservation law of the form [tex]u_t + \Phi_x = f(x,t)[/tex]

    we have

    [tex] \Phi^{'}(u)=\frac{[\Phi(u)]}{} [/tex]

    The flux [tex]\Phi(u)[/tex] for this problem is given by
    [tex]\Phi(u)=-\frac{1}{3}u^3[/tex]
    since
    [tex]\Phi_x=\Phi^{'}(u)u_x[/tex]

    The discontinuity occurs at x=0 and x=1. For the discontinuity at x=0 we have

    [tex]-[u(0)]^2 = \frac{[\Phi(u)]}{} = \frac{\frac{1}{3}((-\frac{1}{2})^3-1^3)}{-\frac{1}{2}-1}=\frac{3}{4}[/tex]

    and for x=1,

    [tex]-[u(1)]^2 = \frac{[\Phi(u)]}{} = \frac{\frac{1}{3}(1^3-(-\frac{1}{2})^3)}{1-\frac{1}{2}}=\frac{3}{4}[/tex]


    I am not sure whether I am on the right track or not, and not sure how I obtain the shock solution from this. Any advice or help on this would be appreciated.
     
  2. jcsd
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