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Quasilinear PDE problem

  1. Oct 2, 2011 #1
    1. The problem statement, all variables and given/known data

    Solve

    [itex]\frac{\partial \phi}{\partial t} + \phi \frac{\partial \phi}{\partial x} - \infty < x < \infty , t > 0 [/itex]

    subject to the following initial condition

    [itex]\phi (x,0) = \left\{ \begin{array}{c}
    1,\; x<0\\
    1-x,\;0\leq x<1\\
    0,\; x\geq1\end{array}\right.[/itex]

    2. Relevant equations

    see 3

    3. The attempt at a solution

    Solving the PDE via method of characteristics, the characteristic lines are:

    [itex]x = \phi t + s[/itex]

    [itex]x < 0 : t = x - s[/itex]

    [itex]0 \leq x < 1 : t = \frac{x-s}{1-s}[/itex]

    [itex]x \geq 1 : x = s[/itex]

    My question is that I don't know where to find a shock. All characteristics originating in the region [itex] 0 \leq x < 1 [/itex] cross over at (1,1), but characteristics also cross over at x = 1.
     
  2. jcsd
  3. Oct 16, 2011 #2
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