1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook