- #1
QuantumJG
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Homework Statement
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]
Homework Equations
see 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.