(adsbygoogle = window.adsbygoogle || []).push({}); 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.

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# Homework Help: Quasilinear PDE problem

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