My question is about solving a Quasilinear PDE without a shock

[QuantumJG]

Homework Statement

Solve

\frac{\partial \phi}{\partial t} + \phi \frac{\partial \phi}{\partial x} - \infty < x < \infty , t > 0

subject to the following initial condition

\phi (x,0) = \left\{ \begin{array}{c}<br /> 1,\; x&lt;0\\<br /> 1-x,\;0\leq x&lt;1\\<br /> 0,\; x\geq1\end{array}\right.

Homework Equations

see 3

The Attempt at a Solution

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

x = \phi t + s

x &lt; 0 : t = x - s

0 \leq x &lt; 1 : t = \frac{x-s}{1-s}

x \geq 1 : x = s

My question is that I don't know where to find a shock. All characteristics originating in the region 0 \leq x &lt; 1 cross over at (1,1), but characteristics also cross over at x = 1.

 

[Coelum]

Assuming you mean (please check your post)

\Phi_t+\Phi\Phi_x=0,

then Professor Strang discusses this equation and the propagation of the shock here:

http://ocw.mit.edu/courses/mathemat...ure-8-convection-diffusion-conservation-laws/