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

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

Homework Equations

see 3

The Attempt at a Solution

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

$x = \phi t + s$

$x < 0 : t = x - s$

$0 \leq x < 1 : t = \frac{x-s}{1-s}$

$x \geq 1 : x = s$

My question is that I don't know where to find a shock. All characteristics originating in the region $0 \leq x < 1$ 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/