# Shock solution by conservation law

1. Apr 20, 2010

### lhy56839

1. The problem statement, all variables and given/known data

I need to find the shock solution for the initial value problem

$$u_t-u^2 u_x =0$$

with

$$u(x,0)=g(x)=\begin{cases}-\frac{1}{2}\quad x\leq 0 \\ 1\quad 0<x<1 \\ \frac{1}{2} \quad x\geq 0\end{cases}$$

2. Relevant equations

3. The attempt at a solution

Using the conservation law of the form $$u_t + \Phi_x = f(x,t)$$

we have

$$\Phi^{'}(u)=\frac{[\Phi(u)]}{}$$

The flux $$\Phi(u)$$ for this problem is given by
$$\Phi(u)=-\frac{1}{3}u^3$$
since
$$\Phi_x=\Phi^{'}(u)u_x$$

The discontinuity occurs at x=0 and x=1. For the discontinuity at x=0 we have

$$-[u(0)]^2 = \frac{[\Phi(u)]}{} = \frac{\frac{1}{3}((-\frac{1}{2})^3-1^3)}{-\frac{1}{2}-1}=\frac{3}{4}$$

and for x=1,

$$-[u(1)]^2 = \frac{[\Phi(u)]}{} = \frac{\frac{1}{3}(1^3-(-\frac{1}{2})^3)}{1-\frac{1}{2}}=\frac{3}{4}$$

I am not sure whether I am on the right track or not, and not sure how I obtain the shock solution from this. Any advice or help on this would be appreciated.

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