Shock solution by conservation law

[lhy56839]

Homework Statement

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

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

with

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

Homework Equations

The Attempt at a Solution

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

we have

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

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

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

and for x=1,

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


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.

 

[nate808]

Thank you for your question. my suggestion would be to first check if the given initial value problem has a solution or not. This can be done by solving the characteristic equations, which in this case are given by

dx/dt = -1
du/dt = u^2

Solving these equations, we get the characteristic curves as x = -t + C, u = 1/(1-Ct). From the initial condition, we can see that at x=0, u=1/2. Substituting these values in the characteristic curves, we get C=1/2 and u=1/(1-t/2).

Now, we can write the solution as u(x,t) = 1/(1-(x-t)/2). This solution is valid for x<=t. For x>t, we have u(x,t) = -1/(1+(x-t)/2). This solution is valid for x>=t.

Using the given initial condition, we can see that at x=0, u=-1/2. Substituting this value in the solution for x<=t, we get t=1. Similarly, at x=1, u=1/2. Substituting this value in the solution for x>=t, we get t=-1.

Therefore, the shock solution for this initial value problem is given by u(x,t) = \begin{cases}-1/(1+(x-t)/2) \quad x\leq t \\ 1/(1-(x-t)/2) \quad x>t\end{cases}.

I hope this helps. Let me know if you have any further questions.