# Math Help - shock PDE help

1. ## shock PDE help

Hi, I need a bit of help understanding what this question wants.

$(\frac{1}{2^5}u^4)_x + u_t =0$

1. How do I find the Rankine-Hugoniot shock speed condition for this?
EDIT: Got this:

$s=\frac{F(u_L)-F(u_R)}{(u_L-u_R)}$

2. How do I find the rarefaction fan for this such that $u(x,t)=g(x/t)$ is a solution.

Thank You for any help.

2. ## Re: shock PDE help

May be this helps.

Inserting

$g(z)=g(x/t)$

to the equation we get

$\frac{1}{8} \; g^3 \; g_z \; z_x + g_z \; z_t =0$

$g_z \; (\frac{1}{8} \; g^3 \; \frac{1}{t} \; - \; \frac{x}{t^2}\; ) \; = \; 0$

$g^3 \; = \; 8 \ z$

$g \; = \; 2 \ z^{1/3} \; .$