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I need help solving $$u_t+[u(1-u)]_x=0$$ with initial conditions $$u(x,0) = \begin{cases} 0.75 & |x|<0.5\\ 0 & else \end{cases} $$ I am trying to use the method of characteristics but I am having trouble with the rarefaction wave part.

First, we can rewrite the PDE as $$u_t+(1-2u)u_x=0$$ So, we have $$\frac{dx}{dt}=1-2u$$ and we know that characteristics are constant because $$\frac{d u(x(t),t)}{dt}=\frac{\partial u}{\partial t} + \frac{\partial u}{\partial x}\frac{dx}{dl t} = u_t+u_x(1-2u)=0$$

So, we have $$\frac{d}{dt}\frac{dx}{d t}=1-2u(x,t)=1-2u(\xi,0)$$ and solving for this gives $$x=(1-2u)t+\xi$$

So, I have that there are characteristics that have slope $1-2u(\xi,0)$ so the slope is 1 for $x\leq-0.5$ and $x\geq -0.5$ and we have slope -2 for $-0.5<x<0.5$.

So, there is a shock at $x=-0.5$. The propagation can be found as $$s'=\frac{0(1-0)-0.75(1-0.75)}{0-0.75}=1/4$$ So the characteristic propagates as $$\frac{dx}{dt}=1/4$$ $$x(0)=-0.5$$ and we get $x(t)=-0.5+t/4$.

Then there is a rarefaction fan at $x=0.5$. I think that it starts at $x=-t/2 +0.5$ and ends at $x=t+0.5$. I am having a hard time figuring out what the equation is for the fan though.

So, my solution is $$u(x,t) = \begin{cases} 0 & x<-0.5+t/4\\ 0.75 & -0.5+t/4\leq x\leq -t/2+0.5\\ rarefaction & -t/2+0.5<x<t+0.5\\ 0 & x\geq t+0.5 \end{cases} $$ I would really appreciate help on how to define the rarefaction fan part. I have tried variations of $\frac{x-0.5}{t}$ and $\frac{t+x-0.5}{t-0.5}$ but they don't seem to work right.

Thank you.

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    $\begingroup$ You've written $u_x + (u(1 - u))_x = 0$; shouldn't there be a $t$-deriviative in there somewhere; otherwise, it's an ODE . . . $\endgroup$ – Robert Lewis Mar 26 at 4:37
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    $\begingroup$ Is it supposed to be $u_t + (1-2u)u_x = 0$? $\endgroup$ – Dylan Mar 26 at 4:39
  • $\begingroup$ @Dylan Sorry about that. You are right. I will edit it. $\endgroup$ – MathIsHard Mar 26 at 4:42
  • $\begingroup$ @RobertLewis you are definitely correct. There is a t derivative. Sorry about the typo there. I fixed it in the post $\endgroup$ – MathIsHard Mar 26 at 4:43
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I think I figured this out after working on this since yesterday afternoon almost nonstop. We need that the rarefaction satisfies $0.75$ at $x=-0.5t+0.5$ and we need that it also satisfies $0$ at $x=t+0.5$ since these are the initial densities to the left and right of the fan.

So with a bit of algebra, one can find that the rarefaction wave should behave like $$-0.5\bigg(\frac{x-0.5}{t}-1\bigg)$$ since then,

$$-0.5\bigg(\frac{(-0.5t+0.5)-0.5}{t}-1\bigg)=0.75$$ and $$-0.5\bigg(\frac{(t+0.5)-0.5}{t}-1\bigg)=0.$$

I jumped for joy when I got this :D

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    $\begingroup$ Nice work!----- $\endgroup$ – Dylan Mar 27 at 9:29

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