## Find exact solution of transport equation. [closed]

Hi. I have the PDE

$u_t+\gamma u_x = 0$

$u(x,0) = 0$ for $x > 0$

$u(0,t) = 1$ for $t \geq 0$

I know that the solution is of the form $$u(x,t)=f(x-\gamma t)$$ where $f(x) = 0$ for $x>0$, so we also see that $f(-\gamma t)=1$ for $t\geq 0$, but I can't find an analytical solution to this problem. Can anyone please help?

Yes, this is homework, and I'm not searching for just the answer, I need help to find it and to understand it.

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I think you will find that this question will be closed as soon as people wake up on the West Coast :) Having said that, you might wish to look into the method of characteristics. A good place to start would be the relevant wikipedia page: en.wikipedia.org/wiki/Method_of_characteristics – José Figueroa-O'Farrill Mar 22 2010 at 11:57
I know of this method, and this is what led me to $u(x,t)=f(x-\gamma t$, but I still can't find an expression for u(x,t) since the boundary values are constants. So it's that part I need help with. So what I know is $f(x)=1$ if $x>0$ and $f(0)=1$, and so it stops. – martiert Mar 22 2010 at 12:39
Some characteristics emanate from the $x$ axis, others from the $t$ axis. You get $f(-\gamma t)=0$ only for $t\ge0$. I am casting the initial vote to close now. – Harald Hanche-Olsen Mar 22 2010 at 13:47
Unfortunately, this site isn't for homework help. See mathoverflow.net/faq – jc Mar 22 2010 at 14:39