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.