## Partial differential equation

Solve the following PDE if for $0\leq x < 1$ we have:

$U_{t}$ + $U_{x}$ = - $\frac{1}{1-x}$ $U$

$U(x,0)$ $=$ $0$ for $x\geq 0$

$U(0,t)$ $=$ $1-e^{-t}$ for $t>0$

and for $x>1$

$U_{t}$ + $U_{x}$ = 0

With the same conditions as before, that is:

$U(x,0)$ $=$ $0$ for $x\geq 0$

$U(0,t)$ $=$ $1-e^-{t}$ for $t>0$