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

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

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

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

and for $\displaystyle x>1$

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

With the same conditions as before, that is:

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

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

Express your answer explicitly in two parts.