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

Express your answer explicitly in two parts.