# Green's function for homogeneous PDE

Hi there, could anyone help me on this particularly frustrating problem I am having... I have a linear parabolic homogeneous PDE in two variables with a boundary condition that is a piecewise function.

I can solve the pde (with a homogeneous BC) however trying to impose the actual BC makes it seem impossible. I think that using Green's functions will help - as I then have a convolution of the green's function with the BC - but I am finding it difficult to find any literature on this case, i.e. homogen pde & nonhomogen BCs.

Could anyone point me in the right direction? Below is the pde...

$$\frac{\partial u}{\partial t}+ a x \frac{\partial^2 u}{\partial x^2} + b \frac{\partial u}{\partial x}=0$$

with BC:
$$u(x,T) = x-\rho \,\, \mbox{for}\,\, x>\rho$$
$$\qquad =0$$ otherwise

Thanks!