I'd like to know if this linear non-homogeneous PDE can be solved using Green's function
$$\frac{d^2u(x,y)}{dy^2}+\frac{d^2u(x,y)}{dx^2}+F\frac{du(x,y)}{dy}-\frac{u(x,y)}{\lambda^2}=-\frac{u_0(y)}{\lambda^2}-\frac{\phi}{\lambda^2} $$ with the following non-homogeneous BC: $$ \frac{du}{dy}=S_y(u-u_0(y))\quad, \quad y=-1/2, \\ u=u_0(y)\quad, \quad y=0, \\ \frac{du}{dx}=\mp S_x u\quad, \quad x=\pm1/2, \\ $$where $u_0(y)$ is a function of $y$ only (and known), and $F,S_x,S_y,\lambda,\phi$ are constants.
Any help would be greatly appreciated.