Green function for Homogeneous diffusion equation in the semi infinite domain with inhomogenous neumann boundary conditions

Question

The Initial Boundary Value-Problem in the semi infinite domain $0 \leq x < \infty$ is given by $$ k u_{xx} - u_{t} = 0 \\ u(x,0) = u_0 \\ -u_x (0,t) = f(t) \\ -u_x (x \rightarrow \infty , t) = constant \, . $$ I know that the greens function to this problem should be $$ G(x,t) = \frac{2\sqrt{kt}}{\sqrt{\pi}} e^{-\frac{x^2}{4kt}} - x \text{erfc}\left( \frac{x}{2\sqrt{kt}} \right) \, . $$ My question is, how do you come up with this solution? The first term is nothing else than the transient solution or fundamental solution for the heat equation in the whole domain. But how one comes to the second part is a mystery to me. I actually thought that Green's functions only apply to homogeneous boundary conditions.