I've recently been studying Green's functions for elliptic PDEs and I'm struggling with how to make sense of it and especially how to apply it to tangible examples.
I've been asked to find a solution to:
$\nabla^2u = f $ in $\{(x,y)|\ x>0\ ,\ y>0\}$
$ \frac{\partial u}{\partial x} = g(y) $ on $x=0$
$ \frac{\partial u}{\partial y} = h(x)$ on $y=0$
using Green's functions. I know all the properties that the Green's function should have but I'm just struggling to see how that can be actually used in this example. Any help would be greatly appreciated.