I have been recently learning (or trying to learn) about Green's functions. For some reason, it is a concept which I am having great difficulty understanding. I have a question here and I was wondering if someone would be able to give me some tips on what to do? Any help would be much appreciated
Consider a domain $D = \{(x,y) |x>0, y>0\}$, Let x$=(x,y)$ and $\xi=(\xi_x,\xi_y)$.
Find the Greens Function, $G($x, $\xi)$, such that $\nabla^2G = \delta($x$-\xi),\space x \in D$,
subject to $G(0,y,\xi) = 0$, for $y>0$
and $\frac{\partial G}{\partial y}(x,0,\xi) = 0$, for $x>0$
Use this to solve $\nabla^2u=0$, $x\in D$ subject to u tends to zero as $|x| \rightarrow \infty$
$u(0,y) = h(y)$ for $y>0$
and $\frac{\partial u}{\partial y}(x,0) = g(x)$ for $x>0$
I believe that I have found the appropriate Green's function but now I am unsure how to use this to find what $u$ is when there are both Dirichlet and Neumann boundary conditions
The Green's function that I found was: $\frac{1}{4\pi}\ln\big|\frac{((x-\xi_x)^2+(y-\xi_y)^2)((x-\xi_x)^2+(y+\xi_y)^2)}{((x+\xi_x)^2+(y+\xi_y)^2)((x+\xi_x)^2+(y-\xi_y)^2)}\big|$
And then I used the following formula: $u(\xi_x,\xi_y) = -\int_0^\infty Gg(x)|_{y=0}\space dx\space- \int_0^\infty h(y)\frac{\partial G}{\partial n}|_{n=0}\space dy$ And got my answer as:
$-\frac{1}{2\pi}\int_0^\infty g(\alpha)\ln\frac{(\alpha-x)^2+y^2}{(\alpha+x)^2+y^2}d\alpha + \frac{1}{2\pi}\int_0^\infty h(\beta) \frac{(\beta - x)^2}{(\beta -x)^2+y^2}d\beta$
I would much appreciate it if someone would be able to tell me if I am correct or if I am on the right path
I've read many PDF lecture notes from the first page of google but I am still finding it difficult to understand what to do.