Hi,
I am one exam away from finishing my undergraduate degree which is in a few days, but I am struggling with a particular concept which is Green's functions for PDE's. I really struggle with this for some reason. I have a number of questions that I will post with my attempt at a solution. Here is my first question:


Let $\displaystyle \mathcal{R}$ be a bounded region in $\displaystyle \mathbb{R}^2$. Let $\displaystyle G$ be a Green's function such that


$\displaystyle \displaystyle \bigtriangledown G = \delta(x-x_0) \ \, x \in \ \mathcal{R} \ , G=0 , \ x \in \partial \mathcal{R}$

Show that the solution of the equation

$\displaystyle \displaystyle \bigtriangledown^2 u = f(x) \, x \in \ \mathcal{R} , u = h(x), x \in \partial \mathcal{R} $

is

$\displaystyle \displaystyle u(x) \int \int_{\mathcal{R}} G(x;x_0)f(x_0) dA_0 + \oint_{\partial \mathcal{R}} h(x_0) \bigtriangledown_{x_0} G(x;x_0) \cdot n ds_0$


Here is my attempt at a solution....but I am sure if my working is sufficient....


$\displaystyle \displaystyle \int \int (u \bigtriangledown^2 G - G \bigtriangledown^2 u ) dA_0 = \oint (u \bigtriangledown G - G \bigtriangledown u) \cdot n \ ds $

(this is one of Green's Identities)

$\displaystyle \displaystyle \int \int u \delta (x - x_0) - G f dA_0 = \oint (h \bigtriangledown G - G \bigtriangledown u) \cdot n \ ds_0$

therefore

$\displaystyle \displaystyle u(x) \int \int_{\mathcal{R}} G(x;x_0)f(x_0) dA_0 + \oint_{\partial \mathcal{R}} h(x_0) \bigtriangledown_{x_0} G(x;x_0) \cdot n \ ds_0$