Part (a): Identify the Green’s function for the Dirichlet problem on the domain $x > y > 0$ (i.e. the region in the first quadrant below the line $y = x$) by using image points.
$∇^2G(x, y|ξ, η) = δ(x − ξ)δ(y − η)$ on $y > 0$ for all $x$
$G(x, y = 0|ξ, η) = 0$
$G(x, y = x|ξ, η) = 0$
$G → 0$ as $r → ∞$
where $r^2=x^2+y^2$.
Part (b): Express the solution of the Dirichlet wedge problem
$∇^2u = 0$ on $y > 0$
$u(x, y = 0) = f(x)$
$u(x, y = x) = g(x)$
$u$ bounded as $r → ∞$
in terms of the Green’s function in the previous part. We’ll assume $f(0) = g(0)$. Note the
Green’s function in part (a) might be slightly cumbersome so just refer to it as $G(x, y|ξ, η)$
for the purposes of part (b).
Attempt:
For part (a) I attempted to write the image points in polar coordinates. I was having a hard time understanding how to cancel the effect of the points on the boundary $y=x$.
For part (b) I think you start with the equation $$\iint_\mathcal{D} [u∇^2G-G∇^2u]dA=\oint_{\mathcal{C}+\mathcal{C}_R}\left[u \frac{\partial G}{\partial n}-G \frac{\partial u}{\partial n}\right]ds$$.
