Method of Images - Green's Function

[OliviaB]

Use the method of images to find a Green's function for the problem in the attached image.

Demonstrate the functions satisfies the homogenous boundary condition.

 

[OliviaB]

It's taken me ages, and I really am struggling to understand this, so I am not sure if this is correct but here goes...

\bigtriangledown^2 G = \delta(\underline{x} - \underline{x}_0) \frac{\partial G(x,0)}{\partial y} = 0 for y \geq 0 \ - \infty < x < \infty

Let r = |\underline{x} - \underline{x}_0 | = \sqrt{(x - x_0)^2 + (y - y_0)^2}

be the distance between \underline{x} and \underline{x}_0

Then \bigtriangledown^2 G = \delta(\underline{x} - \underline{x}_0) becomes

\bigtriangledown^2 G = \frac{1}{r} \frac{\partial}{\partial r} \Big( r \frac{\partial}{\partial r} \Big) = 0

everywhere (although not at r = 0) and subject to

\iint\limits_{\infty} \bigtriangledown G dV = \iint\limits_{\infty} \delta (0) dV = 1

which gives

G(r) = A \ln r + B

A = \frac{1}{2 \pi}

Choosing B = 0

G = \frac{1}{2 \pi} \ln r = \frac{1}{2 \pi} \ln |\underline{x} - \underline{x}_0|

I don't think this is the method of images though...(Headbang)