I have been working through the book "A Guide to First-Passage Processes" and wanted to branch out on my own doing a calculation similar to what occurs in chapter 6. My basic problem comes from the solution to the Helmholtz equation for a sphere, with $k^{2} < 0$. I solved my basic case, where my BC are only radial, and vanish at the origin and at r=R (size of the sphere), but when I move to a more complicated (interesting) geometry, I start running into pretty awful problems with my Green's function.
The Problem I have solutions to the 'imaginary' Helmholtz equation with azimuthal symmetry.
$$ (\nabla^{2} + k^{2})\psi = 0, k^{2} \le 0 $$
Which has solutions of the form (taking out the imaginary part and putting it in the argument as $ix = kr$), and spherical bessel functions:
$$ \psi(r, \theta) = \sum\limits_{l=0}^{\infty}[A_{l}j_{l}(ix)+B_{l}y_{l}(ix)]P_{l}(cos(\theta)) $$
Fine so far. Now comes the bad part with the boundary conditions. I have a boundary where the Green's function ($\psi$) vanishes (Dirichlet) on a straight line along the z cartesian axis, written as a delta function:
$$ G(\frac{1}{2\pi r^{2}}\delta(x)(\delta(cos\theta - 1) + (cos\theta + 1))) = 0 $$
And then a Neumann boundary condition at the surface of the sphere (R):
$$ \nabla G \cdot \hat{r} = 0 $$
The second boundary condition is easy to deal with. The first one, now with delta functions in $\theta$ is hard. When we just look at the angular portion, I was looking at:
$$ a(x,x') = \sum\limits_{l=0}^{\infty}N_{l}(x')P_{l}(x), x = cos(\theta) $$
Then the angular boundary conditions give me $$ a(\pm 1,x') = 0 = \sum\limits_{l=0}^{\infty}N_{l}(x')(\pm 1)^{l} $$
Which drives all of the $N_{l}(x') = 0$. I have been looking at my copy of Jackson, Riley Hobson and Bence, and also looking at google, but I can't necessarily seem to find a solution. I am going to read up on Singular Liouville Problems, since that seems to be related to the angular portion. My main questions are:
- What should I do for the angular portion of my Green's function? I can make it more physical and have a rod of radius a from -R to R along the z-axis, which changes my boundary conditions, and couples the spherical bessel functions to the legendre polynomials.
- When dealing with my multivariable Green's function, is it okay to write the two parts separately using different coefficients i.e. 4 coefficients in general for the two solutions $(r \lt r_{0}, r \gt r_{0}, \theta \lt \theta_{0}, \theta \gt \theta_{0})$, or do I have to move to 8 coefficients to deal with both the radial and angular portions? I am pretty sure the former is ok, if the angular and radial parts can be separated.