Laplace equation in an annulus with Neumann BCs

[bschnei]

Homework Statement

Solve Laplace's equation inside a circular annulus (ring) (a < r < b) subject to the BCs:

$$\frac{\partial{u}}{\partial{r}}\left({a,}\theta\right)&={f}\left(\theta\right)$$

$$\frac{\partial{u}}{\partial{r}}\left({b,}\theta\right)&={g}\left(\theta\right)$$

Homework Equations

$$\nabla^{2}{u}&=0$$

I also believe because of the physics of a ring, we have additional BCs:

$${u}\left({r,-}\pi\right)&={u}\left({r,}\pi\right)$$

$$\frac{\partial{u}}{\partial{\theta}}\left({r,-}\pi\right)&=\frac{\partial{u}}{\partial{\theta}}\left({r,}\pi\right)$$

The Attempt at a Solution

I have used separation of variables to arrive at a general solution:

$${u}\left({r,}\theta\right)&={A}_0+{B}_0\ln{r}+\sum^{\infty}_{n=1}{r}^{n}\left({A}_{n}\cos{n}\theta+{B}_{n}\sin{n}\theta\right)+{r}^{-n}\left({C}_{n}\cos{n}\theta+{D}_{n}\sin{n}\theta\right)$$

This solution takes into account the periodic boundary conditions specified above under "Relevant Equations", but I am struggling to correctly apply the Neumann BCs specified by the problem in order to arrive at equations for the arbitrary constants.

I try to take the partial derivative of u(r,theta) with respect to r and arrive at:

$$\frac{\partial{u}}{\partial{r}}&=\frac{{B}_0}{r}+\sum^{\infty}_{n=1}{n}{r}^{n-1}\left({A}_{n}\cos{n}\theta+{B}_{n}\sin{n}\theta\right)-{nr}^{-n-1}\left({C}_{n}\cos{n}\theta+{D}_{n}\sin{n}\theta\right)$$

I don't know how I can apply Fourier series (assuming that is what I'm supposed to do) to solve for each of the coefficients (A,B,C,D).

 

[Nino MMP]

I am aware that I need 4 equations to solve for the 4 coefficients, but I'm not sure how to obtain those equations. Any help would be greatly appreciated.