Laplace's equation on an annulus with Nuemann BCs

[nathan12343]

Homework Statement

Solve Laplace's equation inside a circular annulus $(a<r<b)$ subject to the boundary conditions $\frac{\partial{u}}{\partial{r}}(a,\theta) = f(\theta)\text{, }\frac{\partial{u}}{\partial{r}}(b,\theta) = g(\theta)$

Homework Equations

Assume solutions of the form $u(r,\theta) = G(r)\phi(\theta)$. This leads to an equation for phi and G, both of which can be solved by substitution:

$$\frac{d^2\phi}{{d\theta}^2} = -\lambda^2\phi$$
$$\phi = A\cos{\lambda\theta} + B\sin{\lambda\theta}$$

$$r^2\frac{d^2G}{{dr}^2} + r\frac{dG}{dr} - n^2G = 0$$
$$G = c_{1n}r^{-n} + c_{2n}r^n\text{ for } n\ne0$$
$$G = c_{10} + c_{20}\ln(r)\text{ for } n=0$$

The Attempt at a Solution

Periodic boundary conditions require that u and its derivative with respect to theta be continuous between -pi and pi. Then means $\lambda = n$ for all n greater than or equal to zero. We can write,

$$u(r,\theta) = &\, A_{01} + A_{02}\ln(r) + \sum_{n=1}^{\infty}(A_{n1}r^n + A_{n2}r^{-n})\cos{n\theta} + (B_{n1}r^n + B_{n2}r^{-n})\sin{n\theta}$$
$$\frac{\partial{u(r,\theta)}}{\partial{r}} = &\, A_{02}r^{-1} + \sum_{n=1}^{\infty}(nA_{n1}r^{n-1} + -nA_{n2}r^{-n-1})\cos{n\theta} + (nB_{n1}r^{n-1} - nB_{n2}r^{-n-1})\sin{n\theta}$$

I know that I can set everything in parentheses in front of the cosines and sines above equal to some constant when I set r=a,b to enforce the boundary conditions on the edge of the annulus. This leaves me with two equations in two unknowns for all the A's and B's with n greater than 1. My problem is how to set [itex]A_{02}[/tex] such that it will work at both boundaries. It seems like I really need two constants to match to the boundary for when $$n=0$$. Am I missing something?

Thanks for your help!

 

[nathan12343]

Please disregard this - I meant to post in Calculus & Beyond.

 

[Architeuthis Dux]

Your approach to solving Laplace's equation on an annulus with Neumann boundary conditions is correct. However, you are correct in noticing that you need two constants to match the boundary conditions for n=0. This is because the solution for G in this case involves a logarithm term, which cannot be matched to a constant value at both boundaries. Instead, you will need to match the derivative of G at both boundaries, which will give you two equations in two unknowns (A_{02} and c_{20}). This will allow you to solve for both constants and obtain a complete solution for u(r,\theta). Good job on recognizing this issue and working towards a solution!