Laplace's equation on a ring

[A_B]

Homework Statement

Solve Laplace's equation on a ring in the plane with r_1<r<r_2. And arbitrary functions on the edges of the ring.

The Attempt at a Solution

After separation of variables the solution to the radial factor is an often-seen problem. It's eigenvalues are λ=n² the eigenfunctions Θ(θ)~sin(nθ) + cos(nθ).

The differential equation for the radial part becomes

r²R" + rR' - n²R = 0

I don't know how to solve this thing but my book gives solutions:

n=0: r → 1 and r → ln(r)
n>0: r → r^n and r → r^(-n)

In my book the functions ln(r) and r^(-n) are thrown away because they are singular in r=0. But for this problem r=0 is not part of the domain, so I should keep all these solutions.

Now I wonder, since ln(r) and r^(-n) can both be written as a Taylor series on the interval [r_1,r_2], can I throw them away anyways?

If so, the solution will be of the form

u(r,θ) = Ʃ A_n r^n cos(nθ) + Ʃ B_n r^n sin(nθ)

Boundary condtions

u(r_1,θ) = Ʃ A_n r_1^n cos(nθ) + Ʃ B_n r_1^n sin(nθ)
u(r_2,θ) = Ʃ A_n r_2^n cos(nθ) + Ʃ B_n r_2^n sin(nθ)

But this seems to be fixing A_n and B_n twice, since both are Fourier series.
Thanks
A_B

 

[mighty2000]

Dear A_B,

Thank you for your post. Solving Laplace's equation on a ring in the plane with arbitrary functions on the edges can be challenging, but it can also be a very interesting problem.

Firstly, I agree with your approach of separating the variables and solving for the radial part first. However, I would like to point out that the solutions given in your book for n=0 and n>0 are only valid for the case where r_1=0 and r_2=1. Since we have a ring with r_1<r<r_2, we need to modify these solutions accordingly.

For n=0, the solution will be of the form u(r,θ) = A + B ln(r), where A and B are constants determined by the boundary conditions. This solution is valid for all values of r within the ring, including r=0.

For n>0, the solution will be of the form u(r,θ) = A_n r^n + B_n r^(-n), where A_n and B_n are constants determined by the boundary conditions. As you mentioned, these solutions are singular at r=0, but they can still be used within the ring, as long as we exclude r=0 from our domain.

Therefore, the general solution for Laplace's equation on a ring with arbitrary functions on the edges will be of the form

u(r,θ) = A + B ln(r) + Ʃ (A_n r^n + B_n r^(-n)) cos(nθ) + Ʃ (C_n r^n + D_n r^(-n)) sin(nθ)

where A, B, A_n, B_n, C_n, and D_n are constants determined by the boundary conditions.

I hope this helps. Keep up the good work in your studies!