The Extinguisher

2014-02-13, 12:12 PM

So there's a bunch of mathy people here, I figure someone should be able to help me. I found an answer, but it's going on five pages for an assignment that has only been <2 pages per question and it's not done yet, so I'm thinking I'm doing something wrong here.

I need to solve Laplace's equation on an annulus. This is what I'm given

Urr + (1/r)Ur + (1/r2)Uθθ = 0

0≤θ≤2π , R1 < r < R2

U(R1,θ) = f(θ) , U(R2,θ) = g(θ)

U(r,0) = U(r,2π) , Uθ(r,0) = Uθ(r,2π)

So I use separation of variables, and with my periodic condition, get

U(r,θ) = Sum from n=1 to ∞ [Ancos(nθ) + Bnsin(nθ)][Cnrn + Dnr-n]

I can't get rid of any other terms right? r doesn't go to zero, so r-n is a valid solution, which is the trick for most disks.

From here, I can use the boundary conditions to find the coefficients. Plugging them in and doing Fourier stuff to it, I get four equations for the four different unknowns. But like I said, I'm on page five, and I still haven't actually found the coefficients yet. So I imagine there's an easier way I'm just missing.

Any help would be greatly appreciated.

I need to solve Laplace's equation on an annulus. This is what I'm given

Urr + (1/r)Ur + (1/r2)Uθθ = 0

0≤θ≤2π , R1 < r < R2

U(R1,θ) = f(θ) , U(R2,θ) = g(θ)

U(r,0) = U(r,2π) , Uθ(r,0) = Uθ(r,2π)

So I use separation of variables, and with my periodic condition, get

U(r,θ) = Sum from n=1 to ∞ [Ancos(nθ) + Bnsin(nθ)][Cnrn + Dnr-n]

I can't get rid of any other terms right? r doesn't go to zero, so r-n is a valid solution, which is the trick for most disks.

From here, I can use the boundary conditions to find the coefficients. Plugging them in and doing Fourier stuff to it, I get four equations for the four different unknowns. But like I said, I'm on page five, and I still haven't actually found the coefficients yet. So I imagine there's an easier way I'm just missing.

Any help would be greatly appreciated.