Possion equation in circular annulus-II

[smoger]

I previosuly asked about possion equation in circular annulus.
I found the general soultion of the problem myself. but i meet a another problem and still can not find the solution.

The problem is :

∂²A/∂r² + 1/r ∂A/∂r + 1/r² ∂²A/∂θ²= f(r,θ) in the circular annulus (a<r<b)

where f(r,θ)=1/r + X_n/r cos(nθ)

The boundary condition is :

1. ∂A/∂θ=0 (at r=a)
2. ∂A/∂r =0 (at θ=±β)


I solved the problem as ,

Seperating variables as :
A(r,θ)=G(r)+R(r)T(θ) , assuming that
G(r) should satisfy 1/r , and R(r)T(θ) should satisfy X_n/r cos(nθ) term in f(r,θ)
Then the problem for G(r) is :

∂²G(r)/∂r² + 1/r ∂G(r)/∂r = 1/r (1)

and Considering the boundary coundition 2,

∂G(r)/∂r =0 at θ=±β (2)

But substituding (2) in (1) becomes 0.
How can i solve the problem? thanks in advance

 

[jvicens]

Thank you for sharing your progress on solving the Poisson equation in a circular annulus. It seems like you have made some good progress and have a solid understanding of the problem. However, I can see why you are facing a difficulty in solving the problem with the given boundary conditions.

Firstly, let's take a closer look at the boundary condition 2, where ∂G(r)/∂r = 0 at θ=±β. This condition implies that the function G(r) is independent of θ at the boundaries. However, in your solution, you have assumed that G(r) satisfies the term 1/r in the equation. This means that G(r) should also have a dependence on θ, which contradicts the boundary condition.

To solve this problem, you can try separating the variables as A(r,θ) = R(r)T(θ) instead of A(r,θ) = G(r) + R(r)T(θ). This will allow you to satisfy both boundary conditions, as R(r) will have a dependence on θ and T(θ) will have a dependence on r.

I hope this helps and good luck with solving the problem. Don't hesitate to reach out if you have any further questions or need any clarification. Keep up the good work!