Help with PDE in circular annulus(poisson eq)

[smoger]

what is the general solution of the poisson equation :

∂²A/∂r² + 1/r ∂A/∂r + 1/r² ∂²A/∂θ² = f(r,θ)

the function f(r,θ) is :
f(r,θ)=1/r (Ʃ X_ncos(nθ) + Y_nsin(nθ))

where the boundary is :

I(a<r<b, 0<θ<2pi)

the boundary condition is the netural boundary on (r=a) expressed as :

∂A/∂r=0 (r=a)


How can i find the A(r,θ)? i can not find any books related to this.
Most of them only consider laplace equation where f(r,θ)=0
someone help me.

 

[Thaakisfox]

Use the Green function for Neumann boundary conditions.

 

[AlephZero]

Using the idea of separating the variables, you should be able to see from the PDE that

A(r,θ) = Cn(r) cos(nθ) + Sn(r) sin(nθ)

is a solution for the right hand side terms (1/r)(Xn cos(nθ) + Yn sin(nθ))

That will give you ordinary differential equations to solve for Cn(r) and Sn(r).

 

[ivan_f_costa]

A(r,θ)=∫∫f(ρ,θ') g(r,θ,ρ,θ') dρ dθ' + cte' from eq. 5.0.19 Ref.1.
where
g(r,θ,ρ,θ') = -ln{[r^2 + ρ^2 - 2rρ cos(θ-θ')] [b^2 + (rρ/b)^2 - 2rρ cos(θ-θ')]}/4∏ + r^2/(4∏b^2)
from third line of page 68 of Ref.2.

Ref.1. https://rs5tl5.rapidshare.com/#!download|5tl6|1682244|Duffy_D._Green_s_functions_with_applications__CRC__2001__T__404s_.djvu|2827|R~0|0|0

Ref.2. http://www1.maths.leeds.ac.uk/~kersale/Teach/M3414/Notes/chap4.pdf