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/#!down...u|2827|R~0|0|0

Ref.2. http://www1.maths.leeds.ac.uk/~kersa...otes/chap4.pdf