Register to reply

Help with PDE in circular annulus(poisson eq)

by smoger
Tags: circular annulus, pde, poisson eq
Share this thread:
Dec28-11, 10:25 PM
P: 3
what is the general solution of the poisson equation :

2A/∂r2 + 1/r ∂A/∂r + 1/r22A/∂θ2 = f(r,θ)

the function f(r,θ) is :
f(r,θ)=1/r (Ʃ Xncos(nθ) + Ynsin(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.
Phys.Org News Partner Science news on
'Office life' of bacteria may be their weak spot
Lunar explorers will walk at higher speeds than thought
Philips introduces BlueTouch, PulseRelief control for pain relief
Dec29-11, 12:29 PM
P: 263
Use the Green function for Neumann boundary conditions.
Dec29-11, 01:28 PM
Sci Advisor
HW Helper
P: 7,288
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).

Feb28-12, 01:06 PM
P: 1
Help with PDE in circular annulus(poisson eq)

A(r,θ)=∫∫f(ρ,θ') g(r,θ,ρ,θ') dρ dθ' + cte' from eq. 5.0.19 Ref.1.
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.



Register to reply

Related Discussions
Potential Due to Annulus Introductory Physics Homework 0
CDF of the ratio of Poisson and possibly-Poisson R.V. Set Theory, Logic, Probability, Statistics 0
Analytic in an annulus Calculus & Beyond Homework 0
Laplace's equation on an annulus Precalculus Mathematics Homework 14
Annulus Area Mechanical Engineering 5