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Help with PDE in circular annulus(poisson eq) 
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#1
Dec2811, 10:25 PM

P: 3

what is the general solution of the poisson equation :
∂^{2}A/∂r^{2} + 1/r ∂A/∂r + 1/r^{2} ∂^{2}A/∂θ^{2} = f(r,θ) the function f(r,θ) is : f(r,θ)=1/r (Ʃ X_{n}cos(nθ) + Y_{n}sin(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. 


#2
Dec2911, 12:29 PM

P: 263

Use the Green function for Neumann boundary conditions.



#3
Dec2911, 01:28 PM

Engineering
Sci Advisor
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Thanks
P: 7,105

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). 


#4
Feb2812, 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.
where g(r,θ,ρ,θ') = ln{[r^2 + ρ^2  2rρ cos(θθ')] [b^2 + (rρ/b)^2  2rρ cos(θθ')]}/4∏ + r^2/(4∏b^2) from third line of page 68 of Ref.2. Ref.1. https://rs5tl5.rapidshare.com/#!down...u2827R~000 Ref.2. http://www1.maths.leeds.ac.uk/~kersa...otes/chap4.pdf 


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