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Help with PDE in circular annulus(poisson eq)

  1. Dec 28, 2011 #1
    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.
     
    Last edited: Dec 28, 2011
  2. jcsd
  3. Dec 29, 2011 #2
    Use the Green function for Neumann boundary conditions.
     
  4. Dec 29, 2011 #3

    AlephZero

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    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).
     
  5. Feb 28, 2012 #4
    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 [Broken]

    Ref.2. http://www1.maths.leeds.ac.uk/~kersale/Teach/M3414/Notes/chap4.pdf
     
    Last edited by a moderator: May 5, 2017
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