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MathGroup Archive 2005

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Re: Urgent

  • To: mathgroup at smc.vnet.net
  • Subject: [mg58550] Re: Urgent
  • From: Roland Franzius <roland.franzius at uos.de>
  • Date: Thu, 7 Jul 2005 05:35:39 -0400 (EDT)
  • Organization: Universitaet Hannover
  • References: <dag0g8$5a3$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Bruno Vicente wrote:
> I need help solving the Laplace equation in three dimensions using Mathematica. Due to the geometric configuration I used     oblate and prolate spheroidal coordinates.
> You can see the equations I'm using in:
> 
> http://mathematica.no.sapo.pt/index.html
> 

Particular solutions are possible as series of products of exponentials 
in phi, Legendre polynomials of the first L^m_n kind of cos theta and 
the second kind Q^m_n of cosh eta, also expressible by spherical 
harmonics Y_lm(cos theta) * Q^m_n (cosh eta).

Make a product ansatz

<< Calculus`VectorAnalysis`

SetCoordinates[OblateSpheroidal[theta, eta, phi, a]]

F=F1[theta]*F2[eta]*F3[phi]]

Laplacian[ F ]/F //
FullSimplify //
Expand

and separate the terms depending on one variable only. You have to solve 
three eigenvalue ODE's and then you select the fourier sum over the 
eigenvalue-indexed products to fit your boundary values.

-- 

Roland Franzius


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