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MathGroup Archive 2006

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Re: Poisson equation

  • To: mathgroup at smc.vnet.net
  • Subject: [mg64303] Re: Poisson equation
  • From: Roland Franzius <roland.franzius at uos.de>
  • Date: Fri, 10 Feb 2006 02:13:41 -0500 (EST)
  • Organization: Universitaet Hannover
  • References: <dsetae$jal$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Uli Wuerfel schrieb:
> Dear Experts,
> 
> I am tying to solve the Poisson equation in cylindrical coordinates:
> Laplace Operator[phi]==0
> I tried to do this with the boundary conditions being that the potential
> phi euqals 1 on the bottom and the walls of the cylinder.
> Unfortunately, Mathematica does not accept it.
> I also tried to make it time-dependent and taking then the steady-state 
> value of the solution but this either does not work.
> Must be something wrong in general.

Mathematica has no method at hand to solve boundary value problems of 
the Laplace equation using DSolve or NDSolve. Diffusionapproxiamtion 
doesn't work either in three or more variables and/or in presence of 
edges with discontinuities of the boundary values.

Do it by Fourier fitting the boundary values. The fundamental solutions 
to Lap=0 in cylindrical rotational symmetric problems are

f_p,q=e^(i q z) (a_q J_0(i q r) + b_q Y_0(i q r)

or

f_p,q=e^(q z) (a_q  J_0(q r) + b_q Y_0(q r)

Special solutions to interpolate between different boundary values are
(a + b z) (1 + c log r)

-- 

Roland Franzius


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