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Re: Poisson equation with boundary conditions on rectangle

  • To: mathgroup at smc.vnet.net
  • Subject: [mg87336] Re: Poisson equation with boundary conditions on rectangle
  • From: Oliver Ruebenkoenig <ruebenko at uni-freiburg.de>
  • Date: Tue, 8 Apr 2008 05:35:37 -0400 (EDT)
  • References: <frvma1$i4d$1@smc.vnet.net> <ftcp1t$l00$1@smc.vnet.net>

Hi Michael,

On Mon, 7 Apr 2008, Michael Debono wrote:

> On Mar 21, 7:58 am, Benjamin Hell <hell... at gmx.de> wrote:
>> Hi,
>> I'm currently trying to solve the following pde with rectangle boundary (I better already use mathematica code here):
>>
>> The equation (Poisson equation):
>> D[u[y, z], y, y] + D[u[y, z], z, z] + 1 == 0
>>
>> The boundary conditions on the rectangle(y in [0,0.1] and z in [-0.4,0.4]):
>> u[y, 0.4] == 0, u[y, -0.4] == 0, u[0, z] == 0,
>> Derivative[1, 0][u][0.1, z] == 0
>>
>> I tried the following in mathematica:
>> eqn = D[u[y, z], y, y] + D[u[y, z], z, z] + 1 == 0; //defining equation
>> NDSolve[{eqn, u[y, 0.4] == 0, u[y, -0.4] == 0, u[0, z] == 0, Derivative[1, 0][u][0.1, z] == 0 }, u, {y, 0, 0.1}, {z, -0.4, 0.4}] //using NDSolve to solve the boundary problem
>>
>> But, as you might guess, I get an error using NDSolve the way above: "NDSolve::ivone: Boundary values may only be specified for one independent variable. Initial values may only be specified at one value of the other independent variable."
>> When I lookup the description of that error I realize that in my example the following error condition matches: "This input generates a message because the equations specify values for the solution on all sides of the solution region."
>> Does this mean, that mathematica can't approximate the solution to my problem with NDSolve or am I doing something wrong?
>> So overall question is: How can I solve my problem using mathematica without writing my own numerical pde solver for my problem?
>>
>> Thanks in advance!
>
> I too have a problem like yours...I found this link which may help
> you:
> http://documents.wolfram.com/mathematica/Built-inFunctions/AdvancedDocumentation/LinearAlgebra/LinearAlgebraInMathematica/Examples/AdvancedDocumentationLinearAlgebra6.3.html
> It uses a finite difference method..even tough I don't really know how
> it works lol!
>
>

If you are interested in how the FDM works have a look at a tutorial i 
wrote some time ago.

http://www.imtek.uni-freiburg.de/simulation/mathematica/IMSweb/imsTOC/Lectures%20and%20Tips/Simulation%20I/index.html

Oliver

Oliver Ruebenkoenig, <ruebenko AT uni-freiburg.de>


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