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Re: How to specify boundary conditions on all 4 sides of a plate for a steady state heat equation (PDE) using NDSolve? (Laplace equation)

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  • Subject: [mg59781] Re: How to specify boundary conditions on all 4 sides of a plate for a steady state heat equation (PDE) using NDSolve? (Laplace equation)
  • From: "Nasser Abbasi" <nma at 12000.org>
  • Date: Sun, 21 Aug 2005 03:51:28 -0400 (EDT)
  • References: <ddpt58$orc$1@smc.vnet.net> <ddscf6$ai1$1@smc.vnet.net> <de6m0a$ch8$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

"Nasser Abbasi" <nma at 12000.org> wrote in message 
news:de6m0a$ch8$1 at smc.vnet.net...
>
> "Jens-Peer Kuska" <kuska at informatik.uni-leipzig.de> wrote in message
> news:ddscf6$ai1$1 at smc.vnet.net...
>>
>> Hi,
>>
>> NDSolve[] is for intial-boundary value problems
>> n+1 and not
>> for pure boundary value problems.
>> You can use the tim depend equation and integrate
>> it until the solution
>> does not change any more.
>>
>> Regards
>>  Jens
>


> Well, I think I'll go back to using direct numerical/mesh based
> methods.
>
> Those will work for any type of linear PDE (in general), and even if
> one must write more code to do that, one can get more control on 
> what
> is going on, and one does not have the limitations imposed by 
> NDSolve.
>
> For fun, here is the solution to 1-D diffusion PDE using FTCS scheme
> (forward time centered space) using Mathematica code. I just hacked
> this quickly, and would like to go over it to again to make it more 
> of
> a 'functional' style, but I find it hard to avoid using the For loop
> sometimes, but I do use the Table command a lot, so I think the code
> is functional :)
>
> http://12000.org/my_notes/mma_matlab_control/e65/HTML/e65.htm
>
> compare the above to the solution given by using NDSolve shown below
> (much less code ofcourse)
>
> http://12000.org/my_notes/mma_matlab_control/e57/HTML/e57.htm
>
> Next, I'll solve the 2D steady state heat equation (Laplace PDE) 
> using
> such direct method, may be using a different scheme.
>

I've coded the solution to the 2-D steady state heat equation based on 
a scheme called SOR (simultaneous over-relaxation), which is an 
improved version of Gauss-Seidel scheme, which is in turn is an 
improved version of the Jacobi method. These algorithms are described 
in details in 'Numerical Methods for physics' by Garcia (nice book 
btw, that was our text book for a course I took few years ago).

The Mathemetica implementation was straight forward and I have to say 
that Mathematica 3D plots look really nice !

http://12000.org/my_notes/mma_matlab_control/e61/HTML/e61.htm

Nasser





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