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Topic: Laplace equation with gradient boundary conditions
Replies: 4   Last Post: Dec 2, 2011 6:36 AM

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Mark McClure

Posts: 193
Registered: 9/12/07
Re: Laplace equation with gradient boundary conditions
Posted: Nov 30, 2011 2:48 AM
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On Tue, Nov 29, 2011 at 7:03 AM, Tom Wolander <ultimni@hotmail.com> wrote:

> I have bought Mathematica 8 a week ago and this is my first post on this board.
> My main purpose for the purchase was to work on PDEs, specifically on the
> heat equation. As one of the first tests I wanted to solve a steady state
> temperature distribution on a rectangular domain with a radiative boundary
> condition on one face (flux=0 on the other 3). I made sure to have
> continuity in corners.


First, you should know that V8 does *not* include a finite element
solver - probably the best tool to use for this type of problem. All
signs seem to indicate that NDSolve will provide access to a finite
element solver by V9, which would hopefully be released sometime next
year. For PDEs, NDSolve use the so-called numerical method of lines,
which requires one dynamic variable. What one can do, is to set up a
hyperbolic equation that converges to the steady state solution you
describe. Clearly, this is not terribly efficient but it's good
enough for government work. This technique can deal with a wide
variety of types of boundary conditions - radiation type conditions
are no problem. Inconsistent boundary conditions can also be dealt
with but, of course, this will affect error estimates.

I have taught a full-semester undergraduate PDE course several times
using Mathematica and have a web page for the last time I taught it:
http://facstaff.unca.edu/mcmcclur/class/Spring11PDE/

This page has quite a few Mathematica demos with explanations on how
to use NDSolve and other tools. Specifically, the third demo link
titled "Heat conduction on a square" describes a situation close to
yours. In that example, the boundary conditions are not continuous;
the technique should work better if your boundary conditions are
continuous.

I'm really looking forward to V9,
Mark McClure




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