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Re: Laplace equation with gradient boundary conditions
Posted:
Nov 30, 2011 2:48 AM


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 socalled 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 fullsemester 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



