Re: Laplace equation with gradient boundary conditions

*To*: mathgroup at smc.vnet.net*Subject*: [mg123292] Re: Laplace equation with gradient boundary conditions*From*: Mark McClure <mcmcclur at unca.edu>*Date*: Wed, 30 Nov 2011 03:22:49 -0500 (EST)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*References*: <201111291203.HAA05406@smc.vnet.net>

On Tue, Nov 29, 2011 at 7:03 AM, Tom Wolander <ultimni at 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

**References**:**Laplace equation with gradient boundary conditions***From:*Tom Wolander <ultimni@hotmail.com>