Re: elliptic equation solved with 2 boundaries
- To: mathgroup at smc.vnet.net
- Subject: [mg43381] Re: [mg43294] elliptic equation solved with 2 boundaries
- From: Selwyn Hollis <selwynh at earthlink.net>
- Date: Wed, 27 Aug 2003 04:05:07 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
Sorry, I should have read your reply more carefully... Let's have
another go at this.
On Tuesday, August 26, 2003, at 09:20 AM, Ferdinand.Cap wrote:
> Hi,
> Thank you for your interest in my notebook. The LAPLACE
> equation has complex characteristics and is therfor called an elliptic
> partial differential eqatiuon. The theory of boundary value problems
> (Table 1.1. on page 14 in the book mentioned in the notebook )shows
> that an elliptic partial differential equation can only satisfy one
> boundary value probblem.
That is patent nonsense. First, equations don't satisfy boundary value
problems, solutions do. Second, even the trivial case of a constant
function satisfies both Dirichlet boundary conditions and Neumann
boundary conditions on any domain you care to consider. Voila: two
boundary value problems.
Perhaps you mean that a given elliptic equation (with smooth
coefficients) subject to appropriate boundary conditions has a unique
solution?
> My notebokk demonstrates that under special conditions
> also 2 boundary value problems can be satisfied - run the notebook!
When you say two boundary value problems, do you mean one with a
two-part boundary? It appears from the notebook that that is what you
really mean. I suggest you restudy the theory if you think that is
something out of the ordinary.
> Due to the approoximation method -variational calculus - the LAPLACIAN
> is
> not zero, but numerically small F.C
What do you mean by numerically small? A plot,
Plot3D[Evaluate[D[f[x, y], x, x] + D[f[x, y], y, y]],{x,-2,2},{y,-2,2}]
shows that the Laplacian is HUGE on the region where you seem to be
claiming it's "numerically small". But then again, you never actually
stated the problem you claimed to have solved, so who knows...
-----
Selwyn Hollis
http://www.math.armstrong.edu/faculty/hollis