Re: Laplace equation with gradient boundary conditions
- To: mathgroup at smc.vnet.net
- Subject: [mg123349] Re: Laplace equation with gradient boundary conditions
- From: Tom Wolander <ultimni at hotmail.com>
- Date: Fri, 2 Dec 2011 07:22:39 -0500 (EST)
- Delivered-to: l-mathgroup@mail-archive0.wolfram.com
- References: <201111291203.HAA05406@smc.vnet.net> <jb7rjp$j16$1@smc.vnet.net>
Oliver
It is really an easy exercice of a rectangular wall (axb) isolated on
3 sides and radiating on the 4th.
In steady state f is solution of Laplace = 0 with BC df/dx(a,y)=df/
dx(0,y)=df/dy(x,0)=0 and df/dy(x,b) = S(x) - k.[f(x,b)]^4 . Same
problem is obtained with a convective case just by replacing k.
[f(x,b)]^4 by k.f(x,b).
S(x) is the incoming radiation and can serve to adjust continuity if
that's what one wants to do.
So it has nothing to do with Sommerfeld conditions. The convective
case is just a Robin BC and the radiating case is a generalised Robin.
Andrzej
It is neither clearly nor obviously stated that NDSolve can't solve
the Laplace equation. DSolve even uses it as an example of symbolical
solution. I agree that after having spent hours and/or for somebody
familiar with resolution algorithms it becomes clear but I can assure
everybody that for a first time user it is a long and unpleasant
journey. I would have expected that in the tutorial (part dealing with
classification of PED) there would be a warning with glowing red
characters following the definitions of PED : "Beware ! NDSolve is
unable to solve most of these PED with the exception of : {list of
conditions}".
Mark
Thanks. I have come empirically to the same conclusion and you
explained me why. I have also got some results by looking at the
asymptotic behaviour of the time dependent equation. This is not
something one would do spontaneously when working by hand - the
Laplace equation in 2D is easy so there is no incentive going to the
time dependent case. I agree, will have to wait for Version 9 to deal
with the problems I wanted to deal.
Thanks for the link. Even if I knew the mathematics, I understood a
bit more about Mathematica :)