Re: How to specify boundary conditions on all 4 sides of a plate for a steady state heat equation (PDE) using NDSolve? (Laplace equation)
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- Subject: [mg59688] Re: How to specify boundary conditions on all 4 sides of a plate for a steady state heat equation (PDE) using NDSolve? (Laplace equation)
- From: "Nasser Abbasi" <nma at 12000.org>
- Date: Wed, 17 Aug 2005 04:00:47 -0400 (EDT)
- References: <ddpt58$orc$1@smc.vnet.net> <dds9co$91c$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
"James Gilmore" <james.gilmore at yale.edu> wrote in message news:dds9co$91c$1 at smc.vnet.net... > This is a classic mathematical physics BVP. You should approach this > problem > in Mathematica, as you would by hand: use separation of variables, > and then > a fourier expansion to satisfy the boundary conditions. Not if you want to use a numerical solvers such as NDSolve. That is the whole idea of using NDSolve. I know how to solve these by hand, and also by direct numerical approach, I've solved my of these before and more advanced ones when I took some courses at the Math dept at UC Berkeley one year ago, I was just playing around to see if NDSolve can solve BVP and get the same plots I got when I solved this problem by hand using sepration of variables. > There are many books > that explain how to do this. Yes, and my home library contains many fine books on PDE's. I like the Satnley Farlow book, and Mary Boas has excellent chapter on the subject, but a bit short on detailes. Also Richard Haberman applied PDE's is nice, and if you want to see a nice new book with the cover showing solutions of PDE's plots which I am sure was made using Mathematica check Charles MacCluer's BVP and fourier expansions Dover book, but it does not contain any Mathematica code. Another book which uses a CAS system to solve PDE's is by David Betounes called PDE's for computational science, lots of examples and plots. Is there a short list somewhere which makes it clear what kind/class of PDE's Mathematica can solve and not solve directly using NDSolve or even DSolve? And why is it that NDSolve can solve an initial value PDE and not BVP? I wonder if NDSolve will be able to solve a BVP PDE in next version? > > "Nasser Abbasi" <nma at 12000.org> wrote in message > news:ddpt58$orc$1 at smc.vnet.net... >> >> hi; >> >> just for fun, I am trying to solve a steady state heat equation >> i.e. >> laplace equation, for a rectangular plate. >> >> So, I have 4 boundary conditions, one for each side of the plate. >> >> But when I do that, NDSolve says that it is designed to solve >> initial >> conditions problems only? is this really the case? May be I am not >> defining the B.C. correctly for Mathematica? >> >> The code is below, also I've posted it on my web page with the full >> error message. >> >> http://12000.org/my_notes/mma_matlab_control/e61/HTML/e61.htm >> >> I find the error strange, saying that NDSolve can only solve IC >> PDE, >> because I solved 1-D heat equation using IC and BC earlier with no >> problem, see this >> >> http://12000.org/my_notes/mma_matlab_control/e57/HTML/e57.htm >> >> So, I have a feeling that NDSolve can do this, I must be just doing >> something not right. >> >> >> Remove["Global`*"]; >> h = 30; w = 10; temp = 100; >> eq = D[T[x, y], x, x] + D[T[x, y], y, y] == 0; >> bc = {T[0, y] == 0, T[w, y] == 0, T[x, 0] == temp,T[x, h] == 0}; >> sol = NDSolve[{eq, bc}, T[x, y], {x, 0, w}, {y, 0, h}] >> >> >> "Boundary values may only be specified for one independent >> variable. Initial values may only be specified at one value of the >> other independent variable." >> >> Nasser >> >> >> >> >> >> >> > > >
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- Re: Re: How to specify boundary conditions on all 4 sides of a plate for a steady state heat equation (PDE) using NDSolve? (Laplace equation)