PDE with two variables by NDSolve
- To: mathgroup at smc.vnet.net
- Subject: [mg94500] PDE with two variables by NDSolve
- From: "Alexei Boulbitch" <boulbitch at gmx.de>
- Date: Mon, 15 Dec 2008 07:43:25 -0500 (EST)
Dear MathGroup members,
I have a misunderstanding when solving a partial differential equation with M6. It is a rather simple equation for a function z=z(x,y):
pde = \!\(
\*SubscriptBox[\(\[PartialD]\), \({x, 2}\)]\ \(z[x, y]\)\) + \!\(
\*SubscriptBox[\(\[PartialD]\), \({y, 2}\)]\ \(z[x,
y]\)\) - (1 - 3*Exp[-(x^2 + y^2)])*z[x, y] -
Log[(1 + z[x, y])/(1 - z[x, y])] == 0;
with simple boundary conditions fixed on a square 10x10. The function should be there either zero, or something small, but different from zero like 0.01. Let us take zero:
bc = {z[-10, y] == z[10, y] == 0, z[x, -10] == z[x, 10] == 0};
Now application of the NDSolve operator:
sol = NDSolve[{pde, bc}, z, {x, -10, 10}, {y, -10, 10}]
returns the error message:
NDSolve::ivone: Boundary values may only be specified for one \
independent variable. Initial values may only be specified at one \
value of the other independent variable. >>
I understand this response, as if Mathematica does not support boundary conditions fixed for both of the variables.
However, in Help/NDSolve/ Scope/Partial Differential Equations/Example Nr 3 shows the following code for SineGordon:
L = 4;
sol = NDSolve[{D[u[t, x, y], t, t] ==
D[u[t, x, y], x, x] + D[u[t, x, y], y, y] + Sin[u[t, x, y]],
u[t, -L, y] == u[t, L, y], u[t, x, -L] == u[t, x, L],
u[0, x, y] == Exp[-(x^2 + y^2)],
Derivative[1, 0, 0][u][0, x, y] == 0},
u, {t, 0, L/2}, {x, -L, L}, {y, -L, L}]
in which the line u[t, -L, y] == u[t, L, y], u[t, x, -L] == u[t, x, L] fixes very similar type of boundary conditions.
What´s wrong?
Regards, Alexei
--
Alexei BOULBITCH,
Dr. , habil.
IEE S.A.
Department for Material Development
ZAE Weiergewan
11, rue Edmond Reuter
L-5326 Contern, Luxembourg
Tel. ++352-2454-2566
Fax.: ++352 424737201
mobile: 0152-03807245
E-mail: alexei.boulbitch at iee.lu
Internet: http://www.physik.tu-muenchen.de/~aboulbit/
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