Poisson equation with boundary conditions on rectangle

• To: mathgroup at smc.vnet.net
• Subject: [mg86805] Poisson equation with boundary conditions on rectangle
• From: Benjamin Hell <hellben at gmx.de>
• Date: Fri, 21 Mar 2008 01:55:41 -0500 (EST)

```Hi,
I'm currently trying to solve the following pde with rectangle boundary (I better already use mathematica code here):

The equation (Poisson equation):
D[u[y, z], y, y] + D[u[y, z], z, z] + 1 == 0

The boundary conditions on the rectangle(y in [0,0.1] and z in [-0.4,0.4]):
u[y, 0.4] == 0, u[y, -0.4] == 0, u[0, z] == 0,
Derivative[1, 0][u][0.1, z] == 0

I tried the following in mathematica:
eqn = D[u[y, z], y, y] + D[u[y, z], z, z] + 1 == 0; //defining equation
NDSolve[{eqn, u[y, 0.4] == 0, u[y, -0.4] == 0, u[0, z] == 0, Derivative[1, 0][u][0.1, z] == 0 }, u, {y, 0, 0.1}, {z, -0.4, 0.4}] //using NDSolve to solve the boundary problem

But, as you might guess, I get an error using NDSolve the way above: "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."
When I lookup the description of that error I realize that in my example the following error condition matches: "This input generates a message because the equations specify values for the solution on all sides of the solution region."
Does this mean, that mathematica can't approximate the solution to my problem with NDSolve or am I doing something wrong?
So overall question is: How can I solve my problem using mathematica without writing my own numerical pde solver for my problem?