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Coupled Diff Eqs or Poisson Eq, is symbolic solution possible?

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  • Subject: [mg102844] Coupled Diff Eqs or Poisson Eq, is symbolic solution possible?
  • From: Neelsonn <neelsonn at gmail.com>
  • Date: Sun, 30 Aug 2009 06:06:05 -0400 (EDT)

Guys,

(This is the 3rd time I am trying to post this; I apologize for any
duplicate)

I've have just installed Mathematica and have some tasks to accomplish
using it. I've spent some time trying to find similar problems at
Wolfram's website, but not success so far. So I am posting here for
the first time (unless someone, please, point me a similar post or
documentation)

Here's what I need to solve:
(Poisson)

=E2=88=87^2(x,y) = Rs * J(x,y)


for two cases:

i)

=CF=88 = 0 for x = 0, x = a, y = b;

=E2=88=82=CF=88/=E2=88=82y = 0 for y = 0.

and

ii)

=CF=88 = 0 for x = 0, x = a, y = 0, y = b.


Some side notes:

- Physically speaking, for both cases I would like to know how the
electrostatic potential (=CF=88) will be distributed on a rectangular shape
(a,b) when it has grounded electrodes on the three edges (case i) and
grounded electrodes surrounding all four edges (case ii).

- The rectangular shape resembles a resistive material, that comes the
Rs (sheet resistance) and J(x,y) is the current that is going to be
distributed on the surface of this geometry as well. In my case J(x,y)
= exp(V(x,y)). An "arrow plot" showing the current distribution will
be also interesting.

- Eventually, once the solution =CF=88(x,y) is found, the electric field E
(x,y) = - =E2=88=87=CF=88 and the total current flow J = 1/=CF=81 * E, =
where =CF=81 is the
resistivity (ohm.meter), is also interested

----------

Now, my question is: Can Mathematica handle such problem and
boundaries like it is in order to solve it analytically (symbolic)? I
haven't seen, in the examples, problems like this. I wonder if I will
have to decouple

=E2=88=87^2(x,y) = Rs * J(x,y)

into first-order partial differential equations. Then,  a follow-up
question that comes: can Mathematica do that automatically or I should
pose the problem myself? For that, I've seen an example from the
website that uses six first-order differential equations to solve the
kinetics of some chemical reactions. But the problem was solved
numerically and I would like to have an analytical equation as a
result. So is it possible to find such analytical solution in case I
have to use a system of first-order partial diff eqs?

(I am pretty sure that this isn't a difficult problem for those who
Master Mathematica)

A final question or better yet, help needed: I would like to do all
the above for a different shape, not a rectangule or square, but for a
trapezoidal shape. I have no idea how to start and don't know how the
boundaries will look like. I wonder if there is a way to draw such
shape in Mathematica and graphically tells the software to solve it
(like those "FEM softwares"...). That would be very easy! I would
really appreciate any input here.

Thanks
N





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