Student Support Forum: 'Coupled diff eq, is symbolic solution possible?' topicStudent Support Forum > General > "Coupled diff eq, is symbolic solution possible?" |
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:
∇^2(x,y) = Rs * J(x,y)
for two cases:
ψ = 0 for x = 0, x = a, y = b;
∂ψ/∂y = 0 for y = 0.
ψ = 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 (ψ) 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 ψ(x,y) is found, the electric field E(x,y) = - ∇ψ and the total current flow J = 1/ρ * E, where ρ 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
∇^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.