Re: Re: Coupled Diff Eqs or Poisson Eq, is symbolic solution
- To: mathgroup at smc.vnet.net
- Subject: [mg102976] Re: Re: [mg102844] Coupled Diff Eqs or Poisson Eq, is symbolic solution
- From: Alexei Boulbitch <Alexei.Boulbitch at iee.lu>
- Date: Thu, 3 Sep 2009 05:42:21 -0400 (EDT)
Dear Mr. Neelsonn, Your question sounds, as if you do not know what type of solutions can have your equation. If this is the case, why would not you look into a pair of books on Mathematical Physics. A Poisson's equation like the one you described is a classical one, and as much as I remember its solution in a rectangular domain has been treated in many textbooks. Really, the most simple case is its solution in a circle, but the second simplest case - is in the rectangle. Have a look and you will see what kind of solution you can expect and what you cannot. In particular, you will see, if such problems admit simple analytical solutions that you asked about, or - may be - analytical solutions can only be written in a form of infinite series (is it OK with you?), and how such series can look like, or you can write them in an integral form, and so on. I know the answer, but this is your problem and your home task. I would advice you to do it first, and only then to turn to Mathematica. Otherwise you will not understand what you get, even if you get something. Regards, Alexei Dear Prof. Murray, Thank you for your reply. I tried to fix the unreadable codes, please see the message below. Thanks again! ---------- On Sun, Aug 30, 2009 at 8:10 AM, Murray Eisenberg <murray at math.umass.edu> wrote: > Your post is essentially unreadable because of the embedded (hex?) codes > such as "=E2", "=CF", etc. Please use plain ASCII and remove all such > codes. > > Neelsonn wrote: > >> 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) >> >> Div^2 phi(x,y) = Rs * J(x,y) >> >> >> for two cases: >> >> i) >> >> phi 0 for x = 0, x = a, y = b; >> >> dphi/dy = 0 for y = 0. >> >> and >> >> ii) >> >> phi = 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 (phi) 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 phi(x,y) is found, the electric field E >> (x,y) = - (Div) phi and the total current flow J = 1/=rho * E, = >> where rho 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 >> >> Div^2 phi(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 >> >> >> >> >> > -- > Murray Eisenberg murray at math.umass.edu > Mathematics & Statistics Dept. > Lederle Graduate Research Tower phone 413 549-1020 (H) > University of Massachusetts 413 545-2859 (W) > 710 North Pleasant Street fax 413 545-1801 > Amherst, MA 01003-9305 > -- Alexei Boulbitch, Dr., habil. Senior Scientist IEE S.A. ZAE Weiergewan 11, rue Edmond Reuter L-5326 Contern Luxembourg Phone: +352 2454 2566 Fax: +352 2454 3566 Website: www.iee.lu This e-mail may contain trade secrets or privileged, undisclosed or otherwise confidential information. If you are not the intended recipient and have received this e-mail in error, you are hereby notified that any review, copying or distribution of it is strictly prohibited. Please inform us immediately and destroy the original transmittal from your system. Thank you for your co-operation.