Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Question on solving Poisson's Eq using method of Super Position.

  1. Mar 7, 2010 #1
    I am reading the PDE book by Asmar of Poisson Eq. regarding of a rectangle with 0<x<a and 0<y<b. The book use super position of Poisson equation with X(0)=X(a)=Y(0)=Y(b)=0 and regular Dirichlet problem of Laplace equation.

    On the Poisson part, the book just gave the formula:

    [tex] u(x,y) = \sum_{m=1}^{\infty}\sum_{n=1}^{\infty}E_{mn} sin(\frac{m\pi}{a}x) sin(\frac{n\pi}{b}y)[/tex]

    And the book proceed to equate the double differentiate and let it equal to f(x,y):

    [tex] \frac{\partial ^2u}{\partial x^2} + \frac{\partial ^2u}{\partial y^2} = f(x,y)[/tex]


    My question, how to they come up with:

    [tex] u(x,y) = \sum_{m=1}^{\infty}\sum_{n=1}^{\infty}E_{mn} sin(\frac{m\pi}{a}x) sin(\frac{n\pi}{b}y)[/tex]

    Can some one explain how this come by?

    The book just said because [tex] X(0)=X(a)=0 \Rightarrow X=sin(\frac{m\pi}{a}x) \;\;\; and \;\;\; Y(0)=Y(b)=0 \Rightarrow Y=sin(\frac{n\pi}{b}y)[/tex]

    But these are separate event, how do we justify putting them together? Feels a little thin!!! Or is this good enough?
     
    Last edited: Mar 7, 2010
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted



Similar Discussions: Question on solving Poisson's Eq using method of Super Position.
Loading...