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

1. Mar 7, 2010

### yungman

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:

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

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

$$\frac{\partial ^2u}{\partial x^2} + \frac{\partial ^2u}{\partial y^2} = f(x,y)$$

My question, how to they come up with:

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

Can some one explain how this come by?

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

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