1. Poisson's equation

where initial conditions are w(x,0) = w(x,b) = w(0,y) = w(a,y) = 0
poisson's equation is d2w/dx2+d2w/dy2 = f(x,y)

if we assume a 1D eigenfunction expansion:

w(x,y) = Sum(a_n(y) * sin(n*pi*x/a)

Obtain the ode and BCs for each a_n(y).

I know you can split this into a homogeneous solution and a particular solution, but how can you get the particular solution?

Or does someone else have a better method of solving this?

And also, we have to solve this by hand. No Matlab.\

Thanks!

2. And also, we have to solve this by hand. No Matlab.

Love u man!

Ok now... First of all, you need to know the eigenvalue expansion of $\displaystyle f(x,y)=\sum b_n(y) \sin((n\pi x)/a)$. Plug $\displaystyle w(x,y)$
in the equation to get $\displaystyle \sum(-((n\pi)/a)^2a_n(y)+a_n''(y))\sin((n\pi x)/a)=\sum b_n(y) \sin((n\pi x)/a)$, so necessarily $\displaystyle -((n\pi)/a)^2a_n(y)+a_n''(y)=b_n(y) \ \forall n$. The BC are $\displaystyle 0=w(x,0)=\ldots=a_n(0)=w(x,a)=\ldots=a_n(b) \ \forall n$, which readily gives you a Sturm-Liouville problem.

(Do check my calculations. I feel extremely bored today)