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Math Help - Poisson's equation

  1. #1
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    Exclamation 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!
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  2. #2
    Super Member Rebesques's Avatar
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    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 f(x,y)=\sum b_n(y) \sin((n\pi x)/a). Plug w(x,y)
    in the equation to get \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 -((n\pi)/a)^2a_n(y)+a_n''(y)=b_n(y) \ \forall n. The BC are 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)
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