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Math Help - Laplace's equation in the semi-infinite strip 0 < x < a, y > 0

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    Newbie mukmar's Avatar
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    Laplace's equation in the semi-infinite strip 0 < x < a, y > 0

    Problem:

    u_{xx} + u_{yy} = 0, 0 < x < a, y > 0

    u(0, y) = u(a, y) = 0 and u(x, 0) = f(x)

    u bounded as y \to \infty

    Find u(x, y) using separation of variables for general f(x) and write the solution in the form:

    Eq. 1

    u(x,y) = \displaystyle\int_0^a G(x, y, s) f(s) ds \label{1}

    Find u(x, y) in as explicit a form as you can, i.e., sum the series.


    Attempt at solution:

    After using the method of separation of variables I came to the solution:

    u(x, y) = \displaystyle\sum_{n = 1}^\infty c_n e^{-n\pi y / a}\sin{\frac{n\pi x}{a}}

    Where  c_n = \frac{2}{a}\displaystyle\int_0^a f(x) \sin{\frac{n\pi x}{a}} dx

    I'm just not sure how I'm supposed to combine these two pieces of information to be able to express u(x, y) in the form outlined in Eq. 1.

    Any assistance would be greatly appreciated!
    Last edited by mukmar; November 25th 2010 at 03:48 AM.
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