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

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

    Problem:

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

    $\displaystyle u(0, y) = u(a, y) = 0$ and $\displaystyle u(x, 0) = f(x)$

    $\displaystyle u$ bounded as $\displaystyle y \to \infty$

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

    Eq. 1

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

    Find $\displaystyle u(x, y)$ 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:

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

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

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

    Any assistance would be greatly appreciated!
    Last edited by mukmar; Nov 25th 2010 at 02:48 AM.
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