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Math Help - Laplace's equation in the square

  1. #1
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    Laplace's equation in the square

    Let
    T(x, y) satisfy Laplace’s equation in the square 0 < x < ¼ and
    0
    < y < ¼. Let T = 1 on the side y = ¼, 0 < x < ¼, and let T = 0 on
    the other three sides. Show that

    T
    (x, y) = big expression involving sinhs (I can derive this)

    and, by considering three similar problems, that
    T = 1/4at the centre of the square.

    Which other 3 problems would I consider and then what would I do with these to get the solution at the centre of the square?

    many thanks

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  2. #2
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    Quote Originally Posted by James0502 View Post
    Let
    T(x, y) satisfy Laplace’s equation in the square 0 < x < ¼ and
    0
    < y < ¼. Let T = 1 on the side y = ¼, 0 < x < ¼, and let T = 0 on
    the other three sides. Show that

    T
    (x, y) = big expression involving sinhs (I can derive this)

    and, by considering three similar problems, that
    T = 1/4at the centre of the square.

    Which other 3 problems would I consider and then what would I do with these to get the solution at the centre of the square?

    many thanks

    The other three problems are the same d.e. with (1) T= 0 on y=0, 0 on the other three sides, (2)T= 0 on x= 0, 0 on the other three sides, and (3)T= 0 on x= 1/4, 0 on the other three sides. Consider the average of the solutions of those four problems.
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