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Math Help - laplace

  1. #1
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    laplace

    how do you solve a laplace equation given that u(x,0) =0, u(x,1) = sin ^3 (pi x)
    u(x,y)=0 and u(1,y) =0 for 0<x<1, 0<y<1?


    i got the final answer of sum {sin ^3(pi x) sinh(n pi y)} over (sinh(pi x))

    but this seems to look wrong.
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  2. #2
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    Quote Originally Posted by alexandrabel90 View Post
    how do you solve a laplace equation given that u(x,0) =0, u(x,1) = sin ^3 (pi x)
    u(x,y)=0 and u(1,y) =0 for 0<x<1, 0<y<1?


    i got the final answer of sum {sin ^3(pi x) sinh(n pi y)} over (sinh(pi x))

    but this seems to look wrong.
    I think you have a typo. Did you mean
    u(x,y)=0
    u(0,y)=0

    Your answer should not have trig functions rasied to any power as they are not part of the basis!

    Hint 1:
    \sin^3(\pi x)=\frac{3}{4}\sin(\pi x)-\frac{1}{4}\sin(3 \pi x)

    Using the above the only terms that will survive when you expand the x's with will the two given above.

    Please post what you have done.
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  3. #3
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    ya i meant u(o,y) = 0.
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  4. #4
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    Untitled.pdf

    this is my working with the help that you gave. im still confused with how to find the solution
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  5. #5
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    So you have

    u_n(x,y)=c_n\sin(n \pi x)\sinh(n  \pi y)

    Now using the boundary condition we have

    u_n(x,1)=\frac{3}{4}\sin(x)-\frac{1}{4}\sin(3 \pi x)=c_n \sin(n \pi x)\sinh(\pi n)

    Now using the inner product we can solve for the constants. Multiply both sides by
    \sin(n \pi x)
    and integrate you will need a 2 in front of your integral to make the set orthonormal.

    2\int_{0}^{1}\sin(n \pi x)\left( \frac{3}{4}\sin(\pi x)-\frac{1}{4}\sin(3 \pi x)\right)dx=c_n\sinh(\pi n)

    The left hand side is only non zero when n=1 or n=3

    This gives

    c_1=\frac{3}{4\sinh(\pi)} \quad c_3=-\frac{1}{4\sinh(3\pi)}

    all of the other c's are zero so the sum only contains two terms and gives

    u(x,y)=\frac{3}{4\sinh(\pi)}\sin(\pi x)\sinh(y)-\frac{1}{4\sinh(3\pi)}\sin(3 \pi x)\sinh(3y)
    Last edited by TheEmptySet; May 10th 2011 at 02:44 PM. Reason: missing \pi
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  6. #6
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    i think c1 and c2 should be both multiplied by 2 right?
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  7. #7
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    Quote Originally Posted by alexandrabel90 View Post
    i think c1 and c2 should be both multiplied by 2 right?
    No becuse

    2\int_{0}^{1}\sin^2(n \pi x)dx=1
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