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Math Help - Non homogenous laplace PDE on a square

  1. #1
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    Non homogenous laplace PDE on a square

    Hello,
    I've tried to solve the attached PDE, as is shown in the file.
    As is stated in the file, it seems unreasonable that U1 explodes so I hope you'll find the error in what I did.

    Thank you in advance
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  2. #2
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    Quote Originally Posted by zokomoko View Post
    Hello,
    I've tried to solve the attached PDE, as is shown in the file.
    As is stated in the file, it seems unreasonable that U1 explodes so I hope you'll find the error in what I did.

    Thank you in advance
    Is u(x,y) to satisfy that equation inside or outside the square? Typically solutions to Laplace's equation, given the value on some closed curve, have very different solutions inside and outside the curve.
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  3. #3
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    Quote Originally Posted by zokomoko View Post
    Hello,
    I've tried to solve the attached PDE, as is shown in the file.
    As is stated in the file, it seems unreasonable that U1 explodes so I hope you'll find the error in what I did.

    Thank you in advance
    Typically Laplace equation is on some domain D (inside) so

    u_{xx} + u_{yy} = 0\; \text{ in}\; [0,1] \times [0,1] with your BC's on the boundary of this domain.
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  4. #4
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    hey

    I don't think I quite follow, I meant that U1 as a series doesn't converge (for almost all x,y).
    The PDE as you said, is indeed defined inside [0,1]X[1,0].

    I think there must be a mistake because the U1 is supposed to be a convergent series.
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  5. #5
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    Quote Originally Posted by zokomoko View Post
    I don't think I quite follow, I meant that U1 as a series doesn't converge (for almost all x,y).
    The PDE as you said, is indeed defined inside [0,1]X[1,0].

    I think there must be a mistake because the U1 is supposed to be a convergent series.
    It'll converge. You must remember that  k_n involves n and has the form

    k_n = \frac{a_n}{\sinh n \pi } where a_n = 2 \int_0^1 f_1(x) \sin n \pi x\, dx
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