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Laplace's equation

  1. May 23, 2010 #1
    1. The problem statement, all variables and given/known data
    Consider Laplace's equation uxx + uyy = 0 on the region -inf <= x <= inf, 0 <= y <= 1 subject to the boundary conditions u(x,0) = 0, u(x,1) = f(x), limit as x tends to inf of u(x,y) = 0.
    Show that the solution is given by u(x,y) = F-1(sinh(wy)f(hat)/sinh(wy))

    2. Relevant equations

    3. The attempt at a solution
    I used Fourier transforms in x.
    I got u(hat)(w,y) = Aeky + Be-ky
    In Fourier space:
    u(hat)(w,0) = F(0) = 0
    u(hat)(w,1) = f(hat)(w)
    But u(hat) is a function of y. My question is how do I apply the 3rd boundary condition (as this is the limit as x(not y) tends to inf) to u(hat)
  2. jcsd
  3. May 26, 2010 #2
    Is it meant to be limit as y tends to inf not x.
  4. May 26, 2010 #3


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    Gold Member

    You mean [itex]\hat{u}(\omega,y)=Ae^{\omega y}+Be^{-\omega y}[/itex], right? :wink:

    You already used it. If [tex]\lim_{x\to\pm\infty} u(x,y)\neq 0[/itex], its Fourier transform (from [itex]x[/itex] to [itex]\omega[/itex]) might not exist (the integral could diverge).
  5. May 27, 2010 #4
    Very true. That is what I meant.
    But when did I use this boundary condition?
  6. May 27, 2010 #5


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    When you took the FT of [itex]u(x,y)[/itex], and hence assumed that it existed.
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