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Laplace's equation on a rectangle (mixed bndy)

  1. Mar 8, 2009 #1
    1. The problem statement, all variables and given/known data

    I'm having issues with a deceptively simple Laplace problem. If anybody could point me in the right direction it would be fantastic.

    It's just Laplace's equation on the square [0.1]x[0,1] (or any rectangle you like) with a mixed boundary.

    2. Relevant equations

    Uxx+Uyy=0
    Ux(0,y)= a (some constant)
    Ux(1,y)= 0
    U(x,0)=0
    Uy(x,1)=0

    3. The attempt at a solution

    The main thing I've tried is just an ordinary seperable solution, but it's definitely not seperable. I need something different.
     
    Last edited: Mar 9, 2009
  2. jcsd
  3. Mar 9, 2009 #2
    Actually, it is separable. Just let U(x,y)=A(x)B(y).

    EDIT: It seems to me that if a isn't zero, B(y) must be a constant function since by the first condition,
    [tex]U_x(0,y)=A'(0)B(y)=a \Rightarrow B(y)=\frac{a}{A'(0)}=\text{constant}.\qquad (1)[/tex]
    Further, by the third condition, either A(x)=0 for all x or B(0)=0. If A(x)=0, then clearly U(x,y)=0. If B(0)=0 (and a is nonzero), then by (1), B(y)=0 for all y, so U(x,y)=0. Thus, in order for U(x,y) to be non-trivial, a must be zero.
     
    Last edited: Mar 9, 2009
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