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Solving Laplace Equations using this boundary conditions?

  1. Apr 9, 2016 #1
    The equation is Uxx + Uyy = 0
    And domain of solution is 0 < x < a, 0 < y < b
    Boundary conditions:
    Ux(0,y) = Ux(a,y) = 0
    U(x,0) = 1
    U(x,b) = 2

    What I've done is that I did separation of variables:

    Plugging into the equation gives:
    X''Y + XY'' = 0

    X''/X = -Y''/Y = k

    For case k > 0, I saw that it gives no non-trivial solutions.
    For case k = 0, I solved it and found U(x,y) = y/b + 1

    For case k < 0, I'm slightly lost.
    X'' + kX = 0
    Y'' - kX = 0


    Using the X boundary conditions:

    Using the Y boundary condition:
    Using Fourier Series to find the coefficient:

    But the integral just gives Dn = 0, and this doesn't satisfy u(x,0) = 1.

    Can someone explain where I went wrong?


    Attached Files:

  2. jcsd
  3. Apr 9, 2016 #2


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    Homework Helper

    Your working is confusing.

    If you start with [itex]X'' = kX[/itex] and want [itex]k < 0[/itex], you should then be writing [itex]\sin(\sqrt{|k|}x)[/itex] and so forth, or defining [itex]k = -c^2[/itex] where [itex]c \geq 0[/itex].

    You want to be using cosines, which in fact you end up doing, but only after expressly stating that [itex]X_n = A_n\sin(n\pi x/a)[/itex].

    Now the cosine expansion of [itex]1[/itex] on [itex][0,a][/itex] is [itex]\cos(0x)[/itex]. Thus you have correctly determined that all the other coefficients vanish.
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