1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Solving Laplace Equations using this boundary conditions?

Tags:
  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:
    U(x,y)=X(x)Y(y)

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

    Rearranging:
    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

    upload_2016-4-9_12-32-57.png

    Using the X boundary conditions:
    upload_2016-4-9_12-33-41.png

    upload_2016-4-9_12-34-11.png
    Using the Y boundary condition:
    upload_2016-4-9_12-34-47.png
    Using Fourier Series to find the coefficient:
    upload_2016-4-9_12-34-59.png

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

    Can someone explain where I went wrong?

    Thanks!
     

    Attached Files:

  2. jcsd
  3. Apr 9, 2016 #2

    pasmith

    User Avatar
    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted