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!

Homework Help: Homogeneous Laplace's Equation

  1. Jun 27, 2010 #1
    1. The problem statement, all variables and given/known data
    uxx+uyy=0
    u(x,0)=u(x,pie)=0
    u(0,y)=0
    ux(5,y)=3siny-5sin4y


    2. Relevant equations



    3. The attempt at a solution

    Using separable method I get
    Y"-kY= 0 and X"+kX=0

    For Case 1 and Case 2 where k>0 and k=0 there are no eigenvalues

    So Case 3 k<0 gives
    Y=ccos(sqrk x y) + dsin(sqrk x y)
    y=0, then c=0
    y=pie, then dsin(pie x sqrk)=0
    k= -n2, Yn=sin(ny)
    X"-n2X=0, X=ccosh(nx)+dsinh(nx)

    u(x,y)= summation (from n=1 to n=infinity) sin(ny)[cncosh(nx)+dnsinh(nx)]

    u(0,y)= summation (from n=1 to n=infinity) cnsin(ny)

    Using Fourier's Series to expand 0, I got 0 so cn is 0

    ux(x,y)= summation (from n=1 to n=infinity) sin(ny)[dnncosh(nx)]
    ux(x,0)= summation (from n=1 to n=infinity) sin(ny)[dnn]

    Using Fourier's Series to expand 3siny-5sin4y, I also got 0 so dn is 0 as well

    Therefore u(x,y)=0

    However the answer given in the book is

    u(x,y)=(3/cosh5)(siny)(sinhx) - (5/cosh20)(sin4y)(sinh4x)

    I realized that the book didn't used Fourier's Series but compared the coefficients of the two equations instead. But my question is what happened to the n because if we compare the coefficients aren't we supposed to get

    [3/ncosh(5n)] n=1 so we get [3/cosh(5)]
    [-5/nsinh(5n)] n=4 so we get [-5/4sinh(20)] so u(x,y) should be

    [3/cosh(5)](siny)(sinhx) - [5/4sinh(20)](sin4y)(sinh4x)

    Did I do it correctly or is it just a printing error?
     
  2. jcsd
  3. Jun 27, 2010 #2

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member



    They left out a 4 multiplying the cosh(20) in the denominator
    And you meant cosh(20) instead of sinh(20). They just dropped a 4.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook