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HELP : Laplace 2D equation on a square

  1. Oct 20, 2006 #1
    HELP!!: Laplace 2D equation on a square

    Hi everyone,

    I quite get the answer out for this question and i just feel like i have been beating my head into a wall for 4 hours! aghh....

    [​IMG]

    I get stuck when i try to solve the sub-problem which has boundary conditions:

    [​IMG]

    I do the usual thing of seperating varibales on [​IMG] which leads to the equations:

    [​IMG]

    so i solved these equations and applyed the boundary conditions:

    X'(0) = 0, X'(1) = 0, Y(1) = 0

    this gives the equation [​IMG]

    but then if i sub in the non-homogenous BC, [​IMG]

    i get 0 = x

    lol, what have i done this time! :P

    any help would be so very much appreciated! :)

    Cheers
    Sarah
     
  2. jcsd
  3. Oct 20, 2006 #2

    StatusX

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    I don't think you solved for Y correctly. You want Y(1)=0, and you have Y(0)=0 instead.
     
  4. Oct 20, 2006 #3
    ok, well what i did was try to solve

    [​IMG]

    which has general solution

    [​IMG]

    and applying the BC Y(1) = 0 gives

    [​IMG]

    which means that k_1 = 0 and so solution for Y is:

    [​IMG]


    i think this is correct, but yeah, i guess it isnt hey? ;) could you please help me to find where i am going wrong?

    thanks

    sarah
     
  5. Oct 20, 2006 #4

    StatusX

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    Why does that mean k_1=0? cosh(n pi) and sinh(n pi) are both positive numbers.
     
  6. Oct 20, 2006 #5
    isnt it because cosh(n pi) is never zero and sinh(n pi) is equal to 0 for n = 0 and so since we need the whole expression to be equal to 0 therefore k_1 needs to be 0?
     
  7. Oct 20, 2006 #6

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    But you need to pick a k_1, k_2 for each n such that that expression is zero. Note that you can rewrite A sinh(x)+B cosh(x) as C sinh(x+d) for some C and d, just like for sin and cos.
     
  8. Oct 20, 2006 #7
    hmm ok i think i see now....

    ok i get this answer for the solution to the sub-problem now

    [​IMG]

    does that look better? :)
     
  9. Oct 20, 2006 #8

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    I don't think so, but it's hard to tell. Take one step at a time. What is Y(y), and is it zero at y=1? EDIT: I'll be gone for the night, hopefully someone else can help you if you still need it right now.
     
    Last edited: Oct 20, 2006
  10. Oct 20, 2006 #9
    i do i do, lol, any parting hints? ;)
     
  11. Oct 20, 2006 #10
    ok, i get now get
    [​IMG]
     
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