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edr1c

  • 2 years ago

any1 can explain the principles in solving laplace equation in rectangular coordinates? i know i need to solve for 2 2nd order ODE but im not sure how to conclude for whether they are trivial or nontrivial.

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  1. edr1c
    • 2 years ago
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    i let u(x, y) = F(x) G(y) so Uxx = F'' G Uyy = F G" then the equation become F"G + FG" = 0 i separate the variables and equate them to a separation constant, k so the 2 2nd order ODE are F''-kF=0 , G"+kG = 0 how do i proceed with the initial conditions given?

  2. edr1c
    • 2 years ago
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    i need to get 3 cases when k=0, k<0 and k>0. but i dont know how to conclude after applying the initial conditions into the cases.

  3. experimentX
    • 2 years ago
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    do you have relevant links?

  4. TuringTest
    • 2 years ago
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    I always forget how to do this, so here is a hopefully helpful link http://tutorial.math.lamar.edu/Classes/DE/LaplacesEqn.aspx

  5. edr1c
    • 2 years ago
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    let say i have the case where k=0 for F"=0 i got F=Ax+B, and i substitube initial conditions to obtain F be trivial (F=0). for G"=0 i got G=Cx+D G(0)=0 : D=0, G=Cx G(24)=24: C=1, G=x ?? is it possible to obtain nontrivial solution for G when F is trivial?

  6. experimentX
    • 2 years ago
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    hold on for a while ... I'll work out few other probs and try to work on my copy.

  7. edr1c
    • 2 years ago
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    oh kay. thx alot

  8. edr1c
    • 2 years ago
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    my lecturer practically skipped this part ==

  9. Mikael
    • 2 years ago
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    @edr1c The devil is in the not-details, in the boundary conditions. Each set of B.C. allows/disallows other solutions

  10. Mikael
    • 2 years ago
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    The coordinates is in some sense "our imagination" - the equation is not influenced by the coordinates we choose. It IS influenced by the symmetry of its BOUNDARY, and of its BOUNDARY CONDITIONS

  11. Mikael
    • 2 years ago
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    So why do we choose the coordinates ? - To fit the symmetry of the boundary of course !

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