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Solve Laplace equation on rectangle domain

  1. Apr 15, 2016 #1

    Dor

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    1. The problem statement, all variables and given/known data
    I'm having issues with a Laplace problem. actually, I have two different boundary problems which I dont know how to solve analytically.
    I couldn't find anything on this situations and if anybody could point me in the right direction it would be fantastic.
    It's just Laplace's equation on the square [0,a]x[0,b] with a mixed boundary.

    2. Relevant equations

    1. the first one is:

    Uxx+Uyy=0
    Ux(0,y)= 0
    Ux(a,y)= f(y) (some function)
    U(x,0)= 0
    Uy(x,b)= v (some constant)

    1. the second is:

    Uxx+Uyy=0
    Ux(0,y)= 0
    U(a,y)= g(y) (some function)
    U(x,0)= 0
    Uy(x,b)= v (some constant)

    3. The attempt at a solution

    I tried to do ordinary separable solution but I don't really know how to do this in such problems.
     
  2. jcsd
  3. Apr 15, 2016 #2
  4. Apr 16, 2016 #3

    Dor

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    I know the variable separation method.
    But I'm not sure how to do it when in one side of the rectangle there is a Dirichlet boundary U(x,b)=const and in the other one I have Neumann boundary Uy(x,0)=0
     
  5. Apr 16, 2016 #4

    pasmith

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    Homework Helper

    Laplace's equation is linear. You can always write the solution as a sum of solutions satisfying simpler boundary conditions. For example in the second problem you can set [itex]U = U_1 + U_2[/itex] where [itex]U_1[/itex] satisfies [tex]
    U_x(0,y) = 0 \\ U(a,y) = f(y) \\ U(x,0) = 0 \\ U_y(x,b) = 0 [/tex] and [itex]U_2 [/itex] satisfies [tex]
    U_x(0,y) = 0 \\ U(a,y) = 0 \\ U(x,0) = 0 \\ U_y(x,b) = v[/tex]
     
  6. Apr 17, 2016 #5

    Dor

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    I'm not sure how to solve it when I have one side with Dirichlet boundary and the other side with Neumann boundary.
    U ( x , 0 ) = 0
    U y ( x , b ) = 0
    or
    U ( x , 0 ) = 0
    U y ( x , b ) = v
     
  7. Apr 17, 2016 #6

    pasmith

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    Homework Helper

    We're looking for [itex]U = \sum_n X_n(x)Y_n(y)[/itex]

    For [itex]U(x,0) = U_y(x,b) = 0[/itex] you need [itex]Y_n(0) = 0[/itex] and [itex]Y_n'(b) = 0[/itex]. Therefore set [itex]Y_n(y) = \sin k_ny[/itex] so [itex]Y_n(0) = 0[/itex] is satisfied and [itex]k_n[/itex] is determined by [itex]Y_n'(b) = k_n\cos k_nb = 0[/itex].

    For [itex]U(x,0) = 0[/itex] and [itex]U_y(x,b) = v[/itex], you should regard [itex]v[/itex] as a function of [itex]x[/itex] to be expanded as a fourier series; accordingly [itex]Y_n[/itex] will be an exponential function with [itex]Y_n(0) = 0[/itex] and (for convenience) [itex]Y_n'(b) = 1[/itex]. The combination of exponentials which satisfies these is [itex]Y_n(y) = \sinh(k_ny)/\sinh(k_nb)[/itex] where [itex]k_n[/itex] is determined by the boundary conditions on [itex]X_n[/itex].
     
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