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Green's Formula for Laplace Eqaution : For any BC's ?

  1. Apr 28, 2009 #1
    Hi all,
    I don't know anything about Green's Function approach to solve pdes.
    But I want to know if It can be used to solve a laplace eqn ,on a rectangle having all the boundary conditions of the type :

    d[Psi]/dx@some x = k[Psi@some x +f(y)] .

    Will it result in an Integral equation ?

  2. jcsd
  3. May 8, 2009 #2
    If you impose the following on the a rectangle:

    u(0,y) = u(x,L_2) = u(L_1,y) = u(x,0) = 0
    You get a very uneventful solution of [tex] u = (x,y) = 0[/tex]
    Once you start having inhomogeneous boundary conditions with linear combinations of Neuman and Dirichlet conditions you get interesting stuff:
    \alpha_1 u(0,y) + \alpha_2 u_x(0,y)= f_1(x) \\
    \beta_1 u(L_1,y) + \beta_2 u_x(L_1,y)= f_2(x) \\
    \gamma_1 u(x,0) + \gamma_2 u_y(x,0)= g_1(x) \\
    \delta_1 u(x,L_2) + \gamma_2 u_y(x,L_2)= g_2(x) \\
    You should get u1 and u2. By linearity u(x,y)=u1(x,y)+u2(x,y)

    In the end you will get a Green's function
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