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

avinashsahoo

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 ?

Thanks,
A.S

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Winzer

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 $$u = (x,y) = 0$$
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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