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

Laplace equation on a rectangle with inhomogeneous boundary conditions.In summary, A.S is asking if Green's Function approach can be used to solve a Laplace equation on a rectangle with specific boundary conditions. The response is that it can be used and will result in a Green's function that can solve the equation with inhomogeneous boundary conditions.
  • #1
avinashsahoo
4
0
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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  • #2
If you impose the following on the a rectangle:

[tex]
u(0,y) = u(x,L_2) = u(L_1,y) = u(x,0) = 0
[/tex]
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:
[tex]
\alpha_1 u(0,y) + \alpha_2 u_x(0,y)= f_1(x) \\
[/tex]
[tex]
\beta_1 u(L_1,y) + \beta_2 u_x(L_1,y)= f_2(x) \\
[/tex]
[tex]
\gamma_1 u(x,0) + \gamma_2 u_y(x,0)= g_1(x) \\
[/tex]
[tex]
\delta_1 u(x,L_2) + \gamma_2 u_y(x,L_2)= g_2(x) \\
[/tex]
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
 

1. What is Green's formula for Laplace equation?

Green's formula for Laplace equation is a mathematical formula used to solve boundary value problems in partial differential equations, specifically the Laplace equation. It relates the values of a harmonic function at two points to the values of its normal derivative on the boundary between these two points.

2. How is Green's formula derived?

Green's formula is derived using the divergence theorem from vector calculus. It states that the surface integral of the normal component of a vector field over a closed surface is equal to the volume integral of the divergence of that vector field over the region enclosed by the surface.

3. What are the boundary conditions required for Green's formula to be applicable?

Green's formula can be applied to any boundary value problem with Dirichlet boundary conditions, which specify the values of the function on the boundary, or Neumann boundary conditions, which specify the values of the normal derivative on the boundary.

4. How is Green's formula used to solve boundary value problems?

Green's formula allows us to convert a boundary value problem for a partial differential equation into a boundary integral equation, which can be solved by plugging in the given boundary conditions and solving for the unknown function. It is a powerful tool for solving complex boundary value problems in various fields of science and engineering.

5. What are the limitations of Green's formula?

Green's formula can only be applied to linear boundary value problems, where the boundary conditions and the differential equation are both linear. It also assumes that the boundary is smooth and that the function being solved is continuous and has continuous derivatives up to the second order. Additionally, it is not applicable to non-homogeneous boundary conditions.

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