Boundary integral method to solve poisson equation

[ra_forever8]

Suggest how to solve Poisson equation
\begin{equation}
σ ∇^2 V = - I δ(x-x_s) δ(y-y_s) δ(z-z_s) \nonumber
\end{equation}
by using the boundary integration method to calculate the potential $V(r,z)$ with the help of changing the Poisson equation into cylindrical polar co ordinates?
Where V is electric Potential (scalar)[volts]. Solution depends only on $r$ and $σ$ is constant. $r$ is horizontal coordinate(direction) $0\leq r \leq \infty $ and $z$ is vertical co ordinate(direction) $-\infty\leq r \leq \infty $. What boundary conditions are appropriate?=>Firstly, is boundary integration method same as boundary element method?if not what do you mean by boundary integration method?To change the Poisson equation into cylindrical polar co ordinates: $σ =(r,θ,z)$ where, $θ$ is not relevant
so, $σ=(r,z)$ cylindrical polar co ordinates is
\begin{equation}
σ ∇^2 V= \int ^{r= ∞}_ {r=0} \int^ {z=∞} _{ z= - ∞} {δ(r) δ(z-z_s)} dz dr.
\end{equation}
where $r_s$=0 at the top layer r=0.Now, it suggest how to calculate the potential V(r,z) by using boundary integral method.for the boundary conditions i thought to use green function in 3-d dimensional but pretending 2-d in our case.
\begin{equation}
( green function) G_{3D}= \frac{- 1}{4πσ r} = \frac{∂G}{∂r} + \frac{1}{r} ≈ 0
\end{equation}

my second question is can i use Neumann condition as a boundary condition under boundary integration method?No current flux on a surface with normal n: $ σ \frac{∂V}{∂n} =0$ [ Neumann ]my third question is how to solve the problem now?can anyone please help me.

 

[jvicens]

I would recommend the following steps to solve the Poisson equation using the boundary integration method:

1. First, convert the Poisson equation into cylindrical polar coordinates as you have suggested. This will simplify the equation and make it easier to solve.

2. Next, use the boundary integration method to calculate the potential V(r,z). This method involves integrating the equation over the boundary of the region of interest. In this case, the boundary would be the surface defined by r=0 and z=±∞.

3. To calculate the potential, you can use the Green's function in 3D. In 2D, this would simplify to a logarithmic function. You can also use the Neumann boundary condition, which states that there is no current flux on a surface with normal n. This would be appropriate for this problem.

4. Finally, to solve the problem, you can use numerical methods such as finite element or finite difference methods. These methods can help you approximate the solution to the Poisson equation and find the potential V(r,z).

I hope this helps you in solving the Poisson equation using the boundary integration method. Good luck!