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?

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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.