# Green's function for Poisson's equation and the EEG equation with Neumann boundary condition on $n$-balls

## TL;DR

This paper derives the Green's function for Poisson's equation with Neumann boundary conditions on n-balls, using an elementary approach based on the EEG equation's Green's function, extending previous results to arbitrary dimensions.

## Contribution

It provides a simplified derivation of the Green's function for Poisson's equation with Neumann boundary conditions on n-balls, generalizing prior work to any dimension.

## Key findings

- Derived Green's function for Poisson's equation with Neumann boundary conditions on n-balls.
- Connected the Green's function for EEG equation to that of Poisson's equation.
- Extended results to arbitrary dimensions.

## Abstract

We provide an elementary derivation of the Green's function for Poisson's equation with Neumann boundary data on balls of arbitrary dimension, which was recently found in [Sadybekov et al., Eurasian Math. J. 7(2):100-105, 2016]. The underlying idea consists of first computing the Green's function for the electroencephalography (EEG) equation (Poisson's equation with dipole right-hand side) and then deriving the Green's function for Poisson's equation from that.