In this blog post, we will dive into the fascinating world of Green’s functions and their application to solving the Poisson equation. Green’s functions offer a powerful and easy way to solve both ordinary and partial differential equations. Having the Green’s function for a specific differential equation, solving the equation becomes a matter of calculating an integral. In this post, we will focus on the Poisson equation, which describes a wide range of physical phenomena such as the electrostatic potential created by an electric charge distribution or the gravitational potential produced by a mass distribution. We will discuss the Green’s function for the Poisson equation with free boundary conditions, and how it can be used to solve the equation for any charge distribution whatsoever.
Ok, let’s find the Green’s function for the Poisson equation
with the boundary condition Φ(r→∞)=0. It is easy, if we apply basic knowledge of physics!
Consider an electric point charge q located at r′. We know from Coulomb’s law that it will create an electrostatic potential of
