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

On the other hand, the charge distribution of the point charge is zero everywhere except at the point where the charge is located.

A Life Before And After the Dirac Delta Function

Your Daily Dose of Scientific Python

In other word the charge distribution is a Dirac delta function! So we have from simple physical reasoning

But this is exactly the defining equation for a Green’s function! So we now know that the Green’s function for Poisson’s equation with free boundary conditions is

From this we can immediately solve the Poisson equation for any charge distribution ρ whatsoever:

Easy, isn’t it? Basically we are just adding up all the potentials created by the charges ρ( r′ )dV in all the infinitesimal volumes, which is possible as the Poisson equation is linear.

In conclusion, Green’s functions are a powerful tool for solving differential equations, including the Poisson equation. We have seen that the Green’s function for the Poisson equation with free boundary conditions is 1/| r - r’ |, which allows us to find the electrostatic potential created by a given charge distribution by calculating the integral of ρ( r’ )*(1/| r - r’ |)dV. This method is particularly useful as it is a simple solution to an equation that describes a wide range of physical phenomena. We hope this post has given you a feeling for the power of Green’s functions in applications relevant to physics and engineering.