Poisson's equation

Summary

Poisson's equation is the partial differential equation $\Delta u = f$, where $\Delta$ is the Laplacian and $f$ is a given function. With $f = 0$ it is Laplace's equation, so it is the inhomogeneous version of the equation of harmonic functions. It describes a potential created by a source: the gravitational potential of a mass density, or the electrostatic potential of a charge density. On the whole space a solution is given by convolving $f$ with the fundamental solution of the Laplacian. On bounded domains it is solved with Green's functions.