Poisson’s equation

Poisson’s equation is a second-order partial differential equationMathworldPlanetmath which arises in physical problems such as finding the electric potential of a given charge distribution. Its general form in n is


where 2 is the LaplacianDlmfMathworld and ρ:D, often called a source , is a given functionMathworldPlanetmath on some subset D of n. If ρ is identically zero, the Poisson equationMathworldPlanetmath reduces to the Laplace equation.

The Poisson equation is linear, and therefore obeys the superposition principle: if 2ϕ1=ρ1 and 2ϕ2=ρ2, then 2(ϕ1+ϕ2)=ρ1+ρ2. This fact can be used to construct solutions to Poisson’s equation from fundamental solutions, or Green’s functions, where the source distribution is a delta functionPlanetmathPlanetmath.

A very important case is the one in which n=3, D is all of 3, and ϕ(𝐫)0 as |𝐫|. The general solution is then given by

Title Poisson’s equation
Canonical name PoissonsEquation
Date of creation 2013-03-22 13:38:28
Last modified on 2013-03-22 13:38:28
Owner pbruin (1001)
Last modified by pbruin (1001)
Numerical id 6
Author pbruin (1001)
Entry type Definition
Classification msc 35J05
Related topic HelmholtzDifferentialEquation
Related topic LaplacesEquation
Related topic GreensFunction