# Poisson Equation

## Poisson Equation

a partial differential equation of the form Δ*u* = *f*, where Δ is the Laplace operator:

When *n =* 3, the equation is satisfied by the potential *u(x, y, z)* due to a mass distribution with volume density *f(x, y, z)/4π* (in regions where *f = 0, u* satisfies the Laplace equation) and by the potential due to a charge distribution. The density of distribution *f* here must satisfy certain smoothness conditions—for example, continuity of the partial derivatives.

Suppose *f* is nonzero only in a finite region *G* and suppose it is bounded and has continuous first-order partial derivatives. When *n* = 2, the Poisson equation then has the particular solution

and when *n =* 3,

Here, *r(A, P)* is the distance between a variable point of integration *A* and some point *P*. In a more detailed form,

The solution of boundary value problems for the Poisson equation reduces, by means of the substitution *u* = ν + *w*, to the solution of boundary value problems for the Laplace equation Δ*w* = 0. The Poisson equation was first studied by S. D. Poisson in 1812.