Editing Poisson's equation (section)

==Statement of the equation==

Poisson's equation is
<math display="block">\Delta\varphi = f,</math>
where <math>\Delta</math> is the [[Laplace operator]], and <math>f</math> and <math>\varphi</math> are [[real number|real]] or [[complex number|complex]]-valued [[function (mathematics)|functions]] on a [[manifold]]. Usually, <math>f</math> is given, and <math>\varphi</math> is sought. When the manifold is [[Euclidean space]], the Laplace operator is often denoted as {{math|∇<sup>2</sup>}}, and so Poisson's equation is frequently written as
<math display="block">\nabla^2 \varphi = f.</math>

In three-dimensional [[Cartesian coordinate]]s, it takes the form
<math display="block">\left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} \right)\varphi(x, y, z) = f(x, y, z).</math>

When <math>f = 0</math> identically, we obtain [[Laplace's equation]].

Poisson's equation may be solved using a [[Green's function]]:
<math display="block">\varphi(\mathbf{r}) = - \iiint \frac{f(\mathbf{r}')}{4\pi |\mathbf{r} - \mathbf{r}'|}\, \mathrm{d}^3 r',</math>
where the integral is over all of space. A general exposition of the Green's function for Poisson's equation is given in the article on the [[screened Poisson equation]]. There are various methods for numerical solution, such as the [[relaxation method]], an iterative algorithm.