information:extraction, linguistic:paraphrasing
Statement of the equation

Poisson's equation is Δ φ = f, {\displaystyle \Delta \varphi =f,} where Δ {\displaystyle \Delta } is the Laplace operator, and f {\displaystyle f} and φ {\displaystyle \varphi } are real or complex -valued functions on a manifold. Usually, f {\displaystyle f} is given, and φ {\displaystyle \varphi } is sought. When the manifold is Euclidean space, the Laplace operator is often denoted as ∇, and so Poisson's equation is frequently written as ∇ 2 φ = f. {\displaystyle \nabla ^{2}\varphi =f.}

In three-dimensional Cartesian coordinates, it takes the form (∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 + ∂ 2 ∂ z 2) φ (x, y, z) = f (x, y, z). {\displaystyle \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).}

When f = 0 {\displaystyle f=0} identically, we obtain Laplace's equation.

Poisson's equation may be solved using a Green's function: φ (r) = − ∭ f (r ′) 4 π | r − r ′ | d 3 r ′, {\displaystyle \varphi (\mathbf {r})=-\iiint {\frac {f(\mathbf {r} ')}{4\pi |\mathbf {r} -\mathbf {r} '|}}\,\mathrm {d} ^{3}r',} 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.