physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rel ...

, the screened Poisson equation is a

Poisson equation Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with t ...

, which arises in (for example) the

Klein–Gordon equation The Klein–Gordon equation (Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic wave equation, related to the Schrödinger equation. It is second-order in space and time and manifestly Lorentz-covariant. ...

, electric field screening in plasmas, and nonlocal granular fluidity in

granular flow A granular material is a conglomeration of discrete solid, macroscopic particles characterized by a loss of energy whenever the particles interact (the most common example would be friction when grains collide). The constituents that compose gra ...

Statement of the equation

The equation is :

u(\mathbf) = - f(\mathbf),

where

\Delta

is the

Laplace operator In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is t ...

, ''λ'' is a constant that expresses the "screening", ''f'' is an arbitrary function of position (known as the "source function") and ''u'' is the function to be determined. In the homogeneous case (''f''=0), the screened Poisson equation is the same as the time-independent

. In the inhomogeneous case, the screened Poisson equation is very similar to the