Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation...

runs over the whole space. Poisson's equation was first published in the Bulletin de la société philomatique (1813). Poisson's two most important memoirs...

\Delta f=h} This is called Poisson's equation, a generalization of Laplace's equation. Laplace's equation and Poisson's equation are the simplest examples...

The Poisson–Boltzmann equation describes the distribution of the electric potential in solution in the direction normal to a charged surface. This distribution...

zero, the equation reduces to Poisson's equation. Therefore, when λ is very small, the solution approaches that of the unscreened Poisson equation, which...

In mathematics, the discrete Poisson equation is the finite difference analog of the Poisson equation. In it, the discrete Laplace operator takes the place...

The uniqueness theorem for Poisson's equation states that, for a large class of boundary conditions, the equation may have many solutions, but the gradient...

{\displaystyle a(u,v)=\mathbf {v} ^{T}\mathbf {A} \mathbf {u} .} To solve Poisson's equation − ∇ 2 u = f , {\displaystyle -\nabla ^{2}u=f,} on a domain Ω ⊂ R d...

time. The canonical examples of elliptic PDEs are Laplace's Equation and Poisson's Equation. Elliptic PDEs are also important in pure mathematics, where...

Schrödinger–Newton equation, sometimes referred to as the Newton–Schrödinger or Schrödinger–Poisson equation, is a nonlinear modification of the Schrödinger equation with...

.} Then the differential form of Gauss's law for gravity becomes Poisson's equation: ∇ 2 ϕ = 4 π G ρ . {\displaystyle \nabla ^{2}\phi =4\pi G\rho .} This...

In astrophysics, the Lane–Emden equation is a dimensionless form of Poisson's equation for the gravitational potential of a Newtonian self-gravitating...

equation which governs the curvature obtained by the band edges in the space charge region, i.e. the band bending phenomenon, is Poisson’s equation,...

velocity together with an elliptic Poisson's equation for the pressure. On the other hand, the compressible Euler equations form a quasilinear hyperbolic system...

Poisson's equation—or in the vacuum, Laplace's equation. There is considerable overlap between potential theory and the theory of Poisson's equation to...

manifold M {\displaystyle M} . In Newton's theory of gravitation, Poisson's equation reads Δ U = 4 π G ρ {\displaystyle \Delta U=4\pi G\rho \,} where U...

step is to specify the electrostatic potential for ion j by means of Poisson's equation ∇ 2 ψ j ( r ) = − 1 ε 0 ε r ρ j ( r ) {\displaystyle \nabla ^{2}\psi...

}}}\cdot {\frac {\partial f_{\alpha }}{\partial \mathbf {v} }}=0,} and Poisson's equation for self-consistent electric field: ∇ 2 ϕ + ρ ε = 0. {\displaystyle...

differential equations describing physical phenomena. Poisson's equation describes electric and gravitational potentials; the diffusion equation describes...

Functional equation Functional equation (L-function) Constitutive equation Laws of science Defining equation (physical chemistry) List of equations in classical...

Korteweg–de Vries equation Modified KdV–Burgers equation Maxwell's equations Navier–Stokes equations Poisson's equation Primitive equations (hydrodynamics)...

Schamel method the Vlasov equations for the species involved are first solved and only in the second step Poisson's equation to ensure self-consistency...

function arises in systems that can be described by Poisson's equation, a partial differential equation (PDE) of the form ∇ 2 u ( x ) = f ( x ) {\displaystyle...

relationship is a form of Poisson's equation. In the absence of unpaired electric charge, the equation becomes Laplace's equation: ∇ 2 ϕ = 0 , {\displaystyle...

The Nernst–Planck equation is a conservation of mass equation used to describe the motion of a charged chemical species in a fluid medium. It extends...

\dots ,y^{(n)}(x))dx,} then y {\displaystyle y} must satisfy the Euler–Poisson equation, ∂ f ∂ y − d d x ( ∂ f ∂ y ′ ) + ⋯ + ( − 1 ) n d n d x n [ ∂ f ∂ y...

(NG) Newton's law of universal gravitation Gauss's law for gravity Poisson's equation for gravity History of gravitational theory General relativity (GR)...

\end{aligned}}} This equation can be solved together with Poisson's equation numerically. An example of results of solving the drift diffusion equation is shown on...

wave equation, the diffusion equation, and the Schrödinger equation for a free particle. In optics, the Helmholtz equation is the wave equation for the...

respectively. The charge density and electric potential are related by Poisson's equation, which gives − ∇ 2 [ Δ ϕ ( r ) ] = 1 ε 0 [ Q δ ( r ) − e Δ ρ ( r )...