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...

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 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...

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

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

{\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...

.} 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...

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,...

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...

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

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

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...

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

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...

Quebec Poisson distribution, a discrete probability distribution named after Siméon Denis Poisson Poisson's equation, a partial differential equation named...

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

In mathematics, the Euler–Poisson–Darboux(EPD) equation is the partial differential equation u x , y + N ( u x + u y ) x + y = 0. {\displaystyle u_{x,y}+{\frac...

modulus is much higher than the shear modulus, Poisson's ratio is near 0.5. For open-cell polymer foams, Poisson's ratio is near zero, since the cells tend...

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 )...

construct a new image by integrating the gradient, which requires solving Poisson's equation. Processing images in the gradient domain is a two-step process. The...

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

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...

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...

solution to Poisson's equation. Dirichlet's principle states that, if the function u ( x ) {\displaystyle u(x)} is the solution to Poisson's equation Δ u +...

\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...