# Poisson Equation

## Poisson Equation

a partial differential equation of the form Δu = f, where Δ is the Laplace operator:

When n = 3, the equation is satisfied by the potential u(x, y, z) due to a mass distribution with volume density f(x, y, z)/4π (in regions where f = 0, u satisfies the Laplace equation) and by the potential due to a charge distribution. The density of distribution f here must satisfy certain smoothness conditions—for example, continuity of the partial derivatives.

Suppose f is nonzero only in a finite region G and suppose it is bounded and has continuous first-order partial derivatives. When n = 2, the Poisson equation then has the particular solution

and when n = 3,

Here, r(A, P) is the distance between a variable point of integration A and some point P. In a more detailed form,

The solution of boundary value problems for the Poisson equation reduces, by means of the substitution u = ν + w, to the solution of boundary value problems for the Laplace equation Δw = 0. The Poisson equation was first studied by S. D. Poisson in 1812.

References in periodicals archive ?
Elliptic PDEs, Measures and Capacities: From the Poisson Equation to Nonlinear Thomas-Fermi Problems
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26] are motivated by the ghost fluid method [27], and were developed using the variable-coefficient Poisson equation [28].
The authors in [16] present a 2-D discretization of the Poisson equation based on the mimetic divergence and gradient operators.
Our aim is to find the solution of the Dirichlet boundary value problem for the Poisson equation through the Poisson integral formula.
in equation (2) the convection-diffusion equation reduce to Poisson equation.
His major contributions were in probability theory and electrostatics, where he developed the well-known Poisson equation governing the electrostatic potential arising from an arbitrary charge distribution.
Table 2: Multivariable Zero-Inflated Poisson Regression Model Summaries Parameter Standard Model and Variable Estimate Error p-value Model 1: All units ([dagger]), ([double dagger]) (A) Dependent variable = Number of HAPU-any stage N = 656 units Poisson equation covariates LOS time of study 0.
where (13) is the Poisson equation for probability of k impulses arriving in time t.
Equation (13) is the Poisson Equation (12) with the number of occurrences, x, set equal to zero.
Why are some models, like the harmonic oscillator, the Ising model, a few Hamiltonian equations in quantum mechanics, the Poisson equation, or the Lokta-Volterra equations, repeatedly used within and across scientific domains, whereas theories allow for many more modeling possibilities?
Consequently, the scalar potential satisfies the Poisson equation

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