# Poisson Equation

The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## 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 ?
[26] are motivated by the ghost fluid method [27], and were developed using the variable-coefficient Poisson equation [28].
The Nernst-Planck equation must be solved simultaneously with the Poisson equation that directly relates the electric potential V to the electric charge:
The PPGMRES(m) in conjunction with MPMSC scheme has been used for solving the Poisson equation in three space variables subject to Dirichlet boundary conditions.
Equation (11) is of the form of a Poisson equation for [V.sub.2] with respect to the operator [L.sub.0] in the variable y.
The authors in [16] present a 2-D discretization of the Poisson equation based on the mimetic divergence and gradient operators.
The representation formula in Theorem 4 provides the solution to the Dirichlet problem for the Poisson equation.
If we put (Eq.) in equation (2) the convection-diffusion equation reduce to Poisson equation. We examine the behavior of the scheme for different values of (Eq.) , Error is reduces for (Eq.).
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 0.049696 0.005062 <.0001 Licensed HPPD -0.089453 0.021995 <.0001 % Risk assess within 24 hours of admission -0.020248 0.007991 .0113 Logistic equation covariates ([section]) % Patients "at risk" at study -0.026990 0.012461 .0303 % Male patients -0.045280 0.019379 .0195 (B) Dependent variable = Number of HAPU-II+ N = 513 units Poisson equation covariates % Patients "at risk" at study 0.009867 0.002831 .0005 Licensed HPPD -0.090562 0.031979 .0046 Pt.
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.

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