PlanetMath: Poisson's equation

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (232)
Orphanage
Unclass'd
Unproven (519)
Corrections (22)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

Poisson's equation

(Definition)

Poisson's equation is a second-order partial differential equation which arises in physical problems such as finding the electric potential of a given charge distribution. Its general form in dimensions is

$\displaystyle \nabla^2\phi(\mathbf r)=\rho(\mathbf r)$

where $\nabla^2$ is the Laplacian and $\rho:D\to\mathbb{R}$ , often called a source function, is a given function on some subset

of $\mathbb{R}^n$ . If $\rho$ is identically zero, the Poisson equation reduces to the Laplace equation.

The Poisson equation is linear, and therefore obeys the superposition principle: if $\nabla^2\phi_1=\rho_1$ and $\nabla^2\phi_2=\rho_2$ , then $\nabla^2(\phi_1+\phi_2)=\rho_1+\rho_2$ . This fact can be used to construct solutions to Poisson's equation from fundamental solutions, or Green's functions, where the source distribution is a delta function.

A very important case is the one in which , is all of $\mathbb{R}^3$ , and $\phi(\mathbf r)\to 0$ as $\vert\mathbf r\vert\to\infty$ . The general solution is then given by

$\displaystyle \phi(\mathbf r)=-\frac{1}{4\pi}\int_{\mathbb{R}^3}\frac{\rho(\mathbf{r'})}{\vert\mathbf{r}-\mathbf{r'}\vert}\mathrm{d}^3\mathbf{r'}.$

"Poisson's equation" is owned by pbruin.

(view preamble | get metadata)

Keywords:

partial differential equation

Log in to rate this entry.
(view current ratings)

Cross-references: general solution, delta function, Green's functions, solutions, Laplace equation, subset, function, Laplacian, potential, partial differential equation, second-order
There is 1 reference to this entry.

This is version 3 of Poisson's equation, born on 2003-05-22, modified 2004-02-27.
Object id is 4291, canonical name is PoissonsEquation.
Accessed 15521 times total.

Classification:

AMS MSC:

35J05 (Partial differential equations :: Partial differential equations of elliptic type :: Laplace equation, reduced wave equation , Poisson equation)

Pending Errata and Addenda

None.[ View all 1 ]

Discussion

forum policy

No messages.

Interact