# Poisson’s equation

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 $n$ is

$${\nabla}^{2}\varphi (\mathbf{r})=\rho (\mathbf{r})$$ |

where ${\nabla}^{2}$ is the Laplacian^{} and $\rho :D\to \mathbb{R}$, often called a *source *, is a given function^{} on some subset $D$ 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 ${\nabla}^{2}{\varphi}_{1}={\rho}_{1}$ and ${\nabla}^{2}{\varphi}_{2}={\rho}_{2}$, then ${\nabla}^{2}({\varphi}_{1}+{\varphi}_{2})={\rho}_{1}+{\rho}_{2}$. 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 $n=3$, $D$ is all of ${\mathbb{R}}^{3}$, and $\varphi (\mathbf{r})\to 0$ as $|\mathbf{r}|\to \mathrm{\infty}$. The general solution is then given by

$$\varphi (\mathbf{r})=-\frac{1}{4\pi}{\int}_{{\mathbb{R}}^{3}}\frac{\rho ({\mathbf{r}}^{\prime})}{|\mathbf{r}-{\mathbf{r}}^{\prime}|}{\mathrm{d}}^{3}{\mathbf{r}}^{\prime}.$$ |

Title | Poisson’s equation |
---|---|

Canonical name | PoissonsEquation |

Date of creation | 2013-03-22 13:38:28 |

Last modified on | 2013-03-22 13:38:28 |

Owner | pbruin (1001) |

Last modified by | pbruin (1001) |

Numerical id | 6 |

Author | pbruin (1001) |

Entry type | Definition |

Classification | msc 35J05 |

Related topic | HelmholtzDifferentialEquation |

Related topic | LaplacesEquation |

Related topic | GreensFunction |