{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "A0F09CMEJM"
    }
  ],
  "body": "\n## Summary\n\nPoisson's equation is the partial differential equation $\\Delta u = f$, where $\\Delta$ is the [[Laplacian|0E0B6RA8K0]] and $f$ is a given function.\nWith $f = 0$ it is [[Laplace's equation|S8RG6N1PRT]], so it is the inhomogeneous version of the equation of harmonic functions.\nIt describes a potential created by a source: the gravitational potential of a mass density, or the electrostatic potential of a charge density.\nOn the whole space a solution is given by convolving $f$ with the [[fundamental solution of the Laplacian|BSJ8X2XRN9]].\nOn bounded domains it is solved with Green's functions.\n\n## Explanations\n\n### L1 human\n\nLaplace's equation describes a steady state with no sources inside, such as the temperature in a plate with no heater.\nPoisson's equation allows sources: the right-hand side $f$ says how much heat is produced, or how much charge sits, at each point.\nFor example, $u = (x^2 + y^2)/4$ solves $\\Delta u = 1$ in the plane, a constant source everywhere.\nTwo solutions differ by a harmonic function, so on a bounded domain the boundary values pin down a single solution.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "35J05",
      "31B35"
    ],
    "name": "Poisson's equation",
    "same_as": {
      "wikidata": "Q827688",
      "wikipedia": "Poisson's equation"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Poisson%27s_equation"
      }
    ],
    "tag": "DWK5N5SKTK"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/DWK5N5SKTK/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/DWK5N5SKTK.md",
  "msc": [
    "35J05",
    "31B35"
  ],
  "name": "Poisson's equation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/DWK/DWK5N5SKTK.md",
  "status": null,
  "tag": "DWK5N5SKTK",
  "uri": "m2://DWK5N5SKTK",
  "url": "https://tamnd.github.io/math2/wiki/poissons-equation/",
  "writing": {
    "tag": "DWK5N5SKTK",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}