{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $n \\ge 2$ and let $\\Omega \\subseteq \\R^n$ be open.\nAn *integro-partial differential equation* on $\\Omega$ is an equation for an unknown function $u \\colon \\Omega \\to \\R$ (or $\\C$, or $\\R^m$) such that:\n\n1. at least one partial derivative of $u$ appears in it, and\n2. at least one integral of $u$, or of a derivative of $u$, with respect to one or more of the variables appears in it.\n\nA typical linear form is\n$$\\sum_{\\abs{\\alpha} \\le k} a_\\alpha(x) \\, \\partial^\\alpha u(x) + \\int_{\\Omega} K(x, y) \\, u(y) \\, dy = f(x), \\quad x \\in \\Omega,$$\nwith given coefficients $a_\\alpha$, kernel $K$ and right side $f$.\n\n## Remarks\n\nExample: let $J \\colon \\R \\to [0, \\infty)$ be even and integrable, and consider $\\partial_t u(x, t) = \\int_{\\R} J(x - y) \\big(u(y, t) - u(x, t)\\big) \\, dy$ on $\\R \\times (0, \\infty)$.\nWrite $\\hat J(\\omega) = \\int_{\\R} J(z) \\cos(\\omega z) \\, dz$.\nThen $u(x, t) = e^{(\\hat J(\\omega) - \\hat J(0)) t} \\cos(\\omega x)$ is a solution, because $\\int_{\\R} J(x - y) \\cos(\\omega y) \\, dy = \\hat J(\\omega) \\cos(\\omega x)$ for even $J$.\n\nNon-example: the heat equation $\\partial_t u = \\partial_x^2 u$ contains no integral of $u$, so it fails clause 2 and is a [[partial differential equation|GC5BDZ6QBC]].\nThe equation $x'(t) = \\int_0^t x(s) \\, ds$ has an unknown of one variable only, so it is an [[integro-differential equation|YRKW2KX2CW]] but not a partial one.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "TH4NGC4016",
    "kind": "definition",
    "msc": [
      "45K05",
      "35R09"
    ],
    "name": "Integro-partial differential equation",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Integro-differential_equation"
      }
    ],
    "tag": "Q3C2ZGRHSG",
    "uses": [
      "YRKW2KX2CW",
      "GC5BDZ6QBC"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/Q3C2ZGRHSG/",
  "links": [
    {
      "field": "defines",
      "target": "TH4NGC4016"
    },
    {
      "field": "uses",
      "target": "YRKW2KX2CW"
    },
    {
      "field": "uses",
      "target": "GC5BDZ6QBC"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/Q3C2ZGRHSG.md",
  "msc": [
    "45K05",
    "35R09"
  ],
  "name": "Integro-partial differential equation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/Q3C/Q3C2ZGRHSG.md",
  "status": null,
  "tag": "Q3C2ZGRHSG",
  "uri": "m2://Q3C2ZGRHSG",
  "url": "https://tamnd.github.io/math2/wiki/integro-partial-differential-equation-definition/",
  "writing": {
    "tag": "Q3C2ZGRHSG",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}