{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "8V9J445R7N"
    }
  ],
  "body": "\n## Summary\n\nAn abstract Volterra equation is a Volterra integral equation whose unknown takes values in a [[Banach space|06GZ956657]].\nIts memory term involves a linear operator, often unbounded.\nIts standard form is $u(t) = f(t) + \\int_0^t a(t - s) A u(s) \\, ds$ with a scalar kernel $a$ and a [[closed operator|VM7B38PA87]] $A$.\nIt contains abstract first-order and second-order evolution equations as the cases $a = 1$ and $a(t) = t$.\nOther kernels model heat conduction and viscoelasticity with memory, where $A$ is a differential operator in space.\n\n## Explanations\n\n### L1 human\n\nMany physical systems change in time according to a rule that involves their whole history.\nWhen the state at each time is a function of space, such as a temperature profile, the state is a point of an infinite-dimensional space.\nThe abstract Volterra equation writes the rule as an integral over the past.\nA weight $a$ says how strongly the past is remembered, and an operator $A$ acts in space.\nIf $a = 1$, all of the past counts equally, and the equation is the integrated form of $u' = A u$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "45N05",
      "45D05",
      "47D06"
    ],
    "name": "Abstract Volterra equation",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Volterra_integral_equation"
      }
    ],
    "specializes": [
      "DND8CBEVXT"
    ],
    "tag": "1N7TZCPBF0"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/1N7TZCPBF0/",
  "links": [
    {
      "field": "specializes",
      "target": "DND8CBEVXT"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/1N7TZCPBF0.md",
  "msc": [
    "45N05",
    "45D05",
    "47D06"
  ],
  "name": "Abstract Volterra equation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/1N7/1N7TZCPBF0.md",
  "status": null,
  "tag": "1N7TZCPBF0",
  "uri": "m2://1N7TZCPBF0",
  "url": "https://tamnd.github.io/math2/wiki/abstract-volterra-equation/",
  "writing": {
    "tag": "1N7TZCPBF0",
    "kind": "concept",
    "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
      }
    ]
  }
}