{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "J0MHS4ZCKR"
    }
  ],
  "body": "\n## Summary\n\nThe Fourier series method solves linear partial differential equations on bounded intervals or boxes by expanding the unknown in a [[Fourier series|G12KXDHPPS]].\nEach term of the series is a sine or cosine mode, and the equation acts on each mode separately.\nThe partial differential equation then becomes a family of ordinary differential equations for the coefficients.\nFourier introduced the method to solve the [[heat equation|VP2HN86VAG]], and it led to the theory of Fourier series.\nSee the [[definition|J0MHS4ZCKR]].\n\n## Explanations\n\n### L1 human\n\nA vibrating string or a heated rod can be described as a sum of simple waves, each with its own frequency.\nUnder the heat equation or the wave equation, each simple wave evolves on its own, in a way that is easy to compute.\nSo one splits the initial state into simple waves, lets each of them evolve, and adds them up again.\nFor the heat equation on $(0, \\pi)$ with zero boundary values, the mode $\\sin(kx)$ just decays like $e^{-k^2 t}$.\nHigh frequencies decay fastest, which is why heat smooths out sharp temperature differences.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "35C10",
      "35K05",
      "42A16"
    ],
    "name": "Fourier series method",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Fourier_series"
      }
    ],
    "tag": "JYEV0Q7T4Z"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/JYEV0Q7T4Z/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/JYEV0Q7T4Z.md",
  "msc": [
    "35C10",
    "35K05",
    "42A16"
  ],
  "name": "Fourier series method",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/JYE/JYEV0Q7T4Z.md",
  "status": null,
  "tag": "JYEV0Q7T4Z",
  "uri": "m2://JYEV0Q7T4Z",
  "url": "https://tamnd.github.io/math2/wiki/fourier-series-method/",
  "writing": {
    "tag": "JYEV0Q7T4Z",
    "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
      }
    ]
  }
}