{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "3DJMFT0SQM"
    }
  ],
  "body": "\n## Summary\n\nA similarity solution, or self-similar solution, of a partial differential equation is a solution that is unchanged by a scaling of its variables.\nIts graph at one time is a rescaled copy of its graph at any other time.\nSuch a solution has the form $u(x, t) = t^{-\\alpha} v(x / t^{\\beta})$, so finding it reduces to an equation in fewer variables.\nMany fundamental solutions, [[rarefaction waves|MFN4Q5SH8H]] and blow-up profiles are of this kind.\nSee the [[definition|3DJMFT0SQM]].\n\n## Explanations\n\n### L1 human\n\nDrop ink into still water: the blot grows and fades, but its shape stays the same, only stretched.\nA similarity solution is a solution with this property.\nIf one measures distance in units that grow like a power of time, the picture stops changing.\nFor the heat equation the blot has width proportional to $\\sqrt{t}$ and height proportional to $t^{-1/2}$ in one dimension.\nThe variable $x / \\sqrt{t}$ is then the only one that matters, and the PDE becomes an ordinary differential equation.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "35C06",
      "35B06",
      "76M55"
    ],
    "name": "Similarity solution",
    "same_as": {
      "wikidata": "Q4056366",
      "wikipedia": "Self-similar solution"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Self-similar_solution"
      }
    ],
    "tag": "579J5K9K5Z"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/579J5K9K5Z/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/579J5K9K5Z.md",
  "msc": [
    "35C06",
    "35B06",
    "76M55"
  ],
  "name": "Similarity solution",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/579/579J5K9K5Z.md",
  "status": null,
  "tag": "579J5K9K5Z",
  "uri": "m2://579J5K9K5Z",
  "url": "https://tamnd.github.io/math2/wiki/similarity-solution/",
  "writing": {
    "tag": "579J5K9K5Z",
    "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
      }
    ]
  }
}