{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $y_\\varepsilon$ solve a [[singularly perturbed|G6J21TGRMM]] problem on an interval $[a, b]$ that has a [[boundary layer|YCFPD0GE5G]] at $a$ of thickness $\\delta(\\varepsilon) \\to 0$.\nThe *method of matched asymptotic expansions* builds an approximation in four steps.\n\n1. Outer expansion: solve the reduced equations order by order in $\\varepsilon$, with the boundary conditions away from $a$, to get $y^{\\mathrm{out}}(x)$.\n2. Inner expansion: set $X = (x - a) / \\delta(\\varepsilon)$ and rewrite the problem in $X$.\n   Solve it order by order with the boundary conditions at $a$, to get $Y^{\\mathrm{in}}(X)$, which may contain free constants.\n3. Matching: choose the free constants so that the two expansions agree in the overlap.\n   At leading order this is Prandtl's rule $\\lim_{X \\to \\infty} Y^{\\mathrm{in}}(X) = \\lim_{x \\to a^+} y^{\\mathrm{out}}(x)$.\n   This common value is the *common part*.\n4. Composite expansion: $y^{\\mathrm{comp}}(x) = y^{\\mathrm{out}}(x) + Y^{\\mathrm{in}}((x - a)/\\delta(\\varepsilon)) - (\\text{common part})$.\n\nEach expansion is an [[asymptotic expansion|Q26QN6GYZX]] valid in its own region, and the composite one is meant to be valid uniformly on $[a, b]$.\n\n## Remarks\n\nExample: $\\varepsilon y'' + y' + y = 0$ on $[0, 1]$, $y(0) = 0$, $y(1) = 1$.\nOuter: $y' + y = 0$ with $y(1) = 1$ gives $y^{\\mathrm{out}} = e^{1 - x}$.\nInner: with $X = x / \\varepsilon$, the leading equation is $Y'' + Y' = 0$ with $Y(0) = 0$, so $Y^{\\mathrm{in}} = A (1 - e^{-X})$.\nMatching gives $A = y^{\\mathrm{out}}(0) = e$, so the common part is $e$.\nComposite: $y^{\\mathrm{comp}} = e^{1 - x} - e^{1 - x/\\varepsilon}$; it satisfies $y(0) = 0$ exactly and $y(1) = 1 - e^{1 - 1/\\varepsilon}$, which is exponentially close to $1$.\n\nNon-example: putting the outer solution and the inner solution side by side without step 3 fails, since the constant $A$ stays undetermined.\nThe layer must also be placed correctly: a trial layer at $x = 1$ leads to inner solutions that grow like $e^{X}$ away from the boundary and cannot match.\n\n## Explanations\n\n### L1 human\n\nSolve the easy problem far from the layer, solve the zoomed-in problem in the layer, and glue the two by making them agree in between.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "QQ1YT0MV2X",
    "kind": "definition",
    "msc": [
      "34E05",
      "34E15"
    ],
    "name": "Method of matched asymptotic expansions",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Method_of_matched_asymptotic_expansions"
      }
    ],
    "tag": "CAYQ1S3RM8",
    "uses": [
      "G6J21TGRMM",
      "YCFPD0GE5G",
      "Q26QN6GYZX"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/CAYQ1S3RM8/",
  "links": [
    {
      "field": "defines",
      "target": "QQ1YT0MV2X"
    },
    {
      "field": "uses",
      "target": "G6J21TGRMM"
    },
    {
      "field": "uses",
      "target": "YCFPD0GE5G"
    },
    {
      "field": "uses",
      "target": "Q26QN6GYZX"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/CAYQ1S3RM8.md",
  "msc": [
    "34E05",
    "34E15"
  ],
  "name": "Method of matched asymptotic expansions",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/CAY/CAYQ1S3RM8.md",
  "status": null,
  "tag": "CAYQ1S3RM8",
  "uri": "m2://CAYQ1S3RM8",
  "url": "https://math2.tamnd.com/wiki/method-of-matched-asymptotic-expansions-definition/",
  "writing": {
    "tag": "CAYQ1S3RM8",
    "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
      }
    ]
  }
}