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  "body": "\n## Summary\n\nThe method of matched asymptotic expansions builds approximate solutions of [[singularly perturbed|G6J21TGRMM]] problems.\nIt uses one expansion away from the thin layers where the solution changes fast, and another expansion inside each layer, in a stretched variable.\nThe two are joined by requiring that they agree in an intermediate region.\nIt is a standard tool for [[boundary layers|YCFPD0GE5G]] in fluid dynamics and for many problems in applied mathematics.\n\n## Explanations\n\n### L1 human\n\nImagine a solution that is smooth almost everywhere but has a steep cliff near one end.\nOne formula describes the flat part well, and a zoomed-in formula describes the cliff well.\nMatching makes the two formulas agree where the cliff meets the flat part.\nAdding them and removing the shared part gives one formula that is good everywhere.\nFor $\\varepsilon y'' + y' + y = 0$, $y(0) = 0$, $y(1) = 1$, this gives $y \\approx e^{1 - x} - e^{1 - x/\\varepsilon}$.\n",
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