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  "body": "\n## Definition\n\nLet $D$ be a set and let $\\varepsilon_0 \u003e 0$.\nFor each $\\varepsilon \\in (0, \\varepsilon_0)$, let $P_\\varepsilon$ be a problem on $D$, such as a [[boundary value problem|TM8VQJWGHZ]] for an [[ordinary differential equation|0NT64JV96A]].\nAssume that the data of $P_\\varepsilon$ have a limit $P_0$ as $\\varepsilon \\to 0$.\nThe problem $P_0$ is the *reduced problem*.\nLet $u_\\varepsilon \\colon D \\to \\R^n$ be a solution of $P_\\varepsilon$.\nThe family is a *regular perturbation* of $P_0$ if $P_0$ has a solution $u_0$ and $u_\\varepsilon \\to u_0$ uniformly on $D$ as $\\varepsilon \\to 0$.\nThe family is a *singular perturbation* if it is not a regular perturbation.\nThis means that $P_0$ has no solution, or that $u_\\varepsilon$ does not converge uniformly on $D$ to a solution of $P_0$.\n\n## Remarks\n\nExample: $P_\\varepsilon$ is $\\varepsilon y'' + y' = 0$ on $D = [0, 1]$ with $y(0) = 0$ and $y(1) = 1$.\nThe solution is $y_\\varepsilon(x) = (1 - e^{-x/\\varepsilon}) / (1 - e^{-1/\\varepsilon})$.\nThe reduced problem $y' = 0$, $y(0) = 0$, $y(1) = 1$ has no solution.\nIndeed $y_\\varepsilon(x) \\to 1$ for each $x \\in (0, 1]$ while $y_\\varepsilon(0) = 0$, so the convergence is not uniform.\n\nNon-example: $y' = -y + \\varepsilon$ on $[0, 1]$ with $y(0) = 1$.\nThe solution $y_\\varepsilon(x) = \\varepsilon + (1 - \\varepsilon) e^{-x}$ converges uniformly to $e^{-x}$, the solution of the reduced problem, so the perturbation is regular.\n\nA small parameter multiplying the highest derivative usually makes the problem singular.\nSingular problems are treated by the method of matched asymptotic expansions and the method of multiple scales.\n\n## Explanations\n\n### L1 human\n\nA perturbation is regular when the solution at $\\varepsilon = 0$ is a good approximation everywhere.\nIt is singular when this fails somewhere, often near the boundary.\n",
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