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  "body": "\n## Example\n\nConsider the [[heat equation|VP2HN86VAG]] on the half-line, for an unknown function $u(x, t)$ with $x \u003e 0$ and $t \u003e 0$:\n$$u_t = u_{xx}, \\qquad u(x, 0) = 0, \\qquad u(0, t) = 1, \\qquad u \\text{ bounded}.$$\nThis models a cold rod, initially at temperature $0$, whose end $x = 0$ is held at temperature $1$ from time $0$ on.\nLet $U(x, s) = \\int_0^\\infty u(x, t) e^{-s t} \\, dt$ be the [[Laplace transform|05XT5AVJBK]] in the time variable, for real $s \u003e 0$.\nSince $u(x, 0) = 0$, the transform of $u_t$ is $s U$, so the equation becomes the ordinary differential equation $U_{xx} = s U$ in $x$.\nIts solutions are $U = A(s) e^{-x \\sqrt{s}} + B(s) e^{x \\sqrt{s}}$.\nBoundedness in $x$ forces $B(s) = 0$, and the boundary condition gives $U(0, s) = \\mathcal{L}\\{1\\}(s) = 1/s$, so $A(s) = 1/s$.\nHence $U(x, s) = e^{-x \\sqrt{s}} / s$.\nThe table pair $\\mathcal{L}\\{\\operatorname{erfc}(x / (2 \\sqrt{t}))\\}(s) = e^{-x \\sqrt{s}} / s$, for fixed $x \u003e 0$, gives the [[inverse transform|0JRW4X1MDC]]:\n$$u(x, t) = \\operatorname{erfc}\\left(\\frac{x}{2 \\sqrt{t}}\\right),$$\nwhere $\\operatorname{erfc}$ is the [[complementary error function|TZ3CT1YA9B]].\nCheck: as $t \\to 0^+$ with $x \u003e 0$ fixed, $\\operatorname{erfc}(x/(2\\sqrt{t})) \\to \\operatorname{erfc}(+\\infty) = 0$; at $x = 0$, $\\operatorname{erfc}(0) = 1$; and $0 \\le u \\le 1$ for $x \\ge 0$, so $u$ is bounded.\n\n## Explanations\n\n### L1 human\n\nThe Laplace transform in time removes the time derivative from the heat equation.\nWhat is left is an ordinary differential equation in space, which one can solve by hand.\nTransforming back gives the temperature profile $\\operatorname{erfc}(x / (2 \\sqrt{t}))$: the heat front spreads like $\\sqrt{t}$.\n",
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