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  "body": "\n## Definition\n\nLet $L \u003e 0$ and let $A$ be a linear differential operator in $x$, with boundary conditions at $0$ and $L$.\nConsider the linear evolution equation $u_t = A u$ for $u \\colon [0, L] \\times [0, \\infty) \\to \\R$.\nAssume that $L^2(0, L)$ has an [[orthonormal basis|0S9GGTR0EX]] $(\\varphi_k)_{k \\ge 1}$ of functions that satisfy the boundary conditions and $A \\varphi_k = -\\mu_k \\varphi_k$ with $\\mu_k \\in \\R$.\nTypically the $\\varphi_k$ are sines or cosines, so that expansions in them are [[Fourier series|G12KXDHPPS]].\nThe *Fourier series method* for the initial value $u(\\cdot, 0) = f$ consists of three steps.\n\n1. Expand $f = \\sum_k b_k \\varphi_k$ with $b_k = \\int_0^L f \\varphi_k \\, dx$.\n2. Seek $u(x, t) = \\sum_k c_k(t) \\varphi_k(x)$, which turns the equation into the ordinary differential equations $c_k' = -\\mu_k c_k$ with $c_k(0) = b_k$.\n3. Solve them, which gives $u(x, t) = \\sum_k b_k e^{-\\mu_k t} \\varphi_k(x)$.\n\nThe same steps apply to second-order equations in time, such as the wave equation $u_{tt} = A u$, with two initial values.\n\n## Remarks\n\nExample: the [[heat equation|VP2HN86VAG]] $u_t = u_{xx}$ on $(0, \\pi)$ with $u(0, t) = u(\\pi, t) = 0$.\nHere $\\varphi_k(x) = \\sqrt{2/\\pi}\\, \\sin(kx)$ and $\\mu_k = k^2$, since $(\\sin kx)'' = -k^2 \\sin kx$.\nFor $f(x) = \\sin x + \\sin 3x$, the method gives $u(x, t) = e^{-t} \\sin x + e^{-9t} \\sin 3x$.\nCheck: $u_t = -e^{-t} \\sin x - 9 e^{-9t} \\sin 3x = u_{xx}$, and $u$ vanishes at $0$ and $\\pi$.\n\nNon-example: the nonlinear equation $u_t = u_{xx} + u^2$ with the same boundary conditions does not decouple.\nThe product of two sine modes contains other modes, so the coefficient equations form an infinite coupled system.\n\nEach term $c_k(t) \\varphi_k(x)$ is a product solution, so the method is [[separation of variables|47NFRVB564]] followed by superposition.\nFor $f \\in L^2(0, L)$ and $\\mu_k \\ge 0$, the series for $u$ converges in $L^2$ for each $t \\ge 0$.\n\n## Explanations\n\n### L1 human\n\nThe method writes the initial state as a sum of basic shapes that the equation only rescales in time.\nEach shape is then evolved by a simple exponential factor, and the pieces are added back together.\n",
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