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  "body": "\n## Definition\n\nLet $\\alpha \u003e 0$ be a real number that is not an integer, let $n = \\ceil{\\alpha}$, let $a \u003c b$ be real numbers, and let $f \\colon [a, b] \\times \\R \\to \\R$ be a function.\nWrite ${}^C D_a^\\alpha$ for the [[Caputo fractional derivative|6VMGQYF8RW]] of order $\\alpha$ with base point $a$.\nThe *fractional differential equation* of Caputo type with right side $f$ is\n$$ {}^C D_a^\\alpha y(t) = f(t, y(t)), \\qquad t \\in (a, b],$$\nusually with initial conditions $y^{(k)}(a) = y_k$ for $k = 0, 1, \\ldots, n - 1$.\nMore generally, a fractional differential equation relates an unknown function to some of its derivatives in the sense of [[fractional calculus|T73XK15SA2]].\nAt least one of these derivatives has non-integer order.\n\n## Remarks\n\nLet $f$ be continuous.\nA continuous function $y$ solves the initial value problem if and only if it solves the Volterra integral equation\n$$y(t) = \\sum_{k=0}^{n-1} \\frac{y_k}{k!} (t - a)^k + \\frac{1}{\\Gamma(\\alpha)} \\int_a^t (t - s)^{\\alpha - 1} f(s, y(s)) \\, ds.$$\nHere $\\Gamma$ is the [[gamma function|0H5SYBSED9]], and the integral is the [[Riemann–Liouville integral|JTKCZ44NN3]] of order $\\alpha$.\nExample: for $0 \u003c \\alpha \u003c 1$, $a = 0$, $f \\equiv 1$ and $y(0) = 0$, the integral equation gives $y(t) = t^\\alpha / \\Gamma(\\alpha + 1)$, and indeed ${}^C D_0^\\alpha t^\\alpha = \\Gamma(\\alpha + 1)$.\nFor $0 \u003c \\alpha \u003c 1$, the equation ${}^C D_0^\\alpha y = \\lambda y$ with $y(0) = 1$ has the solution $y(t) = \\sum_{k \\ge 0} \\lambda^k t^{\\alpha k} / \\Gamma(\\alpha k + 1)$, a Mittag-Leffler function of $\\lambda t^\\alpha$; it reduces to $e^{\\lambda t}$ when $\\alpha = 1$.\nNon-example: the delay equation $y'(t) = y(t - 1)$ is not a fractional differential equation, since it contains only a derivative of integer order.\nEquations with the Riemann–Liouville derivative are also common; their natural initial conditions are on fractional integrals of $y$ at $a$ rather than on $y^{(k)}(a)$.\n",
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