Fractional differential equation

Remarks

Let $f$ be continuous. A continuous function $y$ solves the initial value problem if and only if it solves the Volterra integral equation $$y(t) = \sum_{k=0}^{n-1} \frac{y_k}{k!} (t - a)^k + \frac{1}{\Gamma(\alpha)} \int_a^t (t - s)^{\alpha - 1} f(s, y(s)) \, ds.$$ Here $\Gamma$ is the gamma function, and the integral is the Riemann–Liouville integral of order $\alpha$. Example: for $0 < \alpha < 1$, $a = 0$, $f \equiv 1$ and $y(0) = 0$, the integral equation gives $y(t) = t^\alpha / \Gamma(\alpha + 1)$, and indeed ${}^C D_0^\alpha t^\alpha = \Gamma(\alpha + 1)$. For $0 < \alpha < 1$, 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 $\alpha = 1$. Non-example: the delay equation $y'(t) = y(t - 1)$ is not a fractional differential equation, since it contains only a derivative of integer order. Equations with the Riemann–Liouville derivative are also common; their natural initial conditions are on fractional integrals of $y$ at $a$ rather than on $y^{(k)}(a)$.