Caputo fractional derivative

Definition

Let $a < b$ be real numbers and let $\alpha > 0$ be a real number that is not an integer. Let $n = \ceil{\alpha}$. Let $f \colon [a, b] \to \R$ have $n - 1$ continuous derivatives such that $f^{(n-1)}$ is absolutely continuous, so that $f^{(n)}$ exists almost everywhere and is integrable. The Caputo fractional derivative of order $\alpha$ of $f$ is $${}^{C}D_a^\alpha f(x) = \frac{1}{\Gamma(n - \alpha)} \int_a^x (x - t)^{n - \alpha - 1} f^{(n)}(t) \, dt, \qquad a < x \le b,$$ that is, the Riemann–Liouville integral $I_a^{n - \alpha}$ applied to the higher derivative $f^{(n)}$. Here $\Gamma$ is the gamma function. For an integer $\alpha = n$, one sets ${}^{C}D_a^n f = f^{(n)}$.