CaputoFractionalDerivative

The Caputo fractional derivative, expressed without the explicit use of $n$ and using $\text{frac}(\alpha)$ for the fractional part of $\alpha$, is given by:

$$ D^\alpha f(t) = \frac{1}{\Gamma(1 - \text{frac}(\alpha))} \int_0^t \frac{f^{(\lfloor \alpha \rfloor + 1)}(\tau)}{(t-\tau)^{\text{frac}(\alpha)}} d\tau $$

In this expression, $\lfloor \alpha \rfloor$ represents the integer part of $\alpha$, and $\text{frac}(\alpha)$ is the fractional part of $\alpha$. The function $f^{(\lfloor \alpha \rfloor + 1)}(\tau)$ indicates the derivative of $f$ of order $\lfloor \alpha \rfloor + 1$, the smallest integer strictly greater than $\alpha$.