Caputo fractional derivative

Summary

The Caputo fractional derivative of order $\alpha$ first differentiates a function $n = \ceil{\alpha}$ times and then applies the Riemann–Liouville integral of order $n - \alpha$. This order of operations is the reverse of the Riemann–Liouville derivative. Its main advantage is that the derivative of a constant is zero, and fractional differential equations with it take ordinary initial values $f(a), f'(a), \dots$. It is a standard tool of fractional calculus in applied models.