  In this paper we identify, for small $t$ and a fixed $T>0,$ the order
$\alpha>0$ in the abstract fractional differential equation $$\partial^\alpha
u(t)=Au(t),$$ where the time-fractional derivative $\partial^\alpha$ is
understood in the sense of Caputo and Riemann-Liouville, $A$ is a closed
(possibly unbounded) linear operator in a Banach space $X,$ and $0<\alpha<1$ or
$1<\alpha<2.$
