  In this article, we introduce two families of novel fractional
$\theta$-methods by constructing some new generating functions to discretize
the Riemann-Liouville fractional calculus operator $\mathit{I}^{\alpha}$ with a
second order convergence rate. A new fractional BT-$\theta$ method connects the
fractional BDF2 (when $\theta=0$) with fractional trapezoidal rule (when
$\theta=1/2$), and another novel fractional BN-$\theta$ method joins the
fractional BDF2 (when $\theta=0$) with the second order fractional
Newton-Gregory formula (when $\theta=1/2$). To deal with the initial
singularity, correction terms are added to achieve an optimal convergence
order. In addition, stability regions of different $\theta$-methods when
applied to the Abel equations of the second kind are depicted, which
demonstrate the fact that the fractional $\theta$-methods are
A($\vartheta$)-stable. Finally, numerical experiments are implemented to verify
our theoretical result on the convergence analysis.
