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

Two families of novel second-order fractional numerical formulas and their applications to fractional differential equations