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

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.