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