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

Abstract

Two families of second-order difference formulas called fractional BT- $$\theta $$ and fractional BN- $$\theta $$ for the fractional calculus are proposed by introducing a free parameter $$\theta $$ aiming to generalize the classic fractional BDF2, the fractional trapezoidal rule (FTR) and the second-order generalized Newton-Gregory formula (GNGF2). The error bounds for these two families of novel formulas are analysed indicating that fractional BT- $$\frac{1}{2}$$ (or FTR) is superior to other formulas for integral cases. At least two advantages can be observed for the introduction of $$\theta $$ : (i) The fractional BN- $$\theta $$ exhibits superconvergence for an appropriate $$\theta $$ which permits us to develop more robust third-order schemes for problems such as the diffusion-wave equation, compared with the fractional BDF3. (ii) The fractional BT- $$\theta $$ allows us to take the advantage of FTR that a sub-optimal error bound can be obtained (by taking $$\theta $$ close to $$\frac{1}{2}$$ ) when directly discretizing fractional derivatives. Further, some correction techniques are discussed to overcome the solution initial singularity in numerical tests. The numerical results confirm the correctness of the theoretical analysis and the efficiency of our scheme.