On FBDF5 Method for Delay Differential Equations of Fractional Order with Periodic and Anti-Periodic Conditions

Abstract

In this paper, we study the fractional backward differential formula (FBDF) for the numerical solution of fractional delay differential equations (FDDEs) of the following form: \(\lambda _n {}_0^C D_t^{\alpha _n } y(t - \tau ) + \lambda _{n - 1} {}_0^C D_t^{\alpha _{n - 1} } y(t - \tau ) + \cdots + \lambda _1 {}_0^C D_t^{\alpha _1 } y(t - \tau ) + \lambda _{n + 1} y(t) = f(t), t \in [0,T]\), where \( \lambda _i \in \)\(\mathbb {R}\,(i = 1,\ldots ,n + 1)\,,\,\lambda _{n + 1} \ne 0,\,\, 0 \leqslant \alpha _1 0,\) in Caputo sense. We find the Green’s functions for this equation corresponding to periodic/anti-periodic conditions in term of the Mittag-Leffler type. Our investigation is focused on stability properties of the numerical methods and we determine stability regions for the FDDEs. Finally, some numerical examples are given to show the effectiveness of the numerical method and the results are in excellent agreement with the theoretical analysis