Higher order numerical methods for fractional delay differential equations

Abstract

In this paper, we present a new family of higher-order numerical methods for solving non-linear fractional delay differential equations (FDDEs) along with the error analysis. Further, we solve various non-trivial systems of FDDEs to illustrate their applicability and utility. By using the proposed numerical methods, computational time is reduced drastically. These methods take only 5 to 10 percent of the time required for other methods such as the fractional Adams method (FAM). Furthermore, these methods converge for very small values of fractional derivative while FAM and the new predictor-corrector method (NPCM) introduced by Daftardar-Gejji et al. [1] do not converge. The order of convergence of the proposed methods is \(r+\alpha \), where r is the order of fractional backward difference formulae and \(\alpha \) denotes the order of the fractional derivative. Thus these methods have a higher order of accuracy than FAM or NPCM.