An Extended Predictor–Corrector Algorithm for Variable-Order Fractional Delay Differential Equations

Abstract

This article presents a numerical method based on the Adams–Bashforth–Moulton scheme to solve variable-order fractional delay differential equations (VFDDEs). In these equations, the variable-order (VO) fractional derivatives are described in the Caputo sense. The existence and uniqueness of the solutions are proved under Lipschitz condition. Numerical examples are presented showing the applicability and efficiency of the novel method.

Applications of variable-order fractional operators: a review

“…Chen et al [78] constructed a stable alternating directions implicit scheme for the two-dimensional VO percolation equation. The Adams-Bashforth-Moulton predictor-corrector (ABM-PC) scheme was used in [79,80] to simulate VO-FDEs with time delays. All the above-discussed schemes are convergent to first order, i.e.…”

Natural Transform Decomposition Method for Solving Fractional-Order Partial Differential Equations with Proportional Delay

“…Generally it is tough experience in order to find an exact result of a wide range of differential equations. Nevertheless, numerous effective techniques has been developed to find the approximate result of partial differential equations (PDE's) with proportional delay, such as functional constraint's method [5], iterated pseudo spectral method [6], reduce differential transform method (RDTM) for PDE's by Abazari and Ganji [7], homotopy perturbation method for time fractional PDE's by Sakar et al and F. Shakeri [4,8], variational iteration method (VIM) [9], predictor-corrector algorithm [10], Adams-Bashforth-Moulton algorithm [11], a numerical method for delayed differential equations in fractional order: based on the definition of GL [12], an extended predictor-corrector algorithm for differential equations in variable-order fractional delay [13], Optimal Periodic-Gain fractional delayed state feedback control for linear periodic time-delayed systems [14], spectral collocation methods [15], group analysis method for Burgers equation due to Tanthanuch [16], and the two-dimensional differential transform method the solution of specific class of PDE's [17].…”

A novel predictor-corrector scheme for solving variable-order fractional delay differential equations involving operators with Mittag-Leffler kernel

“…Since the equations described by the VO derivatives are highly complex, difficult to handle analytically, it is therefore advisable to investigate their solutions numerically. Possible numerical implementations of VO fractional derivatives are given in [5]- [23]. In this paper, we considered the fractional derivative with Mittag-Leffler kernel of type Liouville-Caputo and the Liouville-Caputo definition.…”

“…Fractional operators with variable-order. The Liouville-Caputo (C) fractional operator with variable-order α(t) is defined as [23]…”

“…Using ABC fractional derivative, the Figure 1a show the behavior of y(t) with an order α = 1; Figure 1b, shows the phase portrait y(t) vs. y(t − 2) of the system. In this paper, for the Liouville-Caputo fractional derivative with variable-order, we used the numerical scheme developed in [23]. Considering the same conditions above mentioned, the Figure 1c, show the behavior of y(t) and Figure 1d, shows the phase portrait y(t) vs. y(t − 2) of the system.…”