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Article Contents

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

• * Corresponding author: J. F. Gómez-Aguilar

The first author is supported by by CONACyT through the assignment doctoral fellowship

• In this work we present a numerical method based on the Adams-Bashforth-Moulton scheme to solve numerically fractional delay differential equations. We focus in the fractional derivative with Mittag-Leffler kernel of type Liouville-Caputo with variable-order and the Liouville-Caputo fractional derivative with variable-order. Numerical examples are presented to show the applicability and efficiency of this novel method.

Mathematics Subject Classification: Primary: 34A34, 65M12; Secondary: 26A33, 34A08, 65C20, 65P20.

 Citation:

• Figure 1.  Numerical solution of Eq. (26); using ABC derivative, in (a) we show the evolution of $y(t)$ when $\alpha = 1$, in (b) we obtain the phase diagram when $\alpha = 1$. Using Liouville-Caputo derivative, in (c) we show the evolution of $y(t)$ when $\alpha = 1$ and in (d) we obtain the phase diagram when $\alpha = 1$

Figure 2.  Numerical solution of Eq. (26); using ABC derivative, in (a) we show the evolution of $y(t)$ when $\alpha = 0.85$, in (b) we obtain the phase diagram when $\alpha = 0.85$. Using Liouville-Caputo derivative, in (c) we show the evolution of $y(t)$ when $\alpha = 0.85$ and in (d) we obtain the phase diagram when $\alpha = 0.85$

Figure 3.  Numerical solution of Eq. (27). In (a)-(c)-(e) we show the evolution of $y(t)$ using ABC derivative. In (b)-(d)-(f) we show the evolution of $y(t)$ using Liouville-Caputo derivative

Figure 4.  Numerical solution of Eq. (27). In (a)-(c)-(e) we show the phase diagram $y(t)$ vs. $y(t-2)$ using ABC derivative. In (b)-(d)-(f) we show phase diagram $y(t)$ vs. $y(t-2)$ using Liouville-Caputo derivative

Figure 5.  Numerical solution of Eq. (28); using ABC derivative, in (a)-(c) we show the evolution of $y(t)$ and the phase diagram $y(t)$ vs. $y(t-2)$, when $\alpha(t) = \dfrac{1-\cos(2t)}{3}$, respectively; using Liouville-Caputo derivative, in (b)-(d) we show the evolution of $y(t)$ and the phase diagram $y(t)$ vs. $y(t-2)$, when $\alpha(t) = \dfrac{1-\cos(2t)}{3}$, respectively

Figure 6.  Numerical solution of Eq. (29); using ABC derivative, in (a)-(c) we show the evolution of $y(t)$ and the phase diagram $y(t)$ vs. $y(t-2)$, when $\alpha(t) = \dfrac{1-\cos(2t)}{3}$, respectively; using Liouville-Caputo derivative, in (b)-(d) we show the evolution of $y(t)$ and the phase diagram $y(t)$ vs. $y(t-2)$, when $\alpha(t) = \dfrac{1-\cos(2t)}{3}$, respectively

Figures(6)