# American Institute of Mathematical Sciences

• Previous Article
Memorized relaxation with singular and non-singular memory kernels for basic relaxation of dielectric vis-à-vis Curie-von Schweidler & Kohlrausch relaxation laws
• DCDS-S Home
• This Issue
• Next Article
Fractional calculus and applications of family of extended generalized Gauss hypergeometric functions
March  2020, 13(3): 561-574. doi: 10.3934/dcdss.2020031

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

 1 Tecnológico Nacional de México/CENIDET, Interior Internado Palmira S/N, Col. Palmira, C.P. 62490, Cuernavaca Morelos, México 2 CONACyT-Tecnológico Nacional de México/CENIDET, Interior Internado Palmira S/N, Col. Palmira, C.P. 62490, Cuernavaca Morelos, México

* Corresponding author: J. F. Gómez-Aguilar

Received  April 2018 Revised  May 2018 Published  March 2019

Fund Project: 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.

Citation: Antonio Coronel-Escamilla, José Francisco Gómez-Aguilar. A novel predictor-corrector scheme for solving variable-order fractional delay differential equations involving operators with Mittag-Leffler kernel. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 561-574. doi: 10.3934/dcdss.2020031
##### References:

show all references

##### References:
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 $\alpha = 1$
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 $\alpha = 0.85$
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
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
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
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
 [1] Bahaaeldin Abdalla, Thabet Abdeljawad. Oscillation criteria for kernel function dependent fractional dynamic equations. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020443 [2] Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $q$-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440 [3] Stefan Ruschel, Serhiy Yanchuk. The Spectrum of delay differential equations with multiple hierarchical large delays. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 151-175. doi: 10.3934/dcdss.2020321 [4] Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468 [5] Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137 [6] Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319 [7] Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020318 [8] Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264 [9] Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020050 [10] Alessandro Carbotti, Giovanni E. Comi. A note on Riemann-Liouville fractional Sobolev spaces. Communications on Pure & Applied Analysis, 2021, 20 (1) : 17-54. doi: 10.3934/cpaa.2020255 [11] Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047 [12] Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321 [13] Yahia Zare Mehrjerdi. A new methodology for solving bi-criterion fractional stochastic programming. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020054 [14] Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432 [15] Li-Bin Liu, Ying Liang, Jian Zhang, Xiaobing Bao. A robust adaptive grid method for singularly perturbed Burger-Huxley equations. Electronic Research Archive, 2020, 28 (4) : 1439-1457. doi: 10.3934/era.2020076 [16] Maoding Zhen, Binlin Zhang, Vicenţiu D. Rădulescu. Normalized solutions for nonlinear coupled fractional systems: Low and high perturbations in the attractive case. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020379 [17] S. Sadeghi, H. Jafari, S. Nemati. Solving fractional Advection-diffusion equation using Genocchi operational matrix based on Atangana-Baleanu derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020435 [18] Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $p$-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020445 [19] Omid Nikan, Seyedeh Mahboubeh Molavi-Arabshai, Hossein Jafari. Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020466 [20] Xuefeng Zhang, Yingbo Zhang. Fault-tolerant control against actuator failures for uncertain singular fractional order systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 1-12. doi: 10.3934/naco.2020011

2019 Impact Factor: 1.233