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## 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, doi: 10.3934/dcdss.2020031
##### References:
 [1] A. Atangana, Non-validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties, Physica A: Statistical Mechanics and its Applications, 505 (2018), 688-706. doi: 10.1016/j.physa.2018.03.056. Google Scholar [2] A. Atangana and J. F. Gómez Aguilar, Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena, The European Physical Journal Plus, 133 (2018), 166. doi: 10.1140/epjp/i2018-12021-3. Google Scholar [3] A. Atangana, On the stability and convergence of the time-fractional variable order telegraph equation, Journal of Computational Physics, 293 (2015), 104-114. doi: 10.1016/j.jcp.2014.12.043. Google Scholar [4] A. Atangana and J. F. Botha, A generalized groundwater flow equation using the concept of variable-order derivative, Boundary Value Problems, 2013 (2013), 1-11. doi: 10.1186/1687-2770-2013-53. Google Scholar [5] A. Atangana and D. Baleanu, Numerical solution of a kind of fractional parabolic equations via two difference schemes, Abstr. Appl. Anal., 2013 (2013), Art. ID 828764, 8 pp. doi: 10.1155/2013/828764. Google Scholar [6] A. Atangana and D. Baleanu, New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 20 (2016), 763-769. Google Scholar [7] S. Bhalekar, V. Daftardar-Gejji, D. Baleanu and R. Magin, Generalized fractional order bloch equation with extended delay, International Journal of Bifurcation and Chaos, 22 (2012), 1250071. doi: 10.1142/S021812741250071X. Google Scholar [8] W. C. Chen, Nonlinear dynamics and chaos in a fractional-order financial system, Chaos, Solitons and Fractals, 36 (2008), 1305-1314. doi: 10.1016/j.chaos.2006.07.051. Google Scholar [9] C. Coimbra, Mechanics with variable-order differential operators, Ann. Phys., 12 (2003), 692-703. doi: 10.1002/andp.200310032. Google Scholar [10] G. R. J. Cooper and D. R. Cowan, Filtering using variable order vertical derivatives, Computers and Geosciences, 30 (2004), 455-459. Google Scholar [11] J. Dabas and A. Chauhan, Existence and uniqueness of mild solution for an impulsive neutral fractional integro-differential equation with infinite delay, Mathematical and Computer Modelling, 57 (2013), 754-763. doi: 10.1016/j.mcm.2012.09.001. Google Scholar [12] V. Daftardar-Gejji, Y. Sukale and S. Bhalekar, Solving fractional delay differential equations: A new approach, Fractional Calculus and Applied Analysis, 18 (2015), 400-418. doi: 10.1515/fca-2015-0026. Google Scholar [13] V. Daftardar-Gejji, Y. Sukale and S. Bhalekar, A new predictor-corrector method for fractional differential equations, Appl. Math. Comput., 244 (2014), 158-182. doi: 10.1016/j.amc.2014.06.097. Google Scholar [14] V. Daftardar-Gejji and H. Jafari, Analysis of a system of non autonomous fractional differential equations involving Caputo derivatives, J. Math. Anal. Appl., 328 (2007), 1026-1033. doi: 10.1016/j.jmaa.2006.06.007. Google Scholar [15] J. F. Gómez-Aguilar, Analytical and Numerical solutions of a nonlinear alcoholism model via variable-order fractional differential equations, Physica A: Statistical Mechanics and its Applications, 494 (2018), 52-75. doi: 10.1016/j.physa.2017.12.007. Google Scholar [16] M. Kalecki, A macroeconomic theory of business cycle, Econom, 3 (1935), 327-344. Google Scholar [17] M. M. Khader and A. S. Hendy, The approximate and exact solutions of the fractional-order delay differential equations using Legendre seudospectral Method, International Journal of Pure and Applied Mathematics, 74 (2012), 287-297. Google Scholar [18] J. A. Len and S. Tindel, Malliavin calculus for fractional delay equations, Journal of Theoretical Probability, 25 (2012), 854-889. doi: 10.1007/s10959-011-0349-4. Google Scholar [19] Y. Luchko, A New Fractional Calculus Model for the Two-dimensional Anomalous Diffusion and its Analysis, Mathematical Modelling of Natural Phenomena, 11 (2016), 1-17. doi: 10.1051/mmnp/201611301. Google Scholar [20] M. M. Meerschaert and C. Tadjeran, Finite difference approximations for fractional advection dispersion equations, J. Comput. Appl. Math., 172 (2004), 65-77. doi: 10.1016/j.cam.2004.01.033. Google Scholar [21] B. P. Moghaddam and Z. S. Mostaghim, A numerical method based on finite difference for solving fractional delay differential equations, Journal of Taibah University for Science, 7 (2013), 120-127. Google Scholar [22] B. P. Moghaddam and J. A. T. Machado, A stable three-level explicit spline finite difference scheme for a class of nonlinear time variable order fractional partial differential equations, Computers and Mathematics with Applications, 73 (2017), 1262-1269. doi: 10.1016/j.camwa.2016.07.010. Google Scholar [23] B. P. Moghaddam, S. Yaghoobi and J. T. Machado, An extended predictor-corrector algorithm for variable-order fractional delay differential equations, Journal of Computational and Nonlinear Dynamics, 11 (2016), 061001, 7pp. doi: 10.1115/1.4032574. Google Scholar [24] M. L. Morgado, N. J. Ford and P. M. Lima, Analysis and numerical methods for fractional differential equations with delay, Journal of Computational and Applied Mathematics, 252 (2013), 159-168. doi: 10.1016/j.cam.2012.06.034. Google Scholar [25] T. A. Nadzharyan, V. V. Sorokin, G. V. Stepanov, A. N. Bogolyubov and E. Y. Kramarenko, A fractional calculus approach to modeling rheological behavior of soft magnetic elastomers, Polymer, 92 (2016), 179-188. doi: 10.1016/j.polymer.2016.03.075. Google Scholar [26] K. M. Owolabi, Mathematical modelling and analysis of two-component system with Caputo fractional derivative order, Chaos, Solitons and Fractals, 103 (2017), 544-554. doi: 10.1016/j.chaos.2017.07.013. Google Scholar [27] K. M. Owolabi, Robust and adaptive techniques for numerical simulation of nonlinear partial differential equations of fractional order, Communications in Nonlinear Science and Numerical Simulation, 44 (2017), 304-317. doi: 10.1016/j.cnsns.2016.08.021. Google Scholar [28] K. M. Owolabi and A. Atangana, Numerical simulation of noninteger order system in subdiffusive, diffusive, and superdiffusive scenarios, Journal of Computational and Nonlinear Dynamics, 12 (2016), 031010, 7pp. doi: 10.1115/1.4035195. Google Scholar [29] M. A. Ramdan and M. N. Shrif, Numerical solution of system of first order delay differential equations using spline functions, International Journal of Computer Mathematics, 83 (2006), 925-937. doi: 10.1080/00207160601138889. Google Scholar [30] U. Saeed, Hermite wavelet method for fractional delay differential equations, Journal of Difference Equations, 2014 (2014), Article ID 359093, 8 pages. doi: 10.1155/2014/359093. Google Scholar [31] F. Shakeri and M. Dehghan, Solution of delay differential equations via a homotopy perturbation method, Mathematical and Computer Modelling, 48 (2008), 486-498. doi: 10.1016/j.mcm.2007.09.016. Google Scholar [32] J.-J. Shyu, S.-C. Pei and C.-H. Chan, An iterative method for the design of variable fractional-order FIR differintegrators, Signal Process, 89 (2009), 320-327. doi: 10.1016/j.sigpro.2008.09.009. Google Scholar [33] H. G. Sun, W. Chen, C. Li and Y. Q. Chen, Fractional differential models for anomalous diffusion, Physica A, 389 (2010), 2719-2724. doi: 10.1016/j.physa.2010.02.030. Google Scholar [34] H. G. Sun, W. Chen, H. Wei and Y. Q. Chen, A comparative study of constant-order and variable-order fractional models in characterizing memory property of systems, Eur. Phys. J. Spec. Top., 193 (2011), 185-192. doi: 10.1140/epjst/e2011-01390-6. Google Scholar [35] A. A. Tateishi, H. V. Ribeiro and E. K. Lenzi, The role of fractional time-derivative operators on anomalous diffusion, Frontiers in Physics, 5 (2017), 1-9. doi: 10.3389/fphy.2017.00052. Google Scholar [36] L. Tavernini, Continuous-Time Modeling and Simulation, Gordon and Breach, Amsterdam, 1996.Google Scholar [37] A. Tsoularis and J. Wallace, Analysis of logistic growth models, Mathematical Biosciences, 179 (2002), 21-55. doi: 10.1016/S0025-5564(02)00096-2. Google Scholar [38] S. Umarov and S. Steinberg, Variable order differential equations and diffusion with changing modes, Z. Anal. Anwend., 28 (2009), 431-450. doi: 10.4171/ZAA/1392. Google Scholar [39] D. Valrio and J. S. Da Costa, Variable-order fractional derivatives and their numerical approximations, Signal Processing, 91 (2011), 470-483. doi: 10.1016/j.sigpro.2010.04.006. Google Scholar [40] Z. B. Vosika, G. M. Lazovic, G. N. Misevic and J. B. Simic-Krstic, Fractional calculus model of electrical impedance applied to human skin, PloS one, 8 (2013), e59483. doi: 10.1371/journal.pone.0059483. Google Scholar [41] D. R. Will and C. T. Baker, DELSOL.-A numerical code for the solution of systems of delay-differential equations, Applied Numerical Mathematics, 9 (1992), 209-222. doi: 10.1016/0168-9274(92)90016-7. Google Scholar [42] W. Zhen, H. Xia and S. Guodong, Analysis of nonlinear dynamics and chaos in a fractional order financial system with time delay, Computers and Mathematics with Applications, 62 (2011), 1531-1539. doi: 10.1016/j.camwa.2011.04.057. Google Scholar

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##### References:
 [1] A. Atangana, Non-validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties, Physica A: Statistical Mechanics and its Applications, 505 (2018), 688-706. doi: 10.1016/j.physa.2018.03.056. Google Scholar [2] A. Atangana and J. F. Gómez Aguilar, Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena, The European Physical Journal Plus, 133 (2018), 166. doi: 10.1140/epjp/i2018-12021-3. Google Scholar [3] A. Atangana, On the stability and convergence of the time-fractional variable order telegraph equation, Journal of Computational Physics, 293 (2015), 104-114. doi: 10.1016/j.jcp.2014.12.043. Google Scholar [4] A. Atangana and J. F. Botha, A generalized groundwater flow equation using the concept of variable-order derivative, Boundary Value Problems, 2013 (2013), 1-11. doi: 10.1186/1687-2770-2013-53. Google Scholar [5] A. Atangana and D. Baleanu, Numerical solution of a kind of fractional parabolic equations via two difference schemes, Abstr. Appl. Anal., 2013 (2013), Art. ID 828764, 8 pp. doi: 10.1155/2013/828764. Google Scholar [6] A. Atangana and D. Baleanu, New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 20 (2016), 763-769. Google Scholar [7] S. Bhalekar, V. Daftardar-Gejji, D. Baleanu and R. Magin, Generalized fractional order bloch equation with extended delay, International Journal of Bifurcation and Chaos, 22 (2012), 1250071. doi: 10.1142/S021812741250071X. Google Scholar [8] W. C. Chen, Nonlinear dynamics and chaos in a fractional-order financial system, Chaos, Solitons and Fractals, 36 (2008), 1305-1314. doi: 10.1016/j.chaos.2006.07.051. Google Scholar [9] C. Coimbra, Mechanics with variable-order differential operators, Ann. Phys., 12 (2003), 692-703. doi: 10.1002/andp.200310032. Google Scholar [10] G. R. J. Cooper and D. R. Cowan, Filtering using variable order vertical derivatives, Computers and Geosciences, 30 (2004), 455-459. Google Scholar [11] J. Dabas and A. Chauhan, Existence and uniqueness of mild solution for an impulsive neutral fractional integro-differential equation with infinite delay, Mathematical and Computer Modelling, 57 (2013), 754-763. doi: 10.1016/j.mcm.2012.09.001. Google Scholar [12] V. Daftardar-Gejji, Y. Sukale and S. Bhalekar, Solving fractional delay differential equations: A new approach, Fractional Calculus and Applied Analysis, 18 (2015), 400-418. doi: 10.1515/fca-2015-0026. Google Scholar [13] V. Daftardar-Gejji, Y. Sukale and S. Bhalekar, A new predictor-corrector method for fractional differential equations, Appl. Math. Comput., 244 (2014), 158-182. doi: 10.1016/j.amc.2014.06.097. Google Scholar [14] V. Daftardar-Gejji and H. Jafari, Analysis of a system of non autonomous fractional differential equations involving Caputo derivatives, J. Math. Anal. Appl., 328 (2007), 1026-1033. doi: 10.1016/j.jmaa.2006.06.007. Google Scholar [15] J. F. Gómez-Aguilar, Analytical and Numerical solutions of a nonlinear alcoholism model via variable-order fractional differential equations, Physica A: Statistical Mechanics and its Applications, 494 (2018), 52-75. doi: 10.1016/j.physa.2017.12.007. Google Scholar [16] M. Kalecki, A macroeconomic theory of business cycle, Econom, 3 (1935), 327-344. Google Scholar [17] M. M. Khader and A. S. Hendy, The approximate and exact solutions of the fractional-order delay differential equations using Legendre seudospectral Method, International Journal of Pure and Applied Mathematics, 74 (2012), 287-297. Google Scholar [18] J. A. Len and S. Tindel, Malliavin calculus for fractional delay equations, Journal of Theoretical Probability, 25 (2012), 854-889. doi: 10.1007/s10959-011-0349-4. Google Scholar [19] Y. Luchko, A New Fractional Calculus Model for the Two-dimensional Anomalous Diffusion and its Analysis, Mathematical Modelling of Natural Phenomena, 11 (2016), 1-17. doi: 10.1051/mmnp/201611301. Google Scholar [20] M. M. Meerschaert and C. Tadjeran, Finite difference approximations for fractional advection dispersion equations, J. Comput. Appl. Math., 172 (2004), 65-77. doi: 10.1016/j.cam.2004.01.033. Google Scholar [21] B. P. Moghaddam and Z. S. Mostaghim, A numerical method based on finite difference for solving fractional delay differential equations, Journal of Taibah University for Science, 7 (2013), 120-127. Google Scholar [22] B. P. Moghaddam and J. A. T. Machado, A stable three-level explicit spline finite difference scheme for a class of nonlinear time variable order fractional partial differential equations, Computers and Mathematics with Applications, 73 (2017), 1262-1269. doi: 10.1016/j.camwa.2016.07.010. Google Scholar [23] B. P. Moghaddam, S. Yaghoobi and J. T. Machado, An extended predictor-corrector algorithm for variable-order fractional delay differential equations, Journal of Computational and Nonlinear Dynamics, 11 (2016), 061001, 7pp. doi: 10.1115/1.4032574. Google Scholar [24] M. L. Morgado, N. J. Ford and P. M. Lima, Analysis and numerical methods for fractional differential equations with delay, Journal of Computational and Applied Mathematics, 252 (2013), 159-168. doi: 10.1016/j.cam.2012.06.034. Google Scholar [25] T. A. Nadzharyan, V. V. Sorokin, G. V. Stepanov, A. N. Bogolyubov and E. Y. Kramarenko, A fractional calculus approach to modeling rheological behavior of soft magnetic elastomers, Polymer, 92 (2016), 179-188. doi: 10.1016/j.polymer.2016.03.075. Google Scholar [26] K. M. Owolabi, Mathematical modelling and analysis of two-component system with Caputo fractional derivative order, Chaos, Solitons and Fractals, 103 (2017), 544-554. doi: 10.1016/j.chaos.2017.07.013. Google Scholar [27] K. M. Owolabi, Robust and adaptive techniques for numerical simulation of nonlinear partial differential equations of fractional order, Communications in Nonlinear Science and Numerical Simulation, 44 (2017), 304-317. doi: 10.1016/j.cnsns.2016.08.021. Google Scholar [28] K. M. Owolabi and A. Atangana, Numerical simulation of noninteger order system in subdiffusive, diffusive, and superdiffusive scenarios, Journal of Computational and Nonlinear Dynamics, 12 (2016), 031010, 7pp. doi: 10.1115/1.4035195. Google Scholar [29] M. A. Ramdan and M. N. Shrif, Numerical solution of system of first order delay differential equations using spline functions, International Journal of Computer Mathematics, 83 (2006), 925-937. doi: 10.1080/00207160601138889. Google Scholar [30] U. Saeed, Hermite wavelet method for fractional delay differential equations, Journal of Difference Equations, 2014 (2014), Article ID 359093, 8 pages. doi: 10.1155/2014/359093. Google Scholar [31] F. Shakeri and M. Dehghan, Solution of delay differential equations via a homotopy perturbation method, Mathematical and Computer Modelling, 48 (2008), 486-498. doi: 10.1016/j.mcm.2007.09.016. Google Scholar [32] J.-J. Shyu, S.-C. Pei and C.-H. Chan, An iterative method for the design of variable fractional-order FIR differintegrators, Signal Process, 89 (2009), 320-327. doi: 10.1016/j.sigpro.2008.09.009. Google Scholar [33] H. G. Sun, W. Chen, C. Li and Y. Q. Chen, Fractional differential models for anomalous diffusion, Physica A, 389 (2010), 2719-2724. doi: 10.1016/j.physa.2010.02.030. Google Scholar [34] H. G. Sun, W. Chen, H. Wei and Y. Q. Chen, A comparative study of constant-order and variable-order fractional models in characterizing memory property of systems, Eur. Phys. J. Spec. Top., 193 (2011), 185-192. doi: 10.1140/epjst/e2011-01390-6. Google Scholar [35] A. A. Tateishi, H. V. Ribeiro and E. K. Lenzi, The role of fractional time-derivative operators on anomalous diffusion, Frontiers in Physics, 5 (2017), 1-9. doi: 10.3389/fphy.2017.00052. Google Scholar [36] L. Tavernini, Continuous-Time Modeling and Simulation, Gordon and Breach, Amsterdam, 1996.Google Scholar [37] A. Tsoularis and J. Wallace, Analysis of logistic growth models, Mathematical Biosciences, 179 (2002), 21-55. doi: 10.1016/S0025-5564(02)00096-2. Google Scholar [38] S. Umarov and S. Steinberg, Variable order differential equations and diffusion with changing modes, Z. Anal. Anwend., 28 (2009), 431-450. doi: 10.4171/ZAA/1392. Google Scholar [39] D. Valrio and J. S. Da Costa, Variable-order fractional derivatives and their numerical approximations, Signal Processing, 91 (2011), 470-483. doi: 10.1016/j.sigpro.2010.04.006. Google Scholar [40] Z. B. Vosika, G. M. Lazovic, G. N. Misevic and J. B. Simic-Krstic, Fractional calculus model of electrical impedance applied to human skin, PloS one, 8 (2013), e59483. doi: 10.1371/journal.pone.0059483. Google Scholar [41] D. R. Will and C. T. Baker, DELSOL.-A numerical code for the solution of systems of delay-differential equations, Applied Numerical Mathematics, 9 (1992), 209-222. doi: 10.1016/0168-9274(92)90016-7. Google Scholar [42] W. Zhen, H. Xia and S. Guodong, Analysis of nonlinear dynamics and chaos in a fractional order financial system with time delay, Computers and Mathematics with Applications, 62 (2011), 1531-1539. doi: 10.1016/j.camwa.2011.04.057. Google Scholar
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
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