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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:
[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. BhalekarV. Daftardar-GejjiD. 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-GejjiY. 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-GejjiY. 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. MorgadoN. 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. NadzharyanV. V. SorokinG. V. StepanovA. 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. ShyuS.-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. SunW. ChenC. 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. SunW. ChenH. 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. TateishiH. 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. VosikaG. M. LazovicG. 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. ZhenH. 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

show all references

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. BhalekarV. Daftardar-GejjiD. 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-GejjiY. 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-GejjiY. 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. MorgadoN. 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. NadzharyanV. V. SorokinG. V. StepanovA. 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. ShyuS.-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. SunW. ChenC. 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. SunW. ChenH. 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. TateishiH. 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. VosikaG. M. LazovicG. 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. ZhenH. 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

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
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