July  2021, 14(7): 2455-2469. doi: 10.3934/dcdss.2021060

Fractional Adams-Bashforth scheme with the Liouville-Caputo derivative and application to chaotic systems

1. 

Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, Bloemfontein 9300, South Africa

2. 

Department of Mathematical Sciences, Federal University of Technology, PMB 704, Akure, Ondo State, Nigeria

3. 

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

* Corresponding author: kmowolabi@futa.edu.ng (Kolade M. Owolabi)

Received  June 2019 Revised  July 2019 Published  July 2021 Early access  May 2021

Fund Project: A. Atangana would like to thank NRF for their support, J. F. Gómez Aguilar acknowledges the support provided by CONACyT: Cátedras CONACyT para jóvenes investigadores 2014 and SNI-CONACyT

A recently proposed numerical scheme for solving nonlinear ordinary differential equations with integer and non-integer Liouville-Caputo derivative is applied to three systems with chaotic solutions. The Adams-Bashforth scheme involving Lagrange interpolation and the fundamental theorem of fractional calculus. We provide the existence and uniqueness of solutions, also the convergence result is stated. The proposed method is applied to several examples that are shown to have unique solutions. The scheme converges to the classical Adams-Bashforth method when the fractional orders of the derivatives converge to integers.

Citation: Kolade M. Owolabi, Abdon Atangana, Jose Francisco Gómez-Aguilar. Fractional Adams-Bashforth scheme with the Liouville-Caputo derivative and application to chaotic systems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2455-2469. doi: 10.3934/dcdss.2021060
References:
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A. Atangana and I. Koca, Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos Solitons Fractals, 89 (2016), 447-454.  doi: 10.1016/j.chaos.2016.02.012.  Google Scholar

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D. Baleanu, R. Caponetto and J. A. T. Machado, Challenges in fractional dynamics and control theory, J. Vib. Control, 22 (2016), 2151–2152. doi: 10.1177/1077546315609262.  Google Scholar

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H. M. BaskonusT. MekkaouiZ. Hammouch and H. Bulut, Active control of a chaotic fractional order economic system, Entropy, 17 (2015), 5771-5783.   Google Scholar

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J. CaoC. Li and Y. Chen, Compact difference method for solving the fractional reaction-subdiffusion equation with Neumann boundary value condition, Int. J. Comput. Math., 92 (2015), 167-180.  doi: 10.1080/00207160.2014.887702.  Google Scholar

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A. Coronel-EscamillaJ. F. Gómez-AguilarM. G. López-LópezV. M. Alvarado-Martínez and G. V. Guerrero-Ramírez, Triple pendulum model involving fractional derivatives with different kernels, Chaos Solitons Fractals, 91 (2016), 248-261.  doi: 10.1016/j.chaos.2016.06.007.  Google Scholar

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K. M. Owolabi, Mathematical analysis and numerical simulation of patterns in fractional and classical reaction-diffusion systems, Chaos Solitons Fractals, 93 (2016), 89-98.  doi: 10.1016/j.chaos.2016.10.005.  Google Scholar

[31]

K. M. Owolabi and A. Atangana, Numerical approximation of nonlinear fractional parabolic differential equations with Caputo-Fabrizio derivative in Riemann-Liouville sense, Chaos Solitons Fractals, 99 (2017), 171-179.  doi: 10.1016/j.chaos.2017.04.008.  Google Scholar

[32]

K. M. Owolabi, Robust and adaptive techniques for numerical simulation of nonlinear partial differential equations of fractional order, Commun. Nonlinear Sci. Numer. Simul., 44 (2017), 304-317.  doi: 10.1016/j.cnsns.2016.08.021.  Google Scholar

[33]

K. M. Owolabi, Mathematical modelling and analysis of two-component system with Caputo fractional derivative order, Chaos Solitons Fractals, 103 (2017), 544-554.  doi: 10.1016/j.chaos.2017.07.013.  Google Scholar

[34]

K. M. Owolabi and A. Atangana, Analysis and application of new fractional Adams-Bashforth scheme with Caputo-Fabrizio derivative, Chaos Solitons Fractals, 105 (2017), 111-119.  doi: 10.1016/j.chaos.2017.10.020.  Google Scholar

[35]

K. M. Owolabi, Numerical approach to fractional blow-up equations with Atangana-Baleanu derivative in Riemann-Liouville sense, Math. Model. Nat. Phenom., 13 (2018), Paper No. 7, 17 pp. doi: 10.1051/mmnp/2018006.  Google Scholar

[36]

K. M. Owolabi and A. Atangana, Modelling and formation of spatiotemporal patterns of fractional predation system in subdiffusion and superdiffusion scenarios, The European physical Journal Plus, 133 (2018), Article number: 43. doi: 10.1140/epjp/i2018-11886-2.  Google Scholar

[37]

K. M. Owolabi, Modelling and simulation of a dynamical system with the Atangana-Baleanu fractional derivative, The European physical Journal Plus, 133 (2018), Article number: 15. doi: 10.1140/epjp/i2018-11863-9.  Google Scholar

[38]

K. M. Owolabi, Efficient numerical simulation of non-integer-order space-fractional reaction-diffusion equation via the Riemann-Liouville operator, The European Physical Journal Plus, 133 (2018), Article number: 98. doi: 10.1140/epjp/i2018-11951-x.  Google Scholar

[39]

K. M. Owolabi and A. Atangana, Robustness of fractional difference schemes via the Caputo subdiffusion-reaction equations, Chaos Solitons Fractals, 111 (2018), 119-127.  doi: 10.1016/j.chaos.2018.04.019.  Google Scholar

[40]

K. M. Owolabi and Z. Hammouch, Spatiotemporal patterns in the Belousov-Zhabotinskii reaction systems with Atangana-Baleanu fractional order derivative, Phys. A, 523 (2019), 1072-1090.  doi: 10.1016/j.physa.2019.04.017.  Google Scholar

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S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.  Google Scholar

[43]

J. SinghD. KumarZ. Hammouch and A. Atangana, A fractional epidemiological model for computer viruses pertaining to a new fractional derivative, Appl. Math. Comput., 316 (2018), 504-515.  doi: 10.1016/j.amc.2017.08.048.  Google Scholar

[44]

T. A. Sulaimana, M. Yavuz, H. Bulut and H. M. Baskonus, Investigation of the fractional coupled viscous Burgers-equation involving Mittag-Leffler kernel, Phys. A, 527 (2019), 121126, 20 pp. doi: 10.1016/j.physa.2019.121126.  Google Scholar

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M. Yavuz, N. Ozdemir and H. M. Baskonus, Solutions of partial differential equations using the fractional operator involving Mittag-Leffler kernel, The European Physical Journal Plus, 133, (2018), Article number: 215. doi: 10.1140/epjp/i2018-12051-9.  Google Scholar

[46]

M. Yavuz and E. Bonyah, New approaches to the fractional dynamics of schistosomiasis disease model, Phys. A, 525 (2019), 373-393.  doi: 10.1016/j.physa.2019.03.069.  Google Scholar

[47]

X. Zhao and Z.-Z. Sun, A box-type scheme for fractional sub-diffusion equation with Neumann boundary conditions, J. Comput. Phys., 230 (2011), 6061-6074.  doi: 10.1016/j.jcp.2011.04.013.  Google Scholar

[48]

A. T. Azar and S. Vaidyanathan, Advances in Chaos Theory and Intelligent Control, Springer, Switzerland, 2016.  Google Scholar

show all references

References:
[1] A. Atangana, Derivative with a New Parameter: Theory, Methods and Applications, Academic Press, New York, 2016.  doi: 10.1016/B978-0-08-100644-3.00001-5.  Google Scholar
[2] A. Atangana, Fractional Operators With Constant and Variable Order with Application to Geo-Hydrology, Academic Press, London, 2018.   Google Scholar
[3]

A. Atangana and I. Koca, Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos Solitons Fractals, 89 (2016), 447-454.  doi: 10.1016/j.chaos.2016.02.012.  Google Scholar

[4]

A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 20 (2016), 763–769. doi: 10.2298/TSCI160111018A.  Google Scholar

[5]

A. Atangana and K. M. Owolabi, New numerical approach for fractional differential equations, Math. Model. Nat. Phenom., 13 (2018), Paper No. 3, 21 pp. doi: 10.1051/mmnp/2018010.  Google Scholar

[6]

D. Baleanu, R. Caponetto and J. A. T. Machado, Challenges in fractional dynamics and control theory, J. Vib. Control, 22 (2016), 2151–2152. doi: 10.1177/1077546315609262.  Google Scholar

[7]

H. M. BaskonusT. MekkaouiZ. Hammouch and H. Bulut, Active control of a chaotic fractional order economic system, Entropy, 17 (2015), 5771-5783.   Google Scholar

[8]

J. CaoC. Li and Y. Chen, Compact difference method for solving the fractional reaction-subdiffusion equation with Neumann boundary value condition, Int. J. Comput. Math., 92 (2015), 167-180.  doi: 10.1080/00207160.2014.887702.  Google Scholar

[9]

M. Caputo and M. Fabrizio, Applications of new time and spatial fractional derivatives with exponential kernels, Progress in Fractional Differentiation and Applications, 2 (2016), 1-11.  doi: 10.18576/pfda/020101.  Google Scholar

[10]

A. Coronel-EscamillaJ. F. Gómez-AguilarM. G. López-LópezV. M. Alvarado-Martínez and G. V. Guerrero-Ramírez, Triple pendulum model involving fractional derivatives with different kernels, Chaos Solitons Fractals, 91 (2016), 248-261.  doi: 10.1016/j.chaos.2016.06.007.  Google Scholar

[11]

E. Demirci and N. Ozalp, A method for solving differential equations of fractional order, J. Comput. Appl. Math., 236 (2012), 2754-2762.  doi: 10.1016/j.cam.2012.01.005.  Google Scholar

[12]

J. Deng and L. Ma, Existence and uniqueness of solutions of initial value problems for nonlinear fractional differential equations, Appl. Math. Lett., 23 (2010), 676-680.  doi: 10.1016/j.aml.2010.02.007.  Google Scholar

[13]

K. DiethelmN. J. Ford and A. D. Freed, Detailed error analysis for a fractional Adams method, Numer. Algorithms, 36 (2004), 31-52.  doi: 10.1023/B:NUMA.0000027736.85078.be.  Google Scholar

[14]

R. DuW. R. Cao and and Z. Z. Sun, A compact difference scheme for the fractional diffusion-wave equation, Appl. Math. Model., 34 (2010), 2998-3007.  doi: 10.1016/j.apm.2010.01.008.  Google Scholar

[15]

R. Garrappa, On some explicit Adams multistep methods for fractional differential equations, J. Comput. Appl. Math., 229 (2009), 392-399.  doi: 10.1016/j.cam.2008.04.004.  Google Scholar

[16]

J. F. Gómez-Aguilar, L. Torres, H. Yépez-Martínez, D. Baleanu, J. M. Reyes and I. O. Sosa, Fractional Liénard type model of a pipeline within the fractional derivative without singular kernel, Adv. Difference Equ., 2016 (2016), Paper No. 173, 13 pp. doi: 10.1186/s13662-016-0908-1.  Google Scholar

[17]

J. F. Gómez-Aguilar, M. G. López-López, V. M. Alvarado-Martínez, J. Reyes-Reyes and M. Adam-Medina, Modeling diffusive transport with a fractional derivative without singular kernel, Phys. A, 447 (2016), 467–481. doi: 10.1016/j.physa.2015.12.066.  Google Scholar

[18]

J. F. Gómez-Aguilar and Abdon Atangana, New insight in fractional differentiation: Power, exponential decay and Mittag-Leffler laws and applications, The European Physical Journal Plus, 132 (2017). doi: 10.1140/epjp/i2017-11293-3.  Google Scholar

[19]

R. Gorenflo and E. A. Abdel-Rehim, Convergence of the Grünwald-Letnikov scheme for time-fractional diffusion, J. Comput. Appl. Math., 205 (2007), 871-881.  doi: 10.1016/j.cam.2005.12.043.  Google Scholar

[20]

Z. Hammouch and T. Mekkaoui, Control of a new chaotic fractional-order system using Mittag-Leffler stability, Nonlinear Stud., 22 (2015), 565-577.   Google Scholar

[21]

Z. Hammouch and T. Mekkaoui, Chaos synchronization of a fractional nonautonomous system, Nonauton. Dyn. Syst., 1 (2014), 61-71.  doi: 10.2478/msds-2014-0001.  Google Scholar

[22]

X. Hu and L. Zhang, Implicit compact difference schemes for the fractional cable equation, Appl. Math. Model., 36 (2012), 4027-4043.  doi: 10.1016/j.apm.2011.11.027.  Google Scholar

[23]

A. Q. M. KhaliqX. Liang and K. M. Furati, A fourth-order implicit-explicit scheme for the space fractional nonlinear Schrödinger equations, Numer. Algorithms, 75 (2017), 147-172.  doi: 10.1007/s11075-016-0200-1.  Google Scholar

[24]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amersterdam, 2006.  Google Scholar

[25]

V. Lakshmikantham and A. S. Vatsala, Theory of fractional differential inequalities and applications, Commun. Appl. Anal., 11 (2007), 395-402.   Google Scholar

[26]

C. Li and F. Zeng, The finite difference methods for fractional ordinary differential equations, Numer. Funct. Anal. Optim., 34 (2013), 149-179.  doi: 10.1080/01630563.2012.706673.  Google Scholar

[27] C. Li and F. Zeng, Numerical Methods for Fractional Calculus, CRC Press, Taylor and Francis Group, London, 2015.   Google Scholar
[28]

X. LiangA. Q. M. KhaliqH. Bhatt and K. M. Furati, The locally extrapolated exponential splitting scheme for multi-dimensional nonlinear space-fractional Schrödinger equations, Numer. Algorithms, 76 (2017), 939-958.  doi: 10.1007/s11075-017-0291-3.  Google Scholar

[29]

K. M. Owolabi, Numerical solution of diffusive HBV model in a fractional medium, Springer Plus, 5 (2016), 1643. doi: 10.1186/s40064-016-3295-x.  Google Scholar

[30]

K. M. Owolabi, Mathematical analysis and numerical simulation of patterns in fractional and classical reaction-diffusion systems, Chaos Solitons Fractals, 93 (2016), 89-98.  doi: 10.1016/j.chaos.2016.10.005.  Google Scholar

[31]

K. M. Owolabi and A. Atangana, Numerical approximation of nonlinear fractional parabolic differential equations with Caputo-Fabrizio derivative in Riemann-Liouville sense, Chaos Solitons Fractals, 99 (2017), 171-179.  doi: 10.1016/j.chaos.2017.04.008.  Google Scholar

[32]

K. M. Owolabi, Robust and adaptive techniques for numerical simulation of nonlinear partial differential equations of fractional order, Commun. Nonlinear Sci. Numer. Simul., 44 (2017), 304-317.  doi: 10.1016/j.cnsns.2016.08.021.  Google Scholar

[33]

K. M. Owolabi, Mathematical modelling and analysis of two-component system with Caputo fractional derivative order, Chaos Solitons Fractals, 103 (2017), 544-554.  doi: 10.1016/j.chaos.2017.07.013.  Google Scholar

[34]

K. M. Owolabi and A. Atangana, Analysis and application of new fractional Adams-Bashforth scheme with Caputo-Fabrizio derivative, Chaos Solitons Fractals, 105 (2017), 111-119.  doi: 10.1016/j.chaos.2017.10.020.  Google Scholar

[35]

K. M. Owolabi, Numerical approach to fractional blow-up equations with Atangana-Baleanu derivative in Riemann-Liouville sense, Math. Model. Nat. Phenom., 13 (2018), Paper No. 7, 17 pp. doi: 10.1051/mmnp/2018006.  Google Scholar

[36]

K. M. Owolabi and A. Atangana, Modelling and formation of spatiotemporal patterns of fractional predation system in subdiffusion and superdiffusion scenarios, The European physical Journal Plus, 133 (2018), Article number: 43. doi: 10.1140/epjp/i2018-11886-2.  Google Scholar

[37]

K. M. Owolabi, Modelling and simulation of a dynamical system with the Atangana-Baleanu fractional derivative, The European physical Journal Plus, 133 (2018), Article number: 15. doi: 10.1140/epjp/i2018-11863-9.  Google Scholar

[38]

K. M. Owolabi, Efficient numerical simulation of non-integer-order space-fractional reaction-diffusion equation via the Riemann-Liouville operator, The European Physical Journal Plus, 133 (2018), Article number: 98. doi: 10.1140/epjp/i2018-11951-x.  Google Scholar

[39]

K. M. Owolabi and A. Atangana, Robustness of fractional difference schemes via the Caputo subdiffusion-reaction equations, Chaos Solitons Fractals, 111 (2018), 119-127.  doi: 10.1016/j.chaos.2018.04.019.  Google Scholar

[40]

K. M. Owolabi and Z. Hammouch, Spatiotemporal patterns in the Belousov-Zhabotinskii reaction systems with Atangana-Baleanu fractional order derivative, Phys. A, 523 (2019), 1072-1090.  doi: 10.1016/j.physa.2019.04.017.  Google Scholar

[41] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.   Google Scholar
[42]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.  Google Scholar

[43]

J. SinghD. KumarZ. Hammouch and A. Atangana, A fractional epidemiological model for computer viruses pertaining to a new fractional derivative, Appl. Math. Comput., 316 (2018), 504-515.  doi: 10.1016/j.amc.2017.08.048.  Google Scholar

[44]

T. A. Sulaimana, M. Yavuz, H. Bulut and H. M. Baskonus, Investigation of the fractional coupled viscous Burgers-equation involving Mittag-Leffler kernel, Phys. A, 527 (2019), 121126, 20 pp. doi: 10.1016/j.physa.2019.121126.  Google Scholar

[45]

M. Yavuz, N. Ozdemir and H. M. Baskonus, Solutions of partial differential equations using the fractional operator involving Mittag-Leffler kernel, The European Physical Journal Plus, 133, (2018), Article number: 215. doi: 10.1140/epjp/i2018-12051-9.  Google Scholar

[46]

M. Yavuz and E. Bonyah, New approaches to the fractional dynamics of schistosomiasis disease model, Phys. A, 525 (2019), 373-393.  doi: 10.1016/j.physa.2019.03.069.  Google Scholar

[47]

X. Zhao and Z.-Z. Sun, A box-type scheme for fractional sub-diffusion equation with Neumann boundary conditions, J. Comput. Phys., 230 (2011), 6061-6074.  doi: 10.1016/j.jcp.2011.04.013.  Google Scholar

[48]

A. T. Azar and S. Vaidyanathan, Advances in Chaos Theory and Intelligent Control, Springer, Switzerland, 2016.  Google Scholar

Figure 1.  Numerical simulation for Eq. (31). In (a) depicts the classical chaotic attractor and (b-d) shows the classical phase portrait of system (31). In (e) depicts the strange chaotic attractor and (f-h) shows the phase portrait of system (31) for $ p = 0.9 $
Figure 2.  Numerical simulation for Eq. (32). In (a) depicts the classical macroeconomic model with foreign capital investments and (b-d) shows the classical phase portrait of system (32). In (e) depicts the strange chaotic attractor and (f-h) shows the phase portrait of system (32) for $ p = 0.95 $
Figure 3.  Numerical simulation for Eq. (33). In (a)-(c) depicts the time series and the phase portrait of system (33), respectively. In (d)-(f) depicts the time series and the phase portrait of system (33) for $ p = 0.95 $, respectively
Figure 4.  Two and three dimensional projections for the fractional jerk system for $ p = 0.93 $
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