Owolabi, Kolade M.; Atangana, Abdon; Gómez-Aguilar, Jose Francisco Fractional Adams-Bashforth scheme with the Liouville-Caputo derivative and application to chaotic systems. (English) Zbl 1475.65076 Discrete Contin. Dyn. Syst., Ser. S 14, No. 7, 2455-2469 (2021). Summary: 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. Cited in 3 Documents MSC: 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 35R11 Fractional partial differential equations 34A08 Fractional ordinary differential equations 65L05 Numerical methods for initial value problems involving ordinary differential equations Keywords:fractional calculus; Adams-Bashforth scheme; Lagrange interpolation; chaotic systems; Liouville-Caputo derivative PDFBibTeX XMLCite \textit{K. M. Owolabi} et al., Discrete Contin. Dyn. Syst., Ser. S 14, No. 7, 2455--2469 (2021; Zbl 1475.65076) Full Text: DOI References: [1] A. Atangana; I. Koca, Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos Solitons Fractals, 89, 447-454 (2016) · Zbl 1360.34150 [2] 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. [3] A. Atangana and K. M. Owolabi, New numerical approach for fractional differential equations, Math. Model. Nat. Phenom., 13 (2018), Paper No. 3, 21 pp. · Zbl 1406.65045 [4] D. Baleanu, R. Caponetto and J. A. T. Machado, Challenges in fractional dynamics and control theory, J. Vib. Control, 22 (2016), 2151-2152. [5] H. M. Baskonus; T. Mekkaoui; Z. Hammouch; H. Bulut, Active control of a chaotic fractional order economic system, Entropy, 17, 5771-5783 (2015) [6] J. Cao; C. Li; Y. Chen, Compact difference method for solving the fractional reaction-subdiffusion equation with Neumann boundary value condition, Int. J. Comput. Math., 92, 167-180 (2015) · Zbl 1308.65140 [7] M. Caputo; M. Fabrizio, Applications of new time and spatial fractional derivatives with exponential kernels, Progress in Fractional Differentiation and Applications, 2, 1-11 (2016) [8] A. Coronel-Escamilla; J. F. Gómez-Aguilar; M. G. López-López; V. M. Alvarado-Martínez; G. V. Guerrero-Ramírez, Triple pendulum model involving fractional derivatives with different kernels, Chaos Solitons Fractals, 91, 248-261 (2016) · Zbl 1372.70049 [9] E. Demirci; N. Ozalp, A method for solving differential equations of fractional order, J. Comput. Appl. Math., 236, 2754-2762 (2012) · Zbl 1243.34003 [10] J. Deng; L. Ma, Existence and uniqueness of solutions of initial value problems for nonlinear fractional differential equations, Appl. Math. Lett., 23, 676-680 (2010) · Zbl 1201.34008 [11] K. Diethelm; N. J. Ford; A. D. Freed, Detailed error analysis for a fractional Adams method, Numer. Algorithms, 36, 31-52 (2004) · Zbl 1055.65098 [12] R. Du; W. R. Cao; and Z. Z. Sun, A compact difference scheme for the fractional diffusion-wave equation, Appl. Math. Model., 34, 2998-3007 (2010) · Zbl 1201.65154 [13] R. Garrappa, On some explicit Adams multistep methods for fractional differential equations, J. Comput. Appl. Math., 229, 392-399 (2009) · Zbl 1171.65098 [14] 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. · Zbl 1419.35208 [15] 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. · Zbl 1400.82237 [16] 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). · Zbl 1374.34296 [17] R. Gorenflo; E. A. Abdel-Rehim, Convergence of the Grünwald-Letnikov scheme for time-fractional diffusion, J. Comput. Appl. Math., 205, 871-881 (2007) · Zbl 1127.65100 [18] Z. Hammouch; T. Mekkaoui, Control of a new chaotic fractional-order system using Mittag-Leffler stability, Nonlinear Stud., 22, 565-577 (2015) · Zbl 1335.34097 [19] Z. Hammouch; T. Mekkaoui, Chaos synchronization of a fractional nonautonomous system, Nonauton. Dyn. Syst., 1, 61-71 (2014) · Zbl 1298.34093 [20] X. Hu; L. Zhang, Implicit compact difference schemes for the fractional cable equation, Appl. Math. Model., 36, 4027-4043 (2012) · Zbl 1252.74061 [21] A. Q. M. Khaliq; X. Liang; K. M. Furati, A fourth-order implicit-explicit scheme for the space fractional nonlinear Schrödinger equations, Numer. Algorithms, 75, 147-172 (2017) · Zbl 1365.65195 [22] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amersterdam, 2006. · Zbl 1092.45003 [23] V. Lakshmikantham; A. S. Vatsala, Theory of fractional differential inequalities and applications, Commun. Appl. Anal., 11, 395-402 (2007) · Zbl 1159.34006 [24] C. Li; F. Zeng, The finite difference methods for fractional ordinary differential equations, Numer. Funct. Anal. Optim., 34, 149-179 (2013) · Zbl 1267.65094 [25] X. Liang; A. Q. M. Khaliq; H. Bhatt; K. M. Furati, The locally extrapolated exponential splitting scheme for multi-dimensional nonlinear space-fractional Schrödinger equations, Numer. Algorithms, 76, 939-958 (2017) · Zbl 1380.65162 [26] K. M. Owolabi, Numerical solution of diffusive HBV model in a fractional medium, Springer Plus, 5 (2016), 1643. [27] K. M. Owolabi, Mathematical analysis and numerical simulation of patterns in fractional and classical reaction-diffusion systems, Chaos Solitons Fractals, 93, 89-98 (2016) · Zbl 1372.92091 [28] K. M. Owolabi; A. Atangana, Numerical approximation of nonlinear fractional parabolic differential equations with Caputo-Fabrizio derivative in Riemann-Liouville sense, Chaos Solitons Fractals, 99, 171-179 (2017) · Zbl 1422.65178 [29] K. M. Owolabi, Robust and adaptive techniques for numerical simulation of nonlinear partial differential equations of fractional order, Commun. Nonlinear Sci. Numer. Simul., 44, 304-317 (2017) · Zbl 1465.65108 [30] K. M. Owolabi, Mathematical modelling and analysis of two-component system with Caputo fractional derivative order, Chaos Solitons Fractals, 103, 544-554 (2017) · Zbl 1375.35257 [31] K. M. Owolabi; A. Atangana, Analysis and application of new fractional Adams-Bashforth scheme with Caputo-Fabrizio derivative, Chaos Solitons Fractals, 105, 111-119 (2017) · Zbl 1380.65120 [32] 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. · Zbl 1410.65323 [33] 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. [34] 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. [35] 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. [36] K. M. Owolabi; A. Atangana, Robustness of fractional difference schemes via the Caputo subdiffusion-reaction equations, Chaos Solitons Fractals, 111, 119-127 (2018) · Zbl 1395.65026 [37] K. M. Owolabi; Z. Hammouch, Spatiotemporal patterns in the Belousov-Zhabotinskii reaction systems with Atangana-Baleanu fractional order derivative, Phys. A, 523, 1072-1090 (2019) · Zbl 07563440 [38] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993. · Zbl 0818.26003 [39] J. Singh; D. Kumar; Z. Hammouch; A. Atangana, A fractional epidemiological model for computer viruses pertaining to a new fractional derivative, Appl. Math. Comput., 316, 504-515 (2018) · Zbl 1426.68015 [40] 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. · Zbl 07568257 [41] 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. [42] M. Yavuz; E. Bonyah, New approaches to the fractional dynamics of schistosomiasis disease model, Phys. A, 525, 373-393 (2019) · Zbl 07565786 [43] X. Zhao; Z.-Z. Sun, A box-type scheme for fractional sub-diffusion equation with Neumann boundary conditions, J. Comput. Phys., 230, 6061-6074 (2011) · Zbl 1227.65075 [44] A. T. Azar and S. Vaidyanathan, Advances in Chaos Theory and Intelligent Control, Springer, Switzerland, 2016. · Zbl 1350.93001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.