New approach for the chaotic dynamical systems involving Caputo-Prabhakar fractional derivative using Adams-Bashforth scheme

References

• K. Ait Touchent, Z. Hammouch, T. Mekkaoui, and F.B.M. Belgacem, Implementation and convergence analysis of homotopyperturbation coupled with sumudu transform to construct solutions oflocal-fractional PDEs, Fractal. Fract 2(3) (2018), pp. 22.
   Web of Science ®Google Scholar
• H. Alrabaiah, M. Arfan, K. Shah, I. Mahariq, and A. Ullah, A comparative study of spreading of novel corona virus disease by ussing fractional order modified SEIR model, Alex. Eng. J 1 (2021), pp. 573–585.
   Google Scholar
• J. An, E. Van Hese, and M. Baes, Phase-space consistency of stellar dynamical models determined by separable augmented densities, Mon. Not. R. Astron. Soc 422(1) (2012), pp. 652–664.
   Web of Science ®Google Scholar
• A.T. Azar and S. Vaidyanathan, Advances in Chaos Theory and Intelligent Control, Springer, Berlin, Germany, 2016.
   Google Scholar
• E. Balcl, I. Öztürk, and S. Kartal, Dynamical behaviour of fractional order tumor model with Caputo and conformable fractional derivative, Chaos. Solitons. Fractals 123 (2019), pp. 43–51.
   Web of Science ®Google Scholar
• C. Cattani, A review on harmonic wavelets and their fractional extension, J. Adv. Eng. Comput. 2(4) (2018), pp. 224–238.
   Google Scholar
• C. Cattani and G. Pierro, On the fractal geometry of DNA by the binary image analysis, Bull. Math. Biol. 75 (2013), pp. 1544–1570.
   PubMed Web of Science ®Google Scholar
• C. Cattani, J.J. Rushchitsky, and S.V. Sinchilo, Physical constants for one type of nonlinearly elastic fibrous micro-and nanocomposites with hard and soft nonlinearities, Int. Appl. Mech 41(12) (2005), pp. 1368–1377.
   Web of Science ®Google Scholar
• M.H. Cherif, K. Belghaba, and D. Ziane, Homotopy perturbation method for solving the fractional Fisher's equation, Int. J. Anal. Appl. 10(1) (2016), pp. 9–16.
   Google Scholar
• M. D'Ovidio and F. Polito, Fractional diffusion-telegraph equations and their associated stochastic solutions, preprint (2013). Available at ArXiv, pages e-prints 1307.1696. To appear in Theor. Probab. Appl.
   Google Scholar
• J. Danane, K. Allali, and Z. Hammouch, Mathematical analysis of a fractional differential model of HBV infection with antibody immune response, Chaos. Solitons. Fractals 136 (2020), pp. 109787.
   Web of Science ®Google Scholar
• M.H. Derakhshan, M.R. Ahmadi Darani, A. Ansari, and R. Khoshsiar Ghaziani, On asymptotic stability of Prabhakar fractional differential systems, Comput. Methods Differ. Equ. 4(4) (2016), pp. 276–284.
   Web of Science ®Google Scholar
• M.H. Derakhshan and A. Ansari, On Hyers-Ulam stability of fractional differential equations with Prabhakar derivatives, Anal. 38(1) (2018), pp. 37–46.
   Google Scholar
• M.H. Derakhshan and A. Ansari, Fractional Sturm-Liouville problems for Weber fractional derivatives, Int. J. Comput. Math. 96(2) (2019), pp. 217–237.
   Web of Science ®Google Scholar
• M.H. Derakhshan and A. Ansari, Numerical approximation to Prabhakar fractional Sturm-Liouville problem, Comput. Appl. Math. 38(2) (2019), pp. 69.
   Google Scholar
• M.H. Derakhshan, A. Ansari, and M.R. Ahmadi Darani, On asymptotic stability of Weber fractional differential systems, Comput. Methods Differ. Equ. 6(1) (2018), pp. 30–39.
   Web of Science ®Google Scholar
• B. Dumitru, D. Kai, S. Enrico, T. Juan, Fractional Calculus: Models and Numerical Methods, Vol. 3, 2nd ed. World Scientific, New York, 2012.
   Google Scholar
• M. Faraz and G. Haller, The Maxey-Riley equation: existence, uniqueness and regularity of solutions, Nonlinear Anal. Real World Appl. 22 (2015), pp. 98–106.
   Web of Science ®Google Scholar
• W. Gao, P. Veeresha, D.G. Prakasha, H.M. Baskonus, and G. Yel, New numerical results for the time-fractional Phi-four equation using a novel analytical approach, Symmetry 12(3) (2020), pp. 478.
   Google Scholar
• W. Gao, P. Veeresha, D.G. Prakasha, and H. Mehmet Baskonus, Novel dynamic structures of 2019-nCoV with nonlocal operator via powerful computational technique, Biol. 9(5) (2020), pp. 107.
   PubMedGoogle Scholar
• W. Gao, P. Veeresha, D.G. Prakasha, H. Mehmet Baskonus, and G. Yel, A powerful approach for fractional Drinfeld-Sokolov-Wilson equation with Mittag-Leffler law, Alex. Eng. J. 58(4) (2019), pp. 1301–1311.
   Web of Science ®Google Scholar
• R. Garra, R. Gorenflo, F. Polito, and Ž. Tomovski, Hilfer-Prabhakar derivatives and some applications, Appl. Math. Comput. 242 (2014), pp. 576–589.
   Web of Science ®Google Scholar
• R. Garrappa, F. Mainardi, and M. Guido, Models of dielectric relaxation based on completely monotone functions, Fract. Calc. Appl. Anal. 19(5) (2016), pp. 1105–1160.
   Web of Science ®Google Scholar
• A. Giusti and I. Colombaro, Prabhakar-like fractional viscoelasticity, Commun. Nonlinear. Sci. Numer. Simul. 56 (2018), pp. 138–143.
   Web of Science ®Google Scholar
• R. Gorenflo, A.A. Kilbas, F. Mainardi, and S. Rogosin, Mittag-Leffler Functions, Theory. Appl Springer Monographs in Mathematics, Springer, Berlin, 2014.
   Google Scholar
• K. Górska, A. Horzela, L. Bratek, K.A. Penson, and G. Dattoli, The probability density function for the Havriliak-Negami relaxation, preprint (2016). Available at arXiv:1611.06433.
   Google Scholar
• M. Hamarsheh, A.I. Ismail, and Z. Odibat, An analytic solution for fractional order Riccati equations by using optimal homotopy asymptotic method, Appl. Math. Sci. 10(23) (2016), pp. 1131–1150.
   Google Scholar
• T. Hou and H. Leng, Numerical analysis of a stabilized Crank-Nicolson/Adams-Bashforth finite difference scheme for Allen-Cahn equations, Appl. Math. Lett. 102 (2020), pp. 106150.
   Google Scholar
• W. Hundsdorfer and J.G. Verwer, Numerical Solution of Time-dependent Advection-diffusion-reaction Equations, Vol. 33, Springer Science and Business Media, New York, 2013.
   Google Scholar
• E. İlhan and İ.O. Kıymaz, A generalization of truncated M-fractional derivative and applications to fractional differential equations, Appl. Math. Nonlinear. Sci. 5(1) (2020), pp. 171–188.
   Google Scholar
• Y. Kao, C. Gao, and D. Wang, Global exponential stability of reaction-diffusion hop-field neural network swith continuously distributed delays, Math. Appl. 21 (2008), pp. 457–462.
   Google Scholar
• B. Karaagac, Two step Adams Bashforth method for time fractional Tricomi equation with non-local and non-singular Kernel, Chaos Solitons Fractal 128 (2019), pp. 234–241.
   Web of Science ®Google Scholar
• P. Karunakar and S. Chakraverty, Solutions of time-fractional third-and fifth-order Korteweg-de-Vries equations using homotopy perturbation transform method, Eng. Comput. 36(7) (2019), pp. 2309–2326.
   Google Scholar
• R. Khalil, M.A. Horani, A. Yousef, and M. Sababheh, A new definition of fractional derivative, J. Comp. App. Math. 264 (2014), pp. 65–70.
   Web of Science ®Google Scholar
• A.A. Kilbas, M. Saigo, and R.K. Saxena, Generalized Mittag-Leffler function and generalized fractional calculus operators, Integral. Transform. Spec. Funct. 15(1) (2004), pp. 31–49.
   Web of Science ®Google Scholar
• S. Kumar, A. Kumar, and Z.M. Odibat, A nonlinear fractional model to describe the population dynamics of two interacting species, Math. Methods. Appl. Sci 40(11) (2017), pp. 4134–4148.
   Web of Science ®Google Scholar
• Y. Kuramoto, Chemical Oscillations Waves and Turbulence, Springer-Verlag, New York NY USA, 1984.
   Google Scholar
• C. Li, A. Kumar, S. Kumar, and X.J. Yang, On the approximate solution of nonlinear time-fractional KdV equation via modified homotopy analysis Laplace transform method, J. Nonlinear Sci. Appl. 9 (2016), pp. 5463–5470.
   Google Scholar
• A. Liemert, T. Sandev, and H. Kantz, Generalized Langevin equation with tempered memory kernel, Phys. A (Amsterdam) 466 (2017), pp. 356–369.
   Web of Science ®Google Scholar
• J. Lu, Global exponential stability and periodicity of reaction-diffusion delayed recurrent neural networks with Dirichlet boundary conditions, Chaos Solitons Fractal 35 (2008), pp. 116–125.
   Web of Science ®Google Scholar
• J. Luo, Fixed points and exponential stability of mild solutions of stochastic partial differential equations with delays, J. Math. Anal. Appl. 34 (2008), pp. 753–760.
   Google Scholar
• P. Miskinis, The Havriliak-Negami susceptibility as a nonlinear and nonlocal process, Phys. Scr. T136 (2009), pp. 014019.
   Google Scholar
• E. Oliveira and J. Sousa, Hyers-Ulam stability for a class of fractional integro-differential equations, Results Math. 73 (2018), pp. 1–16.
   Web of Science ®Google Scholar
• K.M. Owolabi and A. Atangana, Analysis and application of new fractional Adams-Bashforth scheme with Caputo-Fabrizio derivative, Chaos Solitons Fractals 105 (2017), pp. 111–119.
   Web of Science ®Google Scholar
• K.M. Owolabi and A. Atangana, On the formulation of Adams-Bashforth scheme with Atangana-Baleanu-Caputo fractional derivative to model chaotic problems, Chaos Interdiscip. J. Nonlinear. Sci. 29(2) (2019), pp. 023111.
   PubMedGoogle Scholar
• K.M. Owolabi and Z. Hammouch, Spatiotemporal patterns in the Belousov-Zhabotinskii reaction systems with Atangana-Baleanu fractional order derivative, Phys. A. 523 (2019), pp. 1072–1090.
   Web of Science ®Google Scholar
• Y. Pan, Y. He, and A. Mikkola, Accurate real-time truck simulation via semirecursive formulation and Adams-Bashforth-Moulton algorithm, Proc. Acta Mechanica. Sin. 35 (2019), pp. 641–652.
   Web of Science ®Google Scholar
• R.K. Pandey and H.K. Mishra, Homotopy analysis Sumudu transform method for time-fractional third order dispersive partial differential equation, Adv. Comput. Math. 43(2) (2017), pp. 365–383.
   Web of Science ®Google Scholar
• N.A. Pirim and F. Ayaz, A new technique for solving fractional order systems: Hermite collocation method, Appl. Math. 7(18) (2016), pp. 2307–2323.
   Google Scholar
• I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
   Google Scholar
• T.R. Prabhakar, A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama. Math. J. 19 (1971), pp. 7–15.
   Google Scholar
• S. Rathore, D. Kumar, J. Singh, and S. Gupta, Homotopy analysis Sumudu transform method for nonlinear equations, Int. J. Ind. Math. 4(4) (2012), pp. 301–314.
   Google Scholar
• I. Ratti, Comparative study of nonlinear partial differential equation using honotopy perturbation transform method (HPTM) using Hés polynomial and mixture of Elzaki transform and partial differential transform method (PDTM), Int. J. Appl. Eng. Res. 13(18) (2018), pp. 14037–14040.
   Google Scholar
• A. Salari and B. Ghanbari, Existence and multiplicity for some boundary value problems involving Caputo and Atangana-Baleanu fractional derivatives: A variational approach, Chaos Solitons Fractals127 (2019), pp. 312–317.
   Web of Science ®Google Scholar
• J. Singh, D. Kumar, Z. Hammouch, and A. Atangana, A fractional epidemiological model for computer viruses pertaining to a new fractional derivative, Appl. Math. Comput. 316 (2018), pp. 504–515.
   Web of Science ®Google Scholar
• J. Singh, D. Kumar, R. Swroop, and S. Kumar, An efficient computational approach for time-fractional Rosenau-Hyman equation, Neural. Comput. Appl. 30(10) ( ), pp. 3063–3070.
   Web of Science ®Google Scholar
• L. Song and W. Wang, A new improved Adomian decomposition method and its application to fractional differential equations, Appl. Math. Model. 37(3) (2013), pp. 1590–1598.
   Web of Science ®Google Scholar
• A. Stanislavsky and K. Weron, Atypical case of the dielectric relaxation responses and its fractional kinetic equation, Fract. Calc. Appl. Anal. 19(1) (2016), pp. 212–228.
   Web of Science ®Google Scholar
• S.H. Strogatz, Nonlinear Dynamics and Chaos: with Applications to Physics, Biology, Chemistry and Engineering, CRC Press, Boca Raton, 2018.
   Google Scholar
• R. Subashini, C. Ravichandran, K. Jothimani, and M.H. Baskonus, Existence results of Hilfer integro-differential equations with fractional order, Discrete. Contin. Dyn. Syst. Ser. S 13(3) (2018), pp. 911–923.
   Web of Science ®Google Scholar
• S. Ullah, M. Khan, and M. Farooq, A new fractional model for the dynamics of the Hepatitis-B virus using the Caputo-Fabrizio derivative, Eur. Phys. J. Plus 133 (2018), pp. 1–14.
   Web of Science ®Google Scholar
• J. Vahidi, The combined Laplace-homotopy analysis method for partial differential equations, J. Math. Comput. Sci. JMCS 16(1) (2016), pp. 88–102.
   Web of Science ®Google Scholar
• K. Vishal, S. Kumar, and S. Das, Application of homotopy analysis method for fractional Swift Hohenberg equation-revisited, Appl. Math. Modell. 36(8) (2012), pp. 3630–3637.
   Web of Science ®Google Scholar
• A. Yokuş and S. Gülbahar, Numerical solutions with linearization techniques of the fractional Harry Dym equation, Appl. Math. Nonlinear. Sci. 4(1) (2019), pp. 35–42.
   Google Scholar
• Y. Zhang, Time-Fractional generalized equal width wave equations: formulation and solution via variational methods, Nonlinear. Dyn. Syst. Theor. 14(4) (2014), pp. 410–25.
   Google Scholar
• Y.U. Zhang, C. Cattani, and X.J. Yang, Local fractional homotopy perturbation method for solving non-homogeneous heat conduction equations in fractal domains, Entropy 17(10) (2015), pp. 6753–6764.
   Web of Science ®Google Scholar
• X. Zhang, L. Liu, and Y. Wu, The uniqueness of positive solution for a fractional order model of turbulent flow in a porous medium, Appl. Math. Lett. 37 (2014), pp. 26–33.
   Web of Science ®Google Scholar