Numerical investigation for handling fractional-order Rabinovich–Fabrikant model using the multistep approach
In this paper, we present a reliable multistep numerical approach, so-called Multistep Generalized Differential Transform (MsGDT), to obtain accurate approximate form solution for Rabinovich–Fabrikant model involving Caputo fractional derivative subjected to appropriate initial conditions. The solution methodology provides efficiently convergent approximate series solutions with easily computable coefficients without employing linearization or perturbation. The behavior of approximate solution for different values of fractional-order \(\alpha \) is shown graphically. Furthermore, the stability analysis of the suggested model is discussed quantitatively. Simulation of the MsGDT technique is also presented to show its efficiency and reliability. Numerical results indicate that the method is simple, powerful mathematical tool and fully compatible with the complexity of such problems.
KeywordsFractional Rabinovich–Fabrikant model Multistep approach Differential transform method Generalized Taylor expansion
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Conflict of interest
The authors declare that there is no conflict of interests regarding the publication of this paper.
- Al-Smadi M, Freihat A, Khalil H, Momani S, Khan RA (2017) Numerical multistep approach for solving fractional partial differential equations. Int J Comput Methods 14(2):1–15. doi: 10.1142/S0219876217500293
- Ertürk VS, Momani S (2010) Application to fractional integro-differential equations. Stud Nonlinear Sci 1:118–126Google Scholar
- Kolebaje OT, Ojo OL, Akinyemi P, Adenodi RA (2013) On the application of the multistage laplace adomian decomposition method with pade approximation to the Rabinovich–Fabrikant system. Adv Appl Sci Res 4:232–243Google Scholar
- Magin RL (2006) Fractional calculus in bioengineering. Begell House Publisher Inc, ConnecticutGoogle Scholar
- Matignon D (1996) Stability results for fractional differential equations with applications to control processing. In: IMACS, IEEE-SMC Proceedings of the Computational Engineering in Systems and Application Multiconference, vol 2. Lille, pp 963–968Google Scholar
- Millar KS, Ross B (1993) An introduction to the fractional calculus and fractional differential equations. Wiley, New YorkGoogle Scholar
- Momani S, Freihat A, AL-Smadi M (2014) Analytical study of fractional-order multiple chaotic FitzHugh–Nagumo neurons model using multi-step generalized differential transform method. Abstract Appl Anal 2014, Article ID 276279, 1-10Google Scholar
- Rabinovich MI, Fabrikant AL (1979) Stochastic self-modulation of waves in nonequilibrium media. Sov Phys JETP 50:311–317Google Scholar
- Srivastava M, Agrawal S, Vishal K, Das S (2014) Chaos control of fractional order Rabinovich–Fabrikant system and synchronization between chaotic and chaos controlled fractional order Rabinovich–Fabrikant system. Appl Math Model 38:3361–3372Google Scholar
- Tarasov VE (2011) Fractional dynamics: applications of fractional calculus to dynamics of particles, fields and media. Springer, BerlinGoogle Scholar