This paper applies the Euler and the fourth-order Runge–Kutta methods in the analysis of fractional order dynamical systems. In order to illustrate the two techniques, the numerical algorithms are applied in the solution of several fractional attractors, namely the Lorenz, Duffing and Liu systems. The algorithms are implemented with the aid of Mathematica symbolic package. Furthermore, the Lyapunov exponent is obtained based on the Euler method and applied with the Lorenz fractional attractor.
J. C. Butcher, Numerical methods for ordinary differential equations, 3 ed. Wiley, Chichester, 2016.
J. A. Tenreiro Machado and A. M. S. F. Galhano, Evaluation of manipulator direct dynamics using customized Runge-Kutta methods, Syst. Anal. Modell. Simul. 17 (1995), 229–239.
K. Diethelm and N. J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl. 265 (2002), 229–248.
K. Diethelm and A. D. Freed, The FracPECE subroutine for the numerical solution of differential equations of fractional order, in: Orschung und wissenschaftliches Rechnen: Beiträge zum Heinz-Billing-Preis 1998, pp. 57–71, 01 1999.
R. Garrappa, Numerical solution of fractional differential equations: A survey and a software tutorial, Mathematics 6 (2018), 1–23.
G. E. Karniadakis (ed.), Numerical methods, Handbook of fractional calculus with applications 3, DeGruyter, Berlin/Boston,2019.
M. Caputo, Lectures on seismology and rheological tectonics, Lecture Notes, Universitá La Sapienza, Dipartimento di Fisica, Roma, 1992.
A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, vol. 204, North-Holland Mathematics Studies, Elsevier, Amsterdam, 2006.
G. A. Anastassiou and I. K. Argyros, Intelligent numerical methods: Applications to fractional calculus, Studies in computational intelligence, Springer, Cham, 2015.
K. Diethelm, The analysis of fractional differential equations: An application-oriented exposition using differential operators of caputo type, Lecture notes in mathematics, Springer, Heidelberg, 2010.
C. Li and F. Zeng, Numerical methods for fractional calculus, Lecture notes in electrical engineering, Chapman and Hall/CRC, Boca Raton, 2015.
C. Milici, J. T. Machado and G. Drăgănescu, On the fractional Cornu spirals, Commun. Nonlinear Sci. and Numer. Simul. 67 (2019), 315–320.
R. Lyons, A. S. Vatsala and R. A. Chiquet, Picard’s iterative method for Caputo fractional differential equations with numerical results, Mathematics 5 (2017), 2–9.
C. Milici and G. Drăgănescu, Introduction to fractional calculus, Lambert Academic Publishing, Saarbrücken, 2017.
C. Milici, G. Drăgănescu and J. T. Machado, Introduction to fractional differential equations, Springer, Cham, Switzerland, 2018.
E. N. Lorenz, Deterministic nonperiodic flow, J. Atmos. Sci. 20 (1963), 130–141.
R. Caponetto and S. Fazzino, A semi-analytical method for the computation of the Lyapunov exponents of fractional-order systems, Commun. Nonlinear Sci. and Numer. Simul. 18 (2013), 22–27.10.1016/j.cnsns.2012.06.013)| false
The International Journal of Nonlinear Sciences and Numerical Simulation publishes original papers on all subjects relevant to nonlinear sciences and numerical simulation. The journal is directed at researchers in nonlinear sciences, engineers, and computational scientists, economists, and others, who either study the nature of nonlinear problems or conduct numerical simulations of nonlinear problems.