Numerical treatment for a nine-dimensional chaotic Lorenz model with the Rabotnov fractional-exponential kernel fractional derivative

Document Type : Article

Author

- Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh: 11566, Saudi Arabia. - Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt.

Abstract

In this paper, we will present an effective simulation to study the solution behavior of a high dimensional chaos by considering the nine-dimensional
Lorenz system through the Rabotnov fractional-exponential (RFE) kernel fractional derivative. First, we derive an approximate formula of the
fractional-order derivative of a polynomial function $t^{p}$ in terms of the RFE kernel. In this work, we use the spectral collocation method based
on the properties of the shifted Vieta-Lucas polynomials. This procedure converts the given model to a system of algebraic equations. We satisfy the
efficiency and the accuracy of the given procedure by evaluating the residual error function. The results obtained are compared with the results obtained
by using the fourth-order Runge-Kutta method. The results show that the implemented technique is easy and efficient tool to simulate the solution of such models.

Keywords

Main Subjects

References

References:
1. Lorenz, E. "Deterministic nonperiodic flow", J. Atmospheric  Sci., 20, pp. 130-141 (1963). DOI: 10.1175/1520- 0469(1963)020<0130:DNF>2.0.CO;2.
2. Rossler, O.E. "An equation for continuous chaos", Phys. Lett. A, 57(5), pp. 397-398 (1976). DOI: 10.1016/0375-9601(76)90101-8.
3. Ispolatov, I., Madhok, V., Allende, S., et al. "Chaos in high-dimensional dissipative dynamical systems", Scientific Reports, 5, pp. 12-26 (2015). DOI: 10.1038/srep12506.
4. Eftekhari, S.A. and Jafari, A.A. "Numerical simulation of chaotic dynamical systems by the method of differential quadrature", Scientia Iranica, 19(5), pp. 1299-1315 (2012). DOI: 10.1016/j.scient.2012.08.003.
5. Lozi, R., Pogonin, V.A., and Pchelintsev, A.N. "A new  ccurate numerical method of approximation of chaotic solutions of dynamical model equations with quadratic nonlinearities", Chaos, Solitons and Fractals, 91, pp. 108-114 (2016). DOI: 10.1016/j.chaos.2016.05.010.
6. Odibat, Z.M., Bertelle, C., Aziz-Alaoui, M.A., et al. "A multi-step differential transform method and application to non-chaotic or chaotic systems", Computers and Mathematics with Applications, 59(4), pp. 1462- 1472 (2010). DOI: 10.1016/j.camwa.2009.11.005.
7. Zhou, X., Li, J., Wang, Y., et al. "Numerical simulation of a class of hyperchaotic system using Barycentric Lagrange interpolation collocation method", Complexity, 1, pp. 1-13 (2019). DOI: 10.1155/2019/1739785.
8. Khader, M.M. and Babatin, M.M. "Numerical study for improvement the cooling process through a model of Powell-Eyring 
fluid flow over a stratified stretching sheet with magnetic field", Case Studies in Thermal Engineering, 31, pp. 1-12 (2022).
9. Khader, M.M. and Inc, M. "Numerical technique based on the interpolation with Lagrange polynomials to analyze the fractional variable-order mathematical model of the hepatitis C with different types of virus genome", Chaos, Solitons and Fractals, 152, pp. 1-16 (2021). DOI: 10.1016/j.chaos.2021.111333.
10. Abd-Elhameed, W.M. and Youssri, Y.H. "Connection formulae between generalized Lucas polynomials and some Jacobi polynomials: Application to certain types of fourth-order BVPs", Int. J. Appl. Comput. Math., 6(45), pp. 1-19 (2020). DOI: 10.1007/s40819-020-0799-4.
11. Agarwal, F. and El-Sayed, A.A. "Vieta-Lucas polynomials for solving a fractional-order mathematical physics model", Adv. Differ. Equ., 2020, pp. 1-18 (2020).
12. Samko, S.G., Kilbas, A.A., and Marichev, O.I., Fractional Integrals and Derivatives: Theory and Applications, Gordon & Breach, Yverdon (1993).
13. Khader, M.M. "Numerical study of the nano fluid thin film flow past an unsteady stretching sheet with fractional derivatives using the spectral collocation Chebyshev approximation", International Journal of Modern Physics C, 32(2), pp. 1-14 (2021). DOI: 10.1142/S0129183121500261.
14. Khader, M.M., Saad, K.M., Hammouch, Z., et al. "A spectral collocation method for solving fractional KdV and KdV-Burger's equations with non-singular kernel derivatives", Applied Numerical Mathematics, 161, pp. 137-146 (2021). DOI: 10.1016/j.apnum.2020.10.024.
15. Khader, M.M. and Adel, M. "Introducing the windowed Fourier frames technique for obtaining the approximate solution of coupled system of differential equations", Journal of Pseudo-Differential Operators and Applications, 10, pp. 241-256 (2019). DOI: 10.1007/s11868-018-0240-5.
16. Khader, M.M. and Adel, M. "Numerical study of the fractional modeling on SIR equations with constant vaccination rate using GEM", International Journal of Nonlinear Sciences and Numerical Simulation, 20(1), pp. 69-76 (2019). DOI: 10.1515/ijnsns-2018-0187.
17. Podlubny, I., Fractional Differential Equations, Academic Press, San Diego (1999).
18. Khader, M.M. and Srivastava, H.M. "Numerical simulation for the treatment of nonlinear predator-prey equations by using the finite element optimization method", Fractal Fract, 5(56), pp. 1-9 (2021).
19. Gao, W., Ghanbari, B., and Baskonus, H.M. "New numerical simulations for some real world problems with Atangana-Baleanu fractional derivative", Chaos, Solitons & Fractals, 128, pp. 34-43 (2019). DOI: 10.1016/j.chaos.2019.07.037.
20. Toufik, M. and Atangana, A. "New numerical approximation of fractional derivative with non-local and non-singular kernel: Application to chaotic models", European Physical J. Plus, 132, pp. 1-14 (2017). DOI: 10.1140/epjp/i2017-11717-0.
21. Kumar, D., Singh, J., and Baleanu, D. "On the analysis of vibration equation involving a fractional derivative with Mittag-Leffler law", Math. Methods Appl. Sci, 43(1), pp. 443-457 (2020). DOI: 10.1002/mma.5903.
22. Atangana, A. and Baleanu, D. "New fractional derivative with non-local and non-singular kernel", Therm. Sci., 20(2), pp. 736-769 (2016).
23. Kumar, S., Gomez-Aguilar, J.F., Lavin-Delgado, J.E., et al. "Derivation of operational matrix of Rabotnov fractional-exponential kernel and its application to fractional Lienard equation", Alexandria Engineering Journal, 59(2), pp. 2991-2997 (2020). DOI: 10.1016/j.aej.2020.04.036.
24. Horadam, A.F., Vieta Polynomials, The University of New England, Armidaie, Australia (2000). 
25. Youssef, M.Z., Khader, M.M., Al-Dayel, I., et al. "Solving fractional generalized Fisher-Kolmogorov- Petrovsky-Piskunov's equation using compact finite different method together with spectral collocation algorithms", Journal of Mathematics, 15, pp. 1-12 (2022).
26. Reiterer, P., Lainscsek, C., Sch, F., et al. "A ninedimensional Lorenz system to study high-dimensional chaos", J. Phys. A: Math. Gen., 31, pp. 7121-7139 (1998). DOI: 10.1088/0305-4470/31/34/015.
27. He, J.H. "A new approach to nonlinear partial differential equations", Communications in Nonlinear Science and Numerical Simulation, 2(4), pp. 230-235 (1997).DOI: 10.1016/S1007-5704(97)90007-1.
28. Abdou, M.A. and Soliman, A.A. "New applications of variational iteration method", Physica D, 211(2), pp. 1-8 (2005). DOI: 10.1016/j.physd.2005.08.002.
29. Jafari, H. and Alipoor, A. "A new method for calculating general Lagrange multiplier in the variational iteration method", Numerical Methods for Partial Differential Equations, 27(4), pp. 996-1001 (2011). DOI: 10.1002/num.20567.
30. Momani, S. and Abuasad, S. "Application of He's variational iteration method to Helmholtz equation", Chaos, Solitons and Fractals, 27(5), pp. 1119-1123 (2006). DOI: 10.1016/j.chaos.2005.04.113.
31. Odibat, Z. and Momani, S. "The variational iteration method: An ecient scheme for handling fractional partial differential equations in fluid mechanics", Computers and Mathematics with Applications, 58(12), pp. 2199-2208 (2009). DOI: 10.1016/j.camwa.2009.03.009.
32. He, J.H. "Approximate analytical solution for seepage flow with fractional derivatives in porous media", Computer Methods in Applied Mechanics and Engineering, 167(2), pp. 57-68 (1998). DOI: 10.1016/S0045-7825(98)00108-X.
33. Kouagou, J.N., Dlamini, P.G., and Simelane, S.M. "On the multi-domain compact finite difference relaxation method for high dimensional chaos: The nine-dimensional Lorenz system", Alexandria Engineering Journal, 59, pp. 2617-2625 (2020). DOI: 10.1016/j.aej.2020.04.025.
Scientia Iranica
Volume 31, Issue 12
Transactions on Nanotechnology (F)
May and June 2024
Pages 945-957

Files

History

  • Receive Date: 02 September 2022
  • Revise Date: 23 December 2022
  • Accept Date: 11 April 2023

Share

How to cite

Statistics