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Using the generalized Adams-Bashforth-Moulton method for obtaining the numerical solution of some variable-order fractional dynamical models

  • Mohamed M. Khader EMAIL logo

Abstract

This paper is devoted to introduce a numerical treatment using the generalized Adams-Bashforth-Moulton method for some of the variable-order fractional modeling dynamics problems, such as Riccati and Logistic differential equations. The fractional derivative is described in Caputo variable-order fractional sense. The obtained numerical results of the proposed models show the simplicity and efficiency of the proposed method. Moreover, the convergence order of the method is also estimated numerically.


Corresponding author: Mohamed M. Khader, Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt; and Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia,

Acknowledgments

The author is very grateful to the editor and referees for carefully reading the paper and for their comments and suggestions which have improved the paper.

  1. Authors contributions: The paper has one author.

  2. Research funding: Not applicable.

  3. Conflict of interests: The author declares that there is no conflict of interests regarding the publication of this paper.

  4. Availability of data and materials: Not applicable.

References

[1] C. F. Lorenzo and T. T. Hartley, “Variable order and distributed order fractional operators,” Nonlinear Dynam., vol. 29, no. 14, pp. 57–98, 2002, https://doi.org/10.1023/a:1016586905654.10.1023/A:1016586905654Search in Google Scholar

[2] K. B. Oldham and J. Spanier, The Fractional Calculus, New York, Academic Press, 1974.Search in Google Scholar

[3] I. Podlubny, Fractional Differential Equations, New York, Academic Press, 1999.Search in Google Scholar

[4] S. G. Samko, “Fractional integration and differentiation of variable order,” Anal. Math., vol. 21, no. 3, pp. 213–236, 1995, https://doi.org/10.1007/bf01911126.10.1007/BF01911126Search in Google Scholar

[5] H. Sheng, H. G. Sun, C. Coopmans, Y. Q. Chen, and G. W. Bohannan, “A physical experimental study of variable-order fractional integrator and differentiator,” Eur. Phys. J., vol. 193, no. 1, pp. 93–104, 2011, https://doi.org/10.1140/epjst/e2011-01384-4.10.1140/epjst/e2011-01384-4Search in Google Scholar

[6] H. Sheng, H. G. Sun, Y. Q. Chen, and T. S. Qiu, “Synthesis of multi-fractional Gaussian noises based on variable-order fractional operators,” Signal Process., vol. 91, no. 7, pp. 1645–1650, 2011, https://doi.org/10.1016/j.sigpro.2011.01.010.10.1016/j.sigpro.2011.01.010Search in Google Scholar

[7] D. Valerio and J. S. Costa, “Variable-order fractional derivatives and their numerical approximations,” Signal Process., vol. 91, no. 3, pp. 470–483, 2011, https://doi.org/10.1016/j.sigpro.2010.04.006.10.1016/j.sigpro.2010.04.006Search in Google Scholar

[8] M. M. Khader and M. Adel, “Numerical treatment of the fractional modeling on susceptible-infected-recovered equations with a constant vaccination rate by using GEM,” Int. J. Nonlinear Sci. Numer. Stimul., vol. 14, pp. 1–7, 2018.10.1515/ijnsns-2018-0187Search in Google Scholar

[9] K. Diethelm, J. Ford, and A. Freed, “Detailed error analysis for a fractional Adams method,” Numer. Algorithm., vol. 36, pp. 31–52, 2004, https://doi.org/10.1023/b:numa.0000027736.85078.be.10.1023/B:NUMA.0000027736.85078.beSearch in Google Scholar

[10] K. Diethelm, The Analysis of Fractional Differential Equations, Berlin, Germany, Springer, 2010.10.1007/978-3-642-14574-2Search in Google Scholar

[11] S. Ma, Y. Xu and W. Yue, “Numerical solutions of a variable-order fractional financial system,” J. Appl. Math., vol. 2012, p, 14, 2012, Article no. 417942, https://doi.org/10.1155/2012/417942.10.1155/2012/417942Search in Google Scholar

[12] A. M. A. El-Sayed, A. E. M. El-Mesiry, and H. A. A. El-Saka, “On the fractional-order Logistic equation,” Appl. Math. Lett., vol. 20, no. 7, pp. 817–823, 2007, https://doi.org/10.1016/j.aml.2006.08.013.10.1016/j.aml.2006.08.013Search in Google Scholar

Received: 2019-12-20
Accepted: 2020-08-08
Published Online: 2020-09-23
Published in Print: 2021-02-23

© 2020 Walter de Gruyter GmbH, Berlin/Boston

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