Skip to main content
SpringerLink
Log in

A new piecewise-quasilinearization method for solving chaotic systems of initial value problems

  • Research Article
  • Published:
Central European Journal of Physics

Abstract

In this paper, a modification of the successive linearization method (SLM) for solving nonlinear initial value problems is introduced for the first time. The proposed method is based on a novel technique of extending the standard SLM and adapting it to a sequence of multiple intervals. In this new application the method is referred to as the piecewise successive linearization method(PSLM). This new algorithm is applied to chaotic and non-chaotic differential equations that model the Lotka-Volterra, Lorenz, Rössler and Genesio-Tesi systems. A comparative study between the new algorithm and the MATLAB Runge-Kutta based in-built solver (ode45) method is presented. The results demonstrate accuracy and reliability of the proposed PSLM algorithm.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. O. Abdulaziz, N. F. M. Noor, I. Hashim, M. S. M. Noorani, Chaos Soliton. Fract. 36, 1405 (2008)

    Article  ADS  Google Scholar 

  2. A. K Alomari, M.S.M. Noorani, R. Nazar, Phys. Scr. 81, 045005 (2010)

    Article  ADS  Google Scholar 

  3. A. K. Alomari, Comput. Math. Appl. 61, 2528 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. A. K. Alomari, M. S. M. Noorani, R. Nazar, Commun. Nonlinear Sci. 14, 2336 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. A. K. Alomari, M. S. M. Noorani, R. Nazar, C. P. Li, Commun. Nonlinear Sci. 15, 1864 (2010)

    Article  MATH  Google Scholar 

  6. F. G. Awad, P. Sibanda, S. S. Motsa, O. D. Makinde, Comput. Math. Appl. 61, 1431 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spectral Methods in Fluid Dynamics, (Springer-Verlag, Berlin, 1988)

    MATH  Google Scholar 

  8. G. Chen, T. Ueta, Int. J. Bifurcat. Chaos 9(7), 1465 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. M. S. H. Chowdhury, I. Hashim, Nonlinear Anal. Real. 10, 381 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. J. R. Dormand, P. J. Prince, J. Comput. Appl. Math. 6(1), 19 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  11. M. R. Faieghi, H. Delavari, Commun. Nonlinear Sci. 17, 731 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. R. Genesio, A. Tesi, Automatica 28, 531 (1992)

    Article  MATH  Google Scholar 

  13. A. Ghorbani, J. Saberi-Nadjafi, Math. Comput. Model. 54, 131 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. S.M. Goh, M. S. M. Noorani, I. Hashim, Numer. Algorithms 54, 245 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. A. Gökdogan, M. Merdan, A Yildirim, Commun. Nonlinear Sci. 17, 45 (2012)

    Article  Google Scholar 

  16. I. Hashim, M. S. M. Noorani, R. Ahmad, S. A. Bakar, E. S. Ismail, A. M. Zakaria, Chaos Soliton. Fract. 28, 1149 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  17. E. N. Lorenz, J. Atmos. Sci. 20, 130 (1963)

    Article  ADS  Google Scholar 

  18. Z. G. Makukula, P. Sibanda, S. S. Motsa, Bound. Value Probl., 471793 (2010)

  19. M. Merdan, A. Gokdogan and V. S. Erturk, Iran. J. Sci. Technol. A1, 9 (2011)

  20. M. Mossa Al-Sawalha, M. S. M. Noorani, I. Hashim, Chaos Soliton. Fract. 40, 1801 (2009)

    Article  ADS  MATH  Google Scholar 

  21. S. S. Motsa, Int. J. Mod. Sim. Sci. Comp. 2, 355 (2011)

    Article  Google Scholar 

  22. S. S. Motsa, P. Sibanda, S. Shateyi, Math. Method. Appl. Sci. 34, 1406 (2011)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  23. S. S. Motsa, P. Sibanda, Comput. Math. Appl. 63, 1197 (2012)

    Article  Google Scholar 

  24. Z. M. Odibat, C. Bertelle, M. A. Aziz-Alaoui, G. H. E. Duchamp, Comput. Math. Appl. 59, 1462 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. J. I. Ramos, Appl. Math. Comput. 198, 92 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. O. E. Rössler, Phys. Lett. A 57, 397 (1976)

    Article  ADS  Google Scholar 

  27. S. Shateyi, S.S. Motsa, Bound. Value Probl., Article ID 257568 (2010)

  28. N. T. Shawagfeh, G. Adomian, Appl. Math. Comput 76, 251 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  29. L. N. Trefethen, Spectral Methods in MATLAB, (SIAM, Philadelphia, 2000)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X01, Scottsville, Pietermaritzburg, 3209, South Africa

    Sandile S. Motsa

Authors
  1. Sandile S. Motsa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sandile S. Motsa.

About this article

Cite this article

Motsa, S.S. A new piecewise-quasilinearization method for solving chaotic systems of initial value problems. centr.eur.j.phys. 10, 936–946 (2012). https://doi.org/10.2478/s11534-011-0124-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.2478/s11534-011-0124-2

Keywords

Search

Navigation