Skip to main content
SpringerLink
Log in

A Nobel Variation of 16th-Order Iterative Scheme for Models in Blood Stream, Chemical Reactor, and Its Dynamics

  • Published:
Lobachevskii Journal of Mathematics Aims and scope Submit manuscript

Abstract

As science and engineering continue to advance in numerous fields, it has become increasingly difficult to find a fast and accurate solution to non-linear problems in the real world. To resolve these issues, more effective and efficient iterative techniques are required. In this paper, we present a weighted type four-point sixteenth-order iterative scheme of Steffensen–Ostrowski type. Maple 18, a computer algebra system employed to figure out the optimality of the proposed scheme that satisfies the Kung–Traub conjecture and is optimal. The theoretical convergence order of our recently devised scheme was determined to be sixteen. The scheme has applications in numerous fields, such as blood stream dynamics, chemical reactor dynamics, and Kepler’s equation, among others. In addition, numerical testing demonstrates that the developed iterative schemes are more effective than the existing optimal four-point iterative schemes in the domain. Thus, the fourth-step Steffensen–Ostrowski-type weighted family is a more efficient addition to the literature and a superior alternative.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18

REFERENCES

  1. S. Akram, M. Khalid, M. D. Junjua, S. Altaf, and S. Kumar, ‘‘Extension of King’s iterative scheme by means of memory for nonlinear equations,’’ Symmetry 15, 1116 (2023). https://doi.org/10.3390/sym15051116

    Article  Google Scholar 

  2. S. Akram, M. D. Junjua, N. Yasmin, and F. Zafar, ‘‘A family of weighted mean based optimal fourth order methods for solving system of nonlinear equations,’’ Life Sci. J. 11 (7s), 21–30 (2014). https://doi.org/10.1002/mma.5384

    Article  Google Scholar 

  3. D. K. R. Babajee and R. Thukral, ‘‘On a four-point sixteen-order King family of iterative method for solving nonlinear equations,’’ Int. J. Math. Math. Sci. 2012, 979245 (2012). https://doi.org/10.1155/2012/979245

  4. B. Bradie, A Friendly Introduction to Numerical Analysis (Pearson Education, NJ, 2006).

    Google Scholar 

  5. A. Cordero, J. L. Huesoa, Martínez, and J. R. Torregrosa, ‘‘Generating optimal derivative free iterative methods for nonlinear equations by using polynomial interpolation,’’ Math. Comput. Model. 57, 1950–1956 (2013).

    Article  Google Scholar 

  6. M. D. Junjua, S. Akram, T. Afzal, and A. Ali, ‘‘Construction of optimal derivative free iterative method for nonlinear equations using lagrange interpolation,’’ J. Prog. Res. Math. 16, 30–45 (2020).

    MathSciNet  Google Scholar 

  7. V. Kanwar, R. Bala, and M. Kansa, ‘‘Some new weighted eighth-order variants of Steffensen–King’s type family for solving nonlinear equations and its dynamics,’’ SeMA 74, 75–90 (2017). https://doi.org/10.1007/s40324-016-0081-1

    Article  MathSciNet  Google Scholar 

  8. H. T. Kung and J. F. Traub, ‘‘Optimal order of one-point and multipoint iteration,’’ Assoc. Comput. Math. 21, 634–651 (1974).

    MathSciNet  Google Scholar 

  9. N. Rafiq, S. Akram, N. Ahmad Mir, and M. Shams, ‘‘Study of dynamical behaviour and stability of iterative methods for finding distinct and multiple roots of non-linear equation with application in engineering,’’ Math. Probl. Eng. 2020, 3524324 (2020). https://doi.org/10.1155/2020/3524324

  10. S. Sharifi, M. Salimib, S. Siegmundb, and T. Lotfi, ‘‘A new class of optimal four-point methods with convergence order sixteen for solving nonlinear equations,’’ Math. Comput. Simul. 119, 69–90 (2016).

    Article  Google Scholar 

  11. J. R. Sharma and S. Kumar, ‘‘Efficient methods of optimal eighth and sixteenth order convergence for solving nonlinear equations,’’ SeMA 75, 229–253 (2018).

    Article  MathSciNet  Google Scholar 

  12. P. Sivakumar, K. Madhu, and J. Jayaraman, ‘‘Optimal eighth and sixteenth order iterative methods for solving nonlinear equation with basins of attraction,’’ Appl. Math. E-Notes 21, 320–343 (2021).

    MathSciNet  Google Scholar 

  13. J. F. Traub, Iterative Methods for the Solution of Equations (Prentice-Hall, Englewood Cliffs, NJ, 1964).

    Google Scholar 

  14. J. L. Varona, ‘‘Graphic and numerical comparison between iterative methods,’’ Math. Intell. 24, 37–46 (2002).

    Article  MathSciNet  Google Scholar 

  15. E. R. Vrscay and W. J. Gilbert, ‘‘Extraneous fixed points, basin boundaries and chaotic dynamics for Schröder and König rational iteration functions,’’ Numer. Math. 52, 1–16 (1988).

    Article  MathSciNet  Google Scholar 

  16. F. Zafar, S. Akram, N. Yasmin, and M. D. Junjua, ‘‘On the construction of three step derivative free four parametric methods with accelerated order of convergence,’’ J. Nonlin. Sci. Appl. 9, 4542–4553 (2016). https://doi.org/10.22436/jnsa.009.06.92

    Article  MathSciNet  Google Scholar 

Download references

ACKNOWLEDGMENTS

The authors are grateful to the referee for useful comments.

Funding

This work was supported by ongoing institutional funding. No additional grants to carry out or direct this particular research were obtained.

Author information

Authors and Affiliations

  1. Department of Mathematics, Govt. College Women University, 38000, Faisalabad, Pakistan

    Saima Akram

  2. Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, 60000, Multan, Pakistan

    Saima Akram, Hareem Khalid & Maira Khalid

  3. Department of Mathematical Analysis, Bukhara State University, 200100, Bukhara, Uzbekistan

    Tulkin Rasulov & Mutti-Ur Rehman

Authors
  1. Saima Akram
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hareem Khalid
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tulkin Rasulov
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Maira Khalid
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Mutti-Ur Rehman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Saima Akram or Tulkin Rasulov.

Ethics declarations

The authors of this work declare that they have no conflicts of interest.

Additional information

Publisher’s Note.

Pleiades Publishing remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

(Submitted by A. V. Lapin)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Akram, S., Khalid, H., Rasulov, T. et al. A Nobel Variation of 16th-Order Iterative Scheme for Models in Blood Stream, Chemical Reactor, and Its Dynamics. Lobachevskii J Math 44, 5116–5131 (2023). https://doi.org/10.1134/S199508022312003X

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S199508022312003X

Keywords:

Search

Navigation