Skip to main content

CMMSE-2019 mean-based iterative methods for solving nonlinear chemistry problems

A Correction to this article was published on 05 December 2019

This article has been updated

Abstract

The third-order iterative method designed by Weerakoon and Fernando includes the arithmetic mean of two functional evaluations in its expression. Replacing this arithmetic mean with different means, other iterative methods have been proposed in the literature. The evolution of these methods in terms of order of convergence implies the inclusion of a weight function for each case, showing an optimal fourth-order convergence, in the sense of Kung–Traub’s conjecture. The analysis of these new schemes is performed by means of complex dynamics. These methods are applied on the solution of the nonlinear Colebrook–White equation and the nonlinear system of the equilibrium conversion, both frequently used in Chemistry.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Change history

  • 05 December 2019

    The original version of this article unfortunately contained an error in title. Unintentionally, the special issue title was presented in addition to the article’s title.

References

  1. 1.

    O. Ababneh, New Newton’s method with third order convergence for solving nonlinear equations. World Acad. Sci. Eng. Technol. 61, 1071–1073 (2012)

    Google Scholar 

  2. 2.

    S. Amat, S. Busquier, Advances in iterative methods for nonlinear equations, chapter 5. SEMA SIMAI Springer Series. (Springer, Berlin, 2016), vol. 10, pp. 79–111

  3. 3.

    R. Behl, Í. Sarría, R. González, Á.A. Magreñán, Highly efficient family of iterative methods for solving nonlinear models. J. Comput. Appl. Math. 346, 110–132 (2019)

    Article  Google Scholar 

  4. 4.

    B. Campos, J. Canela, P. Vindel, Convergence regions for the Chebyshev-Halley family. Commun. Nonlinear Sci. Numer. Simul. 56, 508–525 (2018)

    Article  Google Scholar 

  5. 5.

    F.I. Chicharro, A. Cordero, J.R. Torregrosa, Drawing dynamical and parameters planes of iterative families and methods. Sci. World J. 780513, 1–11 (2013)

    Article  Google Scholar 

  6. 6.

    F.I. Chicharro, A. Cordero, J.R. Torregrosa, Dynamics of iterative families with memory based on weight functions procedure. J. Comput. Appl. Math. 354, 286–298 (2019)

    Article  Google Scholar 

  7. 7.

    C.F. Colebrook, C.M. White, Experiments with fluid friction in roughened pipes. Proc. R. Soc. Lond. 161, 367–381 (1937)

    Google Scholar 

  8. 8.

    A. Constantinides, N. Mostoufi, Numerical Methods for Chemical Engineers with MATLAB Applications (Prentice-Hall, Englewood Cliffs, 1999)

    Google Scholar 

  9. 9.

    A. Cordero, J. Franceschi, J.R. Torregrosa, A.C. Zagati, A convex combination approach for mean-based variants of Newton’s method. Symmetry 11, 1062 (2019)

    Article  Google Scholar 

  10. 10.

    A. Cordero, J.R. Torregrosa, Variants of Newton’s method using fifth order quadrature formulas. Appl. Math. Comput. 190, 686–698 (2007)

    Google Scholar 

  11. 11.

    H.T. Kung, J.F. Traub, Optimal order of one-point and multipoint iteration. J. Assoc. Comput. Math. 21, 643–651 (1974)

    Article  Google Scholar 

  12. 12.

    T. Lukić, N. Ralević, Geometric mean Newton’s method for simple and multiple roots. Appl. Math. Lett. 21, 30–36 (2008)

    Article  Google Scholar 

  13. 13.

    A. Özban, Some new variants of Newton’s method. Appl. Math. Lett. 17, 677–682 (2004)

    Article  Google Scholar 

  14. 14.

    M. Petković, B. Neta, L. Petković, J. Dz̆unić, Multipoint Methods for Solving Nonlinear Equations (Academic Press, Cambridge, 2013)

    Google Scholar 

  15. 15.

    E. Shashi, Transmission Pipeline Calculations and Simulations Manual, Fluid Flow in Pipes (Elsevier, London, 2015), pp. 149–234

    Book  Google Scholar 

  16. 16.

    M.K. Singh, A.K. Singh, A new-mean type variant of Newton’s method for simple and multiple roots. Int. J. Math. Trends Technol. 49, 174–177 (2017)

    Article  Google Scholar 

  17. 17.

    K. Verma, On the centroidal mean Newton’s method for simple and multiple roots of nonlinear equations. Int. J. Comput. Sci. Math. 7, 126–143 (2016)

    Article  Google Scholar 

  18. 18.

    S. Weerakoon, T.G.I. Fernando, A variant of Newton’s method with third-order convergence. Appl. Math. Lett. 13, 87–93 (2000)

    Article  Google Scholar 

  19. 19.

    Z. Xiaojian, A class of Newton’s methods with third-order convergence. Appl. Math. Lett. 20, 1026–1030 (2007)

    Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Francisco I. Chicharro.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This research was partially supported by PGC2018-095896-B-C22 (MCIU/AEI/FEDER/UE) and Generalitat Valenciana PROMETEO/2016/089.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Chicharro, F.I., Cordero, A., Martínez, T.H. et al. CMMSE-2019 mean-based iterative methods for solving nonlinear chemistry problems. J Math Chem 58, 555–572 (2020). https://doi.org/10.1007/s10910-019-01085-2

Download citation

Keywords

  • Nonlinear systems
  • Iterative method
  • Weight functions
  • Complex dynamics
  • Basin of attraction
  • Chemical applications

Mathematics Subject Classification

  • MSC 65H05