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

  • Francisco I. ChicharroEmail author
  • Alicia Cordero
  • Tobías H. Martínez
  • Juan R. Torregrosa
Original Paper

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.

Keywords

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

Mathematics Subject Classification

MSC 65H05 

Notes

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. 79–111Google Scholar
  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)CrossRefGoogle 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)CrossRefGoogle 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)CrossRefGoogle 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)CrossRefGoogle Scholar
  7. 7.
    C.F. Colebrook, C.M. White, Experiments with fluid friction in roughened pipes. Proc. R. Soc. Lond. 161, 367–381 (1937)CrossRefGoogle 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)CrossRefGoogle 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)CrossRefGoogle Scholar
  12. 12.
    T. Lukić, N. Ralević, Geometric mean Newton’s method for simple and multiple roots. Appl. Math. Lett. 21, 30–36 (2008)CrossRefGoogle Scholar
  13. 13.
    A. Özban, Some new variants of Newton’s method. Appl. Math. Lett. 17, 677–682 (2004)CrossRefGoogle 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. 149–234CrossRefGoogle 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)CrossRefGoogle 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)CrossRefGoogle 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)CrossRefGoogle Scholar
  19. 19.
    Z. Xiaojian, A class of Newton’s methods with third-order convergence. Appl. Math. Lett. 20, 1026–1030 (2007)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Escuela Superior de Ingeniería y TecnologíaUniversidad Internacional de La RiojaLogroñoSpain
  2. 2.Institute for Multidisciplinary MathematicsUniversitat Politècnica de ValènciaValènciaSpain

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