Skip to main content

A new fourth-order family for solving nonlinear problems and its dynamics

Abstract

In this manuscript, a new parametric class of iterative methods for solving nonlinear systems of equations is proposed. Its fourth-order of convergence is proved and a dynamical analysis on low-degree polynomials is made in order to choose those elements of the family with better conditions of stability. These results are checked by solving the nonlinear system that arises from the partial differential equation of molecular interaction.

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
Fig. 9

References

  1. R.C. Rach, J.S. Duan, A.M. Wazwaz, Solving coupled Lane–Emden boundary value problems in catalytic diffusion reactions by the Adomian decomposition method. J. Math. Chem. 52(1), 255–267 (2014)

    Article  CAS  Google Scholar 

  2. R. Singh, G. Nelakanti, J. Kumar, A new efficient technique for solving two-point boundary value problems for integro-differential equations. J. Math. Chem. doi:10.1007/s10910-014-0363-8

  3. M. Mahalakshmi, G. Hariharan, K. Kannan, The wavelet methods to linear and nonlineal reaction–diffusion model arising in mathematical chemistry. J. Math. Chem. 51, 2361–2385 (2013)

    Article  CAS  Google Scholar 

  4. P.G. Logrado, J.D.M. Vianna, Partitioning technique procedure revisited: formalism and first application to atomic problems. J. Math. Chem. 22, 107–116 (1997)

    Article  CAS  Google Scholar 

  5. C.G. Jesudason, I. Numerical nonlinear analysis: differential methods and optimization applied to chemical reaction rate determination. J. Math. Chem. 49, 1384–1415 (2011)

    Article  CAS  Google Scholar 

  6. A. Klamt, Conductor-like screening model for real solvents: a new approach to the quantitative calculation of solvation phenomena. J. Phys. Chem. 99, 2224–2235 (1995)

    Article  CAS  Google Scholar 

  7. A. Klamt, V. Jonas, T. Brger, J.C.W. Lohrenz, Refinement and parametrization of COSMORS. J. Phys. Chem. A 102, 5074–5085 (1998)

    Article  CAS  Google Scholar 

  8. H. Grensemann, J. Gmehling, Performance of a conductor-like screening model for real solvents model in comparison to classical group contribution methods. Ind. Eng. Chem. Res. 44(5), 1610–1624 (2005)

    Article  CAS  Google Scholar 

  9. T. Banerjee, A. Khanna, Infinite dilution activity coefficients for trihexyltetradecyl phosphonium ionic liquids: measurements and COSMO-RS prediction. J. Chem. Eng. Data 51(6), 2170–2177 (2006)

    Article  CAS  Google Scholar 

  10. R. Franke, B. Hannebauer, On the influence of basis sets and quantum chemical methods on the prediction accuracy of COSMO-RS. Phys. Chem. Chem. Phys. 13, 21344–21350 (2011)

    Article  CAS  Google Scholar 

  11. K. Maleknejad, M. Alizadeh, An efficient numerical scheme for solving Hammerstein integral equation arisen in chemical phenomenon. Proc. Comput. Sci. 3, 361–364 (2011)

    Article  Google Scholar 

  12. M. Petković, B. Neta, L. Petković, J. Džunić, Multipoint Methods for Solving Nonlinear Equations (Academic Press, Amsterdam, 2012)

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  15. A.M. Ostrowski, Solution of Equations and Systems of Equations (Prentice-Hall, Englewood Cliffs, 1964)

    Google Scholar 

  16. P. Jarratt, Some fourth order multipoint iterative methods for solving equations. Math. Comput. 20, 434–437 (1966)

  17. R.F. King, A family of fourth order methods for nonlinear equations. SIAM J. Numer. Anal. 10, 876–879 (1973)

    Article  Google Scholar 

  18. A. Cordero, J.L. Hueso, E. Martínez, J.R. Torregrosa, A modified Newton Jarratt’s composition. Numer. Algorithms 55, 87–99 (2010)

    Article  Google Scholar 

  19. S. Amat, S. Busquier, Á.A. Magreñán, Reducing Chaos and Bifurcations in Newton-Type Methods. Abstract and Applied Analysis Volume 2013 (2013), Article ID 726701, 10 pages, doi:10.1155/2013/726701

  20. S. Amat, S. Busquier, S. Plaza, Review of some iterative root-finding methods from a dynamical point of view. Sci. Ser. A Math. Sci. 10, 3–35 (2004)

    Google Scholar 

  21. F. Chicharro, A. Cordero, J.M. Gutiérrez, J.R. Torregrosa, Complex dynamics of derivative-free methods for nonlinear equations. Appl. Math. Comput. 219, 7023–7035 (2013)

    Article  Google Scholar 

  22. C. Chun, M.Y. Lee, B. Neta, J. Džunić, On optimal fourth-order iterative methods free from second derivative and their dynamics. Appl. Math. Comput. 218, 6427–6438 (2012)

    Article  Google Scholar 

  23. Á.A. Magreñán, Different anomalies in a Jarratt family of iterative root-finding methods. Appl. Math. Comput. 233, 29–38 (2014)

    Article  Google Scholar 

  24. A. Cordero, J.R. Torregrosa, P. Vindel, Dynamics of a family of Chebyshev–Halley type methods. Appl. Math. Comput. 219, 8568–8583 (2013)

    Article  Google Scholar 

  25. Á. A. Magreñán, Estudio de la dinámica del método de Newton amortiguado (PhD Thesis). Servicio de Publicaciones, Universidad de La Rioja, (2013). http://dialnet.unirioja.es/servlet/tesis?codigo=38821

  26. P. Blanchard, The dynamics of Newton’s method. Proc. Symp. Appl. Math. 49, 139–154 (1994)

    Article  Google Scholar 

  27. F. Chicharro, A. Cordero, J.R. Torregrosa, Drawing dynamical and parameters planes of iterative families and methods. The Scientific World J. 2013 (Article ID 780153) (2013)

  28. L.B. Rall, Computational Solution of Nonlinear Operator Equations (Robert E. Krieger Publishing Company Inc., New York, 1969)

  29. J.R. Sharma, R.K. Guna, R. Sharma, An efficient fourth order weighted-Newton method for systems of nonlinear equations. Numer. Algorithms 62, 307–323 (2013)

    Article  Google Scholar 

Download references

Acknowledgments

The authors thank to the anonymous referees for their suggestions to improve the paper.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Juan R. Torregrosa.

Additional information

This research was supported by Ministerio de Ciencia y Tecnología MTM2011-28636-C02-\(\{01,02\}\) and Universitat Politècnica de València SP20120474.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Cordero, A., Feng, L., Magreñán, A. et al. A new fourth-order family for solving nonlinear problems and its dynamics. J Math Chem 53, 893–910 (2015). https://doi.org/10.1007/s10910-014-0464-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10910-014-0464-4

Keywords

  • Nonlinear systems
  • Iterative methods
  • Complex dynamics
  • Parameter space
  • Basins of attraction
  • Stability

Mathematics Subject Classfication

  • 65H05