Skip to main content
SpringerLink
Log in

Variational iteration technique for solving a system of nonlinear equations

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

In this paper, we use the variational iteration technique to suggest and analyze some new iterative methods for solving a system of nonlinear equations. We prove that the new method has fourth-order convergence. Several numerical examples are given to illustrate the efficiency and performance of the new iterative methods. Our results can be viewed as a refinement and improvement of the previously known results.

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. Burden R.L., Faires J.D.: Numerical Analysis. PWS Publishing Company, Boston (2001)

    Google Scholar 

  2. Babajee D.K.R., Dauhoo M.Z., Darvishi M.T., Barati A.: A note on the local convergence of iterative methods based on Adomian decomposition method and 3-node quadrature rule. Appl. Math. Comput. 200, 452–458 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Babajee D.K.R., Dauhoo M.Z.: Convergence and spectral analysis of the Frontini–Sormani family of multipoint third order methods from quadrature rule. Numer. Algorithms 53, 467–484 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Babajee D.K.R., Dauhoo M.Z., Darvishi M.T., Karami A., Barati A.: Analysis of two Chebyshev-like third order methods free from second derivatives for solving systems of nonlinear equations. Appl. Math. Comput. 223, 2002–2012 (2010)

    MathSciNet  Google Scholar 

  5. Darvishi M.T., Barati A.: A third-order Newton-type method to solve systems of nonlinear equations. Appl. Math. Comput. 187, 630–635 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Darvishi M.T., Barati A.: Super cubic iterative methods to solve systems of nonlinear equations. Appl. Math. Comput. 188, 1678–1685 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Decker D.W., Kelley C.T.: Newton’s method at singular points. II. SIAM J. Numer. Anal. 17, 465–471 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  8. Frontini M., Sormani E.: Third-order methods from quadrature formulae for solving systems of nonlinear equations. Appl. Math. Comput. 149, 771–782 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Haijun W.: New third-order method for solving systems of nonlinear equations. Numer. Algorithms 50, 271–282 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. He J.H.: Variational iteration method-some recent results and new interpretations. J. Comput. Appl. Math. 207, 3–17 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Inokuti M., Sekine H., Mura T.: General use of the Lagrange multiplier in nonlinear mathematical physics. In: Nemat-Nasser, S. (ed) Variational Methods in the Mechanics of Solids, pp. 156–162. Pergamon Press, New York (1978)

    Google Scholar 

  12. Noor M.A., Noor K.I.: Three-step iterative methods for nonlinear equations. Appl. Math. Comput. 183, 322–327 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Noor M.A., Mohyud-Din S.T.: Variational iteration techniques for solving higher order boundary value problems. Appl. Math. Comput. 189, 1929–1942 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Noor M.A.: New classes of iterative methods for nonlinear equations. Appl. Math. Comput. 191, 128–131 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Noor M.A., Shah F.A.: Variational iteration technique for solving nonlinear equations. J. Appl. Math. Comput. 31, 247–254 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Noor M.A., Waseem M.: Some iterative methods for solving a system of nonlinear equations. Comput. Math. Appl. 57, 101–106 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Noor M.A., Waseem M.: Variants of Newton’s method using fifth-order quadrature formulas: revisited. J. Appl. Math. Inform. 27, 1195–1209 (2009)

    Google Scholar 

  18. Noor M.A.: Some iterative methods for solving nonlinear equations using homotopy perturbation method. Int. J. Comp. Math. 87, 141–149 (2010)

    Article  MATH  Google Scholar 

  19. Noor M.A.: Iterative methods for nonlinear equations using homotopy perturbation technique. Appl. Math. Inf. Sci. 4, 227–235 (2010)

    MathSciNet  MATH  Google Scholar 

  20. Noor M.A., Noor K.I., Waseem M.: Fourth-order iterative methods for solving nonlinear equations. Int. J. Appl. Math. Eng. Sci. 4, 43–52 (2010)

    Google Scholar 

  21. Noor, M.A., Noor, K.I., Al-Said, E., Waseem, M.: Some new iterative methods for nonlinear equations. Math. Probl. Eng., 1–12 (2010). Article ID 198943

  22. Noor M.A., Shah F.A., Noor K.I., Al-Said E.: Variational iteration technique for finding multiple roots of nonlinear equations. Sci. Res. Essays 6, 1344–1350 (2011)

    Google Scholar 

  23. Noor M.A., Al-Said E., Mohyud-Din S.T.: A reliable algorithm for solving tenth-order boundary value problems. Appl. Math. Inf. Sci. 6, 103–107 (2012)

    MathSciNet  Google Scholar 

  24. Noor, M.A., Noor, K.I., Waseem, M.: Applications of quadrature formula for solving systems of nonlinear equations. Internat. J. Modern Phys. B (2012, accepted)

  25. Ortega J.M., Rheinboldt W.C.: Iterative Solution of Nonlinear Equations in several variables. Academic Press, New York (1970)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, COMSATS Institute of Information Technology, Park Road, Islamabad, Pakistan

    Muhammad Aslam Noor & Khalida Inayat Noor

  2. Department of Computer Science, COMSATS Institute of Information Technology, Sahiwal, Pakistan

    Muhammad Waseem

  3. Mathematics Department, College of Science, King Saud University, Riyadh, Saudi Arabia

    Eisa Al-Said

Authors
  1. Muhammad Aslam Noor
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Muhammad Waseem
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Khalida Inayat Noor
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Eisa Al-Said
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Muhammad Aslam Noor.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Noor, M.A., Waseem, M., Noor, K.I. et al. Variational iteration technique for solving a system of nonlinear equations. Optim Lett 7, 991–1007 (2013). https://doi.org/10.1007/s11590-012-0479-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-012-0479-3

Keywords

Search

Navigation