Some Multistep Iterative Methods for Nonlinear Equation Using Quadrature Rule

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Gul Sana, Muhammad Aslam Noor, Khalida Inayat Noor, Some Multistep Iterative Methods for Nonlinear Equation Using Quadrature Rule, Int. J. Anal. Appl., 18 (6) (2020), 920-938.

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Gul Sana, Muhammad Aslam Noor, Khalida Inayat Noor

Abstract

We introduce a sequence of third and fourth order iterative schemes to determine the roots of nonlinear equations by applying quadrature formula and decomposition approach. We also examined the convergence of suggested iterative methods under varied constraint. Various numerical test examples are presented to exhibit the validity, efficiency and implementaion of our algorithms.

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References

  1. S. Abbasbandy, Improving Newton-Raphson method for nonlinear equations by modified Adomian decomposition method., Appl. Math. Comput. 145 (2003), 887-893.
  2. G. Adomian, Nonlinear stochastic systems and applications to physics, Kluwer Academic Publishers, Dordrecht, 1989.
  3. F. Ali, W. Aslam, A. Rafiq, Some new iterative techniques for the problems involving nonlinear equations, Int. J. Comput. Math. 16 (2019),1-18.
  4. E. Babolian, J. Biazar, A.R. Vahidi, Solution of a system of nonlinear equations by Adomian decomposition method, Appl. Math. Comput. 150 (3) (2004), 847-854.
  5. F.A. Shah, M. Darus, I. Faisal, M.A. Shafiq, Decomposition technique and family of efficient schemes for nonlinear equations, Discrete Dyn. Nat. Soc. 2017 (2017), 3794357.
  6. C. Chun, Iterative methods improving Newton's method by the decomposition method, Comput. Math. Appl. 50 (2005), 1559-1568.
  7. C. Chun, Y. Ham, Some fourth-order modifications of Newton's method, Appl. Math. Comput. 197 (2) (2008), 654-658.
  8. A. Cordero, J.R. Torregrosa, Variants of Newton's method using fifth-order quadrature formulas, Appl. Math. Comput. 190 (1) (2007), 686-698.
  9. M.T. Darvishi, A. Barati, A third-order newton-type method to solve systems of nonlinear equations, Appl. Math. Comput. 187 (2007) ,630-635.
  10. V. Daftardar-Gejji and H. Jafari, An iterative method for solving nonlinear functional equations, J. Math. Anal.and Appl. 316 (2) (2006), 753-763.
  11. M. Frontini, E. Sormani, Third-order methods from quadrature formulae for solving systems of nonlinear equations, Appl. Math. Comput. 149 (3) (2004), 771- 782.
  12. E. Halley, A new, exact and easy method for finding the roots of equations generally, and that without any previous reduction, Phil. R. Soc. Lond. 18 (1964), 136-147.
  13. A. Melman, Geometry and convergence of Halley's method, Siam Rev. 39 (4) (1997), 728-735.
  14. J.H. He, Variational iteration method some new results and new interpretations, J. Comput., Appl. Math. 207 (1) (2007), 3-17.
  15. J.H. He, Homotpy perturbation technique, Comp. Math. Appl. Mech. Eng. 178 (3-4) (1999), 257-262.
  16. J.H. He, A new iteration method for solving algebraic equations, Appl. Math. Comput. 135 (2003), 81-84.
  17. H.H. Homeier, On Newton-type methods with cubic convergence, J. Comput. Appl. Math. 176 (2) (2005), 425-432.
  18. V.I. Hasanov, I.G. Ivanov, and G. Nedzhibov, A new modification of Newton method, Appl. Math. Eng. 27 (2002), 278-286.
  19. M.A. Noor, K.I. Noor, S.T. Mohyud-Din, A. Shabbir, An iterative method with cubic convergence for nonlinear equations, Appl. Math. Comput. 183 (2006), 1249-1255.
  20. M.A. Noor, New iterative schemes for nonlinear equations, Appl. Math. Comput. 187 (2007), 937-943.
  21. M.A. Noor, K.I. Noor, E. Al-Said, M. Waseem, Some new iterative methods for nonlinear equations, Math. Probl. Eng. 2010, (2010), 198943.
  22. M.A. Noor, Some iterative methods for solving nonlinear equations using homotopy perturbation method, Int. J. Comput. Math. 87 (2010), 141-149.
  23. M.A. Noor, K.I. Noor, E. Al-Said, M. Waseem, Higher-order iterative algorithms for solving nonlinear equations, World Appl. Sci. J., 16 (2012), 1657-1663.
  24. M.A. Noor, M. Waseem, K.I. Noor, M.A. Ali, New iterative technique for solving nonlinear equations, Appl. Math. Comput. 265 (2015), 1115-1129.
  25. M.A. Noor, Fifth order convergent iterative method for solving nonlinear equation using quadrature formula, J. Math. Control Sci. Appl. 4 (1) (2018), 95-104.
  26. O. Ogbereyivwe, K.O. Muka, On the efficiency of family of quadrature based methods for solving nonlinear equations, Glob. Sci. J. 6 (9) (2018), 149-159.
  27. A.Y. Ozban, Some New Variants of Newton's method, Appl. Math. Lett. 17 (6) (2004), 677-682.
  28. S.M. Kang, A. Rafiq, Y.C. Kwun, A new second-order iteration method for solving nonlinear equations, Abstr. Appl. Anal. 2013 (2013), 48706.
  29. S.M. Kang, M. Saqib, M. Fahad, W. Nazeer, Two new third and fourth order algorithms for resolution of nonlinear scalar equations based on decomposition technique, Far East J. Math. Sci. 101 (3) (2017), 457-471.
  30. M. Saqib, M. Iqbal, Some multi-step iterative methods for solving nonlinear equations, Open J. Math. Sci. 1 (2017), 25-33.
  31. M. Saqib, M. Iqbal, S. Ali, T. Ismail, New fourth and fifth order iterative methods for solving nonlinear equations, Appl. Math. 6 (2015), 1220-1227.
  32. S. Weerakoon, T.G.I. Fernando, A variant of Newton's method with accelerated third-order convergence, Appl. Math. Letter., 13(8) (2000), 87-93.
  33. M. Waseem, M.A. Noor, K.I. Noor, F.A. Shah, K.I. Noor, An efficient technique to solve nonlinear equations using multiplicative calculus, Turk. J. Math. 42 (2018), 679-691.