New higher order iterative methods for solving nonlinear equations

Year 2018, Volume: 47 Issue: 1, 77 - 91, 01.02.2018

Shuliang Huang , Arif Rafiq Muhammad Rizwan Shahzad Faisal Ali

Abstract

In this paper, using the system of coupled equations involving an auxiliary function, we introduce some new efficient higher order iterative methods based on modified homotopy perturbation method. We study the convergence analysis and also present various numerical examples to  demonstrate the validity and efficiency of our methods.

Keywords

Iterative methods , Nonlinear equations , Order of convergence , Auxiliary function , Modified homotopy perturbation method

References

• S. Abbasbandy, Improving Newton-Raphson method for nonlinear equations by modified Adomian decomposition method, Appl. Math. Comput. 145 (2003) 887-893.
• I.K. Argyros, D. Chen, Q. Qian, The Jarrat method in Banach space setting, J. Comput. Appl. Math. 51 (1994) 103-106.
• G. Adomian, Nonlinear Stochastic system and applications to physics, Kluwer Academic Publishers, Dordrecht, 1989.
• E. Babolian and J. Biazar, Solution of nonlinear equations by modified Adomian decomposition method, Appl. Math. Comput. 132 (2002), 167-172.
• C. Chun, Construction of Newton-like iterative methods for solving nonlinear equations, Numer. Math. 104 (2006) 297-315.
• V. Daftardar-Gejji, H. Jafari, An iterative method for solving nonlinear functional equations, J. Anal. 316 (2006), 753-763.
• A. Golbabai, M. Javidi, A third-order Newton type method for nonlinear equations based on modified homotopy perturbation method, Appl. Math. Comput. 191(2007), 199-205.
• Y. Ham, C. Chun, A fifth order iterative method for solving nonlinear equations, Appl. Math. Comput. 194(2007), 287-290
• J. H. He, A new iteratration method for solving algebraic equations, Appl. Math. Comput. 135 (2003) 81-84.
• J. H. He, Homotopy perturbation technique, Comput. Methods Appl. Mech. Eng. 178(3-4) (1999), 257-262.
• J. H. He, A coupling method of a homotopy technique and a perturbation technique for non-linear problems, Int. J. Non-Linear Mech. 35(1) (2000), 37-43.
• J. H. He, The homotopy perturbation method for non-linear oscillators with discontinuities, Appl. Math. Comput. 151(2004), 287-292.
• J. H. He, Application of homotopy perturbation method to nonlinear wave equations, Chaos Solitons Fractals 26(3) (2005), 695-700.
• J. H. He, Asymptotology by homotopy perturbation method, Appl. Math. Comput. 156(3) (2004), 591-596.
• J. H. He, Homotopy perturbation method for solving boundary problems, Phys. Lett. A. 350(1-2) (2006), 87-88.
• J. H. He, Limit cycle and bifurcation of nonlinear problems, Chaos Solitons Fractals 26(3) (2005), 827-833.
• J. H. He, Variational iteration method-some recent results and new interpretations, J. Appl. Math. Comput. 207 (2007) 3-17.
• M. Javidi, Fourth-order and fifth-order iterative methods for nonlinear algebraic equations, Math. Comput. Modelling 50 (2009) 66-71.
• M. A. Noor, New Classes of iterative methods for nonlinear equations, Appl. Math. Comput. 191 (2007) 128-131.
• A.M. Ostrowski, Solution of equations and system of equations, Academic press, New York, 1966.
• M. Rafiullah, A Fifth-order Iterative Method for Solving Nonlinear Equations, Sibirskii Zhurnal Vychislitel’noi Mathematiki. 14(3) (2011), 297-302.
• F.A. Shah, M.A. Noor, Some numerical methods for solving nonlinear equations by using decomposition technique, Appl. Math. Comput. 251 (2015), 378-386.
• J.F. Traub, Iterative Methods for the Solution of Equations, Prentice-Hall Englewood Cliffs, New Jersey, USA, 1964.