New Fourth and Fifth-Order Iterative Methods for Solving Nonlinear Equations

Muhammad Saqib¹, Muhammad Iqbal¹, Shahid Ali¹, Tariq Ismaeel^2,3

¹Department of Mathematics, Lahore Leads University, Lahore, Pakistan.
²Department of Mathematics, GC University, Lahore, Pakistan.
³School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, China.

DOI: 10.4236/am.2015.68114                

Abstract

In this paper, we establish two new iterative methods of order four and five by using modified homotopy perturbation technique. We also present the convergence analysis of these iterative methods. To assess the validity and performance of these iterative methods, we have applied to solve some nonlinear problems.

Keywords

Iterative Methods, Homotopy Perturbation Technique, Order of Convergence, Nonlinear Equations

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Abbasbandy, S. (2003) Improving Newton-Raphson Method for Nonlinear Equations by Modified Adomain Decomposition Method. Applied Mathematics and Computation, 145, 887-893. http://dx.doi.org/10.1016/S0096-3003(03)00282-0
[2]	Adomain, G. (1989) Nonlinear Atochastic Systems and Applications to Physics. Kluwer Academy Publishers, Dordrecht. http://dx.doi.org/10.1007/978-94-009-2569-4
[3]	Chun, C. (2005) Iterative Methods Improving Newton’s Method by Decomposition Method. Applied Mathematics and Computation, 50, 1559-1568. http://dx.doi.org/10.1016/j.camwa.2005.08.022
[4]	Noor, M.A. and Inayat, K. (2006) Three-Step Iterative Methods for Nonlinear Equations. Applied Mathematics and Computation, 183, 322-327. http://dx.doi.org/10.1016/j.amc.2006.05.055
[5]	Noor, M.A., Inayat, K. and Tauseef, M.D. (2006) An Iterative Method with Cubic Convergence for Nonlinear Equations. Applied Mathematics and Computation, 183, 1249-1255. http://dx.doi.org/10.1016/j.amc.2006.05.133
[6]	Babolian, E. and Biazar, J. (2002) On the Order of Convergence of Adomain Method. Applied Mathematics and Computation, 130, 383-387. http://dx.doi.org/10.1016/S0096-3003(01)00103-5
[7]	Daftardar-Gejji, V. and Jafari, H. (2006) An Iterative Method for Solving Nonlinear Functional Equations. Journal of Mathematical Analysis and Applications, 316, 753-763. http://dx.doi.org/10.1016/j.jmaa.2005.05.009
[8]	Javidi, M. (2009) Fourth Order and Fifth Order Iterative Methods for Nonlinear Algebraic Equations. Mathematical and Computer Modelling, 50, 66-71.
[9]	Kang, S.M., et al. (2013) A New Second Order Iteration Method for Solving Nonlinear Equations. Handawi Publishing Company, Abstract and Applied Analysis, 2013, Article ID: 48706.
[10]	He, J.H. (1999) Homotopy Perturbation Technique. Computer Methods in Applied Mechanics and Engineering, 178, 257-262. http://dx.doi.org/10.1016/S0045-7825(99)00018-3