A Hybrid Third-Order Iterative Process To Solve Nonlinear Equations

Year 2020, Volume: 10 Issue: 1, 563 - 575, 01.03.2020

Kadri Doğan

https://doi.org/10.21597/jist.583528

Abstract

In this study, by using the iterative method discussed in (Kang et al., 2013) and adopting a technique given in details (Biazar and Amirteimoori, 2006) introduced a new hybrid third-order iterative method to solve nonlinear equations derived from the Picard-Mann fixed-point iterative method. Some problems have been solved in order to demonstrate the performance of the established iterative method for the solution of the nonlinear equations.

Keywords

Fixed Point Method, Picard-Mann Iterative Method, Newton Iterative Method, Nonlinear Equations, Convergence Analysis

Supporting Institution

Artvin Coruh Univesity

References

• Abbas M, Nazir T, 2014. A new faster iteration process applied to constrained minimization and feasibility problems, Matematicki Vesnik 66(2): 223-234.
• Ashiq A, Qaisar M, Tanveer M, Aslam A, NazeerW, 2015. Modified new third-order iterative method for non-linear equations, Sci.Int.(Lahore), 27(3), 1741-1744, 2015.
• Babolian E, Biazar J, 2002. Solution of nonlinear equations by modified Adomian decomposition method, Appl. Math. Comput. 132 (1): 167–172.
• Berinde V, 2014. Picard iteration converges faster than Mann iteration for a class of quasi-contractive operators, Fixed Point Theory Appl. 2014 (2004):1.
• Biazar J, Amirteimoori A, 2006. An improvement to the fixed point iterative method, Appl. Math. Comput. 182 (1): 567–571.
• Chugh R, Malik P, Kumar V, 2015. On a new faster implicit xed point iterative scheme in convex metric spaces, J. Function Spaces (2015), Article ID 905834.
• Dogan K, Karakaya V, 2018. A study in the fixed point theory for a new iterative scheme and a class of generalized mapings, Creat. Math. Inform. 27(2018), No. 2, 151-160.
• Fukhar-ud-din H, Berinde V, 2016. Iterative methods for the class of quasi-contractive type operators and comparsion of their rate of convergence in convex metric spaces, Filomat 30 (2016) 223-230.
• Isaacson E, Keller HB, 1966. Analysis of Numerical Methods, John Wiley & Sons, Inc., No:1, 106-108 New York, USA.
• Khan SH, 2013. A Picard-Mann hybrid iterative process. Fixed Point Theory Appl. (2013). doi:10.1186/1687-1812-2013-69.
• Kang SM, Rafiq A, and Kwun Y C, 2013. A New Second-Order Iteration Method for Solving Nonlinear Equations, Abstract and Applied Analysis. Volume 2013, Article ID 487062, 4 pages.
• Karakaya V, Dogan K, Gursoy F, Erturk M, 2013. Fixed point of a new three-step iteration algorithm under contractive-like operators over normed spaces, Abstract and Applied Analysis, 2013(2013), 9 pages.
• Karakaya V, Dogan K, 2014. On the convergence and stability results for a new general iterative process, The Scientific World Journal, 2014(2014), 8 pages.
• Karakaya V, Atalan Y, Dogan K, Bouzara N. El Houda, 2017. Some fixed point results for a new three steps iteration process in Banach spaces, Fixed Point Theory, 18(2017), No. 2, 625-640.
• Picard E, 1890. Memoire sur la theorie des equations aux derivees partielles et la methode des approximations successives, J. Math. Pures Appl., 6(1890), 145-210.
• Phuengrattana W, Suantai S, 2013. Comparison of the rate of convergence of various iterative methods for the class of weak contractions in Banach Spaces, Thai J. Math. 11 (2013) 217-226.