An Efficient Four Step Fifteenth Order Method for Solution of Non-Linear Models in Real-World Problems
DOI:
https://doi.org/10.53560/PPASA(61-3)852Keywords:
Real Word Problems, Taylor Series Expansion, Order of Convergence, Fifteenth Order Methods, Basin of AttractionAbstract
Non-linear equations are fundamental to a wide range of practical applications in engineering and applied sciences. This research paper presents a novel iterative scheme—a fifteenth-order approach—designed to effectively solve non-linear problems. The numerical results of the Proposed Scheme are thoroughly compared with those of existing methods. Graphical representations and basin of attraction analysis reveal that the fifteenth-order method achieves superior accuracy and efficiency, surpassing alternative methods in the precise estimation of solutions to non-linear problems.
References
S. Thota and P. Shanmugasundaram. On new sixth and seventh order iterative methods for solving non-linear equations using homotopy perturbation technique. BMC Research Notes 15(1): 1-15 (2022).
Z. Xiaojian. Modified Chebyshev-Halley methods free from second derivative. Applied Mathematics and Computation 203(2): 824-827 (2008).
F.A. Lakho, Z.A. Kalhoro, S. Jamali, A.W. Shaikh, and J. Guan. A three steps seventh order iterative method for solution nonlinear equation using Lagrange Interpolation technique. VFAST Transactions on Mathematics 12(1): 46-59 (2024).
A. Naseem, M.A. Rehman, and J. Younis. A New Root-Finding Algorithm for Solving Real-World Problems and Its Complex Dynamics via Computer Technology. Complexity 2021: 1-10 (2021).
Z. Abbasi, Z.A. Kalhoro, S. Jamali, A.W. Shaikh, and O.A. Rajput. A novel Approach for Real-World Problems Based on Hermite Interpolation Technique and Analysis Using Basins of Attraction. The Sciencetech. 5(3): 112-126 (2024).
M.I. Soomro, Z.A. Kalhoro, A.W. Shaikh, S. Jamali, and O. Ali. Modified Bracketing Iterative Method for Solving Nonlinear Equations. VFAST Transactions on Mathematics 12(1): 105-120 (2024).
T. Eftekhari. A new family of four-step fifteenth-order root-finding methods with high efficiency index. Computational Methods for Differential Equations 3(1): 51-58 (2015).
F. Soleymani and M. Sharifi. On a class of fifteenth-order iterative formulas for simple roots. International Electronic Journal of Pure and Applied Mathematics 3(3): 245-252 (2011).
S. Jamali, Z.A. Kalhoro, A.W. Shaikh, and M.S. Chandio. An iterative, bracketing & derivative-free method for numerical solution of non-linear equations using stirling interpolation technique. Journal of Mechanics of Continua and Mathematical Sciences 16(6): 13-27 (2021).
Z. Kovacs. Understanding convergence and stability of the Newton-Raphson method. Teaching Mathematics and Statistics in Sciences: 1-12 (2011). http://www.model.u-szeged.hu/data/etc/edoc/imp/ZKovacs/ZKovacs.pdf
M. Kumar, A.K. Singh, and A. Srivastava. Various Newton-type iterative methods for solving nonlinear equations. Journal of the Egyptian Mathematical Society 21(3): 334-339 (2013).
S. Jamali, Z.A. Kalhoro, A.W. Shaikh, M.S. Chandio, and S. Dehraj. A novel two point optimal derivative free method for numerical solution of nonlinear algebraic, transcendental Equations and application problems using weight function. VFAST Transactions on Mathematics 10(2): 137-146 (2022).
P. Sivakumar and J. Jayaraman. Some New Higher Order Weighted Newton Methods for Solving Nonlinear Equation with Applications. Mathematical and Computational Applications 24(2): 1-16 (2019).
M. Grau-Sánchez, M. Noguera, À. Grau, and J.R. Herrero. On new computational local orders of convergence. Applied Mathematics Letters 25(12): 2023-2030 (2012).
C.S. Liu and T.L. Lee. A New Family of Fourth-Order Optimal Iterative Schemes and Remark on Kung and Traub’s Conjecture. Journal of Mathematics 2021: 5516694 (2021).
H. fei Ding, Y. xin Zhang, S. fu Wang, and X. ya Yang. A note on some quadrature based three-step iterative methods for non-linear equations. Applied Mathematics and Computation 215(1): 53-57 (2009).
A. Naseem, M.A. Rehman, and T. Abdeljawad. Computational methods for non-linear equations with some real-world applications and their graphical analysis. Intelligent Automation and Soft Computing 30(3): 805-819 (2021).
