Design and multidimensional extension of iterative methods for solving nonlinear problems

[EN] In this paper, a three-step iterative method with sixth-order local convergence for approximating the solution of a nonlinear system is presented. From Ostrowski¿s scheme adding one step of Newton with ¿frozen¿ derivative and by using a divided difference operator we construct an iterative scheme of order six for solving nonlinear systems. The computational efficiency of the new method is compared with some known ones, obtaining good conclusions. Numerical comparisons are made with other existing methods, on standard nonlinear systems and the classical 1D-Bratu problem by transforming it in a nonlinear system by using finite differences. From this numerical examples, we confirm the theoretical results and show the performance of the presented scheme.This research was partially supported by Ministerio de Economia y Competitividad MTM2014-52016-C2-2-P and FONDOCYT 2014-1C1-088 Republica Dominicana.Artidiello, S.; Cordero Barbero, A.; Torregrosa Sánchez, JR.; Vassileva, MP. (2017). Design and multidimensional extension of iterative methods for solving nonlinear problems. Applied Mathematics and Computation. 293:194-203. https://doi.org/10.1016/j.amc.2016.08.034S19420329

Stable high-order iterative methods for solving nonlinear models

[EN] There are several problems of pure and applied science which can be studied in the unified framework of the scalar and vectorial nonlinear equations. In this paper, we propose a sixth-order family of Jarratt type methods for solving nonlinear equations. Further, we extend this family to the multidimensional case preserving the order of convergence. Their theoretical and computational properties are fully investigated along with two main theorems describing the order of convergence. We use complex dynamics techniques in order to select, among the elements of this class of iterative methods, those more stable. This process is made by analyzing the conjugacy class, calculating the fixed and critical points and getting conclusions from parameter and dynamical planes. For the implementation of the proposed schemes for system of nonlinear equations, we consider some applied science problems namely, Van der Pol problem, kinematic syntheses, etc. Further, we compare them with existing sixth-order methods to check the validity of the theoretical results. From the numerical experiments, we find that our proposed schemes perform better than the existing ones. Further, we also consider a variety of nonlinear equations to check the performance of the proposed methods for scalar equations.This research was partially supported by Ministerio de Economia y Competitividad MTM2014-52016-C2-2-P and by Generalitat Valenciana PROMETEO/2016/089.Behl, R.; Cordero Barbero, A.; Motsa, SS.; Torregrosa Sánchez, JR. (2017). Stable high-order iterative methods for solving nonlinear models. Applied Mathematics and Computation. 303:70-88. https://doi.org/10.1016/j.amc.2017.01.029S708830

A new efficient parametric family of iterative methods for solving nonlinear systems

[EN] A bi-parametric family of iterative schemes for solving nonlinear systems is presented. We prove for any value of parameters the sixth-order of convergence of any members of the class. The efficiency and computational efficiency indices are studied for this family and compared with that of the other known schemes with similar structure. In the numerical section, we solve, after discretizating, the nonlinear boundary problem described by the Fisher's equation. This numerical example confirms the theoretical results and show the performance of the proposed schemes.This research was partially supported by both Ministerio de Ciencia, Innovacion y Universidades and Generalitat Valenciana [grant numbers PGC2018-095896-B-C22 and PROMETEO/2016/089], respectively. The authors would like to thank the anonymous reviewers for their helpful comments and suggestions.Chicharro, FI.; Cordero Barbero, A.; Garrido-Saez, N.; Torregrosa Sánchez, JR. (2019). A new efficient parametric family of iterative methods for solving nonlinear systems. The Journal of Difference Equations and Applications. 25(9-10):1454-1467. https://doi.org/10.1080/10236198.2019.1665653S14541467259-10Amat, S., & Busquier, S. (2017). After notes on Chebyshev’s iterative method. Applied Mathematics and Nonlinear Sciences, 2(1), 1-12. doi:10.21042/amns.2017.1.00001Amiri, A. R., Cordero, A., Darvishi, M. T., & Torregrosa, J. R. (2018). Preserving the order of convergence: Low-complexity Jacobian-free iterative schemes for solving nonlinear systems. Journal of Computational and Applied Mathematics, 337, 87-97. doi:10.1016/j.cam.2018.01.004Awawdeh, F. (2009). On new iterative method for solving systems of nonlinear equations. Numerical Algorithms, 54(3), 395-409. doi:10.1007/s11075-009-9342-8Cordero, A., Hueso, J. L., Martínez, E., & Torregrosa, J. R. (2009). A modified Newton-Jarratt’s composition. Numerical Algorithms, 55(1), 87-99. doi:10.1007/s11075-009-9359-zCordero, A., Gómez, E., & Torregrosa, J. R. (2017). Efficient High-Order Iterative Methods for Solving Nonlinear Systems and Their Application on Heat Conduction Problems. Complexity, 2017, 1-11. doi:10.1155/2017/6457532Cordero, A., Jordán, C., Sanabria-Codesal, E., & Torregrosa, J. R. (2018). Highly efficient iterative algorithms for solving nonlinear systems with arbitrary order of convergence p+3, p≥5. Journal of Computational and Applied Mathematics, 330, 748-758. doi:10.1016/j.cam.2017.02.032Grau-Sánchez, M., Grau, À., & Noguera, M. (2011). Ostrowski type methods for solving systems of nonlinear equations. Applied Mathematics and Computation, 218(6), 2377-2385. doi:10.1016/j.amc.2011.08.011Grosan, C., & Abraham, A. (2008). A New Approach for Solving Nonlinear Equations Systems. IEEE Transactions on Systems, Man, and Cybernetics - Part A: Systems and Humans, 38(3), 698-714. doi:10.1109/tsmca.2008.918599Hueso, J. L., Martínez, E., & Teruel, C. (2015). Convergence, efficiency and dynamics of new fourth and sixth order families of iterative methods for nonlinear systems. Journal of Computational and Applied Mathematics, 275, 412-420. doi:10.1016/j.cam.2014.06.010Khalique, C. M., & Mhlanga, I. E. (2018). Travelling waves and conservation laws of a (2+1)-dimensional coupling system with Korteweg-de Vries equation. Applied Mathematics and Nonlinear Sciences, 3(1), 241-254. doi:10.21042/amns.2018.1.00018Sharma, J. R., & Arora, H. (2013). Efficient Jarratt-like methods for solving systems of nonlinear equations. Calcolo, 51(1), 193-210. doi:10.1007/s10092-013-0097-1Soleymani, F., Lotfi, T., & Bakhtiari, P. (2013). A multi-step class of iterative methods for nonlinear systems. Optimization Letters, 8(3), 1001-1015. doi:10.1007/s11590-013-0617-6Wang, X., Zhang, T., Qian, W., & Teng, M. (2015). Seventh-order derivative-free iterative method for solving nonlinear systems. Numerical Algorithms, 70(3), 545-558. doi:10.1007/s11075-015-9960-2Xiao, X. Y., & Yin, H. W. (2015). Increasing the order of convergence for iterative methods to solve nonlinear systems. Calcolo, 53(3), 285-300. doi:10.1007/s10092-015-0149-