Sixth new family method for solving nonlinear equations
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Abstract
Introduction. In this manuscript, we introduce a new multi-step iterative family designed to solve non linear equations.
Materials and Methods. Furthermore, we highly investigate the analysis of convergence from this new iterative family once the multi-step method was defined. We demonstrated that its order of convergence was 6 and its approximated computational order of convergence was 5.98 in majority of numerical experiments.
Results and Discussion. This family is derived from Newtons scheme, but it includes a ‘’fronzen’’ or weight function to add up a third step. Initially, we used as first iterative method the second order Newtons method with the aim to reduce the number of functional evaluations.
Conclusions. Finally, we chose a set of nonlinear equations from recent investigations with the aim to evaluate its efficacy from the new iterative family verifying the amount of iterations to find the approximate solution of the nonlinear equation in comparison to other already proposed sixth order methods achieving a lower computational cost which allow to be an efficient and novel iterative scheme.
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Ali, K. K., Mehanna, M. S., Shaalan, M. A., Nisar, K. S., Albalawi, W., & Abdel-Aty, A.-H. (2023). Analytical and numerical solutions with bifurcation analysis for the nonlinear evolution equation in (2+1)-dimensions. Results in Physics, 49, 106495. https://doi.org/10.1016/j.rinp.2023.106495
Alipour, S., & Mirzaee, F. (2020). An iterative algorithm for solving two dimensional nonlinear stochastic integral equations: A combined successive approximations method with bilinear spline interpolation. Applied Mathematics and Computation, 371, 124947. https://doi.org/10.1016/j.amc.2019.124947
Caporale, A., Vaccaro, M. S., Barretta, R., & Luciano, R. (2025). Nonlocal elastic plate problems via iterative method. Mechanics Research Communications, 150, 104538. https://doi.org/10.1016/j.mechrescom.2025.104538
Caprais, M., & Bergeron, A. (2025a). An iterative scheme to include turbulent diffusion in advective-dominated transport of delayed neutron precursors. Annals of Nuclear Energy, 215, 111251. https://doi.org/10.1016/j.anucene.2025.111251
Caprais, M., & Bergeron, A. (2025b). An iterative scheme to include turbulent diffusion in advective - dominated transport of delayed neutro precursors. 10.
Cordero, A., Hueso, J. L., Martínez, E., & Torregrosa, J. R. (2010). A family of iterative methods with sixth and seventh order convergence for nonlinear equations. Mathematical and Computer Modelling, 52(9–10), 1490–1496. https://doi.org/10.1016/j.mcm.2010.05.033
Erfanifar, R., Sayevand, K., & Esmaeili, H. (2020). A novel iterative method for the solution of a nonlinear matrix equation. Applied Numerical Mathematics, 153, 503–518. https://doi.org/10.1016/j.apnum.2020.03.012
Ibrahim, A. H., & Kumam, P. (2021). Re-modified derivative-free iterative method for nonlinear monotone equations with convex constraints. Ain Shams Engineering Journal, 12(2), 2205–2210. https://doi.org/10.1016/j.asej.2020.11.009
Kodnyanko, V. (2021). Improved bracketing parabolic method for numerical solution of nonlinear equations. Applied Mathematics and Computation, 400, 1–6. https://doi.org/10.1016/j.amc.2021.125995
Kozitskiy, S. B., Trofimov, M. Y., & Petrov, P. S. (2022). On the numerical solution of the iterative parabolic equations by ETDRK pseudospectral methods in linear and nonlinear media. Communications in Nonlinear Science and Numerical Simulation, 108, 106228. https://doi.org/10.1016/j.cnsns.2021.106228
Li, H., & Guo, Y. (2020). Numerical solution of coupled nonlinear Schrödinger equations on unbounded domains. Applied Mathematics Letters, 104, 106286. https://doi.org/10.1016/j.aml.2020.106286
Mohanty, R. K., & Niranjan. (2024). A class of new implicit compact sixth-order approximations for Poisson equations and the estimates of normal derivatives in multi-dimensions. Results in Applied Mathematics, 22, 100454. https://doi.org/10.1016/j.rinam.2024.100454
Nirmala, A., & Kumbinarasaiah, S. (2024). Numerical solution of nonlinear Hunter - Saxton equation, Benjamin - Bona Mahony equation, and Klein - Gordon equation using Hosoya polynomial method. Control and Optimization, 26.
Özban, A. Y., & Kaya, B. (2022). A new family of optimal fourth-order iterative methods for nonlinear equations. Results in Control and Optimization, 8(July). https://doi.org/10.1016/j.rico.2022.100157
Pokusiński, B., & Kamiński, M. (2023). Numerical convergence and error analysis for the truncated iterative generalized stochastic perturbation-based finite element method. Computer Methods in Applied Mechanics and Engineering, 410, 115993. https://doi.org/10.1016/j.cma.2023.115993
Rawani, M. K., Verma, A. K., & Cattani, C. (2023). A novel hybrid approach for computing numerical solution of the time-fractional nonlinear one and two-dimensional partial integro-differential equation. Communications in Nonlinear Science and Numerical Simulation, 118, 106986. https://doi.org/10.1016/j.cnsns.2022.106986
Sharma, E., & Panday, S. (2022). Efficient sixth order iterative method free from higher derivatives for nonlinear equations. Journal of Mathematical and Computational Science, 1–13. https://doi.org/10.28919/jmcs/6950
Solaiman, O. S., Karim, S. A. A., & Hashim, I. (2021). Dynamical comparison of several third-order iterative methods for nonlinear equations. Computers, Materials and Continua, 67(2), 1951–1962. https://doi.org/10.32604/cmc.2021.015344
Thangkhenpau, G., Panday, S., Panday, B., Stoenoiu, C. E., & Jäntschi, L. (2024). Generalized high-order iterative methods for solutions of nonlinear systems and their applications. AIMS Mathematics, 9(3), 6161–6182. https://doi.org/10.3934/math.2024301
Usman, M., Iqbal, J., Khan, A., Ullah, I., Khan, H., Alzabut, J., & Alkhawar, H. M. (2025). A new iterative multi-step method for solving nonlinear equation. MethodsX, 15(March). https://doi.org/10.1016/j.mex.2025.103394
Zhanlav, T., & Otgondorj, K. (2021). Higher order Jarratt-like iterations for solving systems of nonlinear equations. Applied Mathematics and Computation, 395, 125849. https://doi.org/10.1016/j.amc.2020.125849