This study presents two efficient iterative methods for solving nonlinear equations using Adomian decomposition technique and a quadrature rule. The proposed methods are free of second derivatives and exhibit a higher order of convergence through a predictor-corrector scheme. Their effectiveness is verified by testing them on various mathematical models and comparing their performance against several well-known methods. Graphical analysis is also presented to validate the numerical results. MSC2020: 65H05, 65B99

In this study, we suggest and analyze two new one-parameter families of an efficient iterative methods free from higher derivatives for solving nonlinear equations based on Newton theorem of calculus and Bernstein quadrature formula, Bernoulli polynomial basis, Taylor’s expansion and some numerical techniques. We prove that the new iterative methods reach orders of convergence ten with six and eight with four functional evaluations per iteration, which implies that the efficiency index of the new iterative methods is (10)1/6 1.4678 and (8)1/4 1.6818 respectively. Numerical examples are provided to show the efficiency and performance of our iterative methods, compare to Newton’s method and other relevant methods.