New Family of Multi-Step Iterative Methods Based on Homotopy Perturbation Technique for Solving Nonlinear Equations

Abstract

This research aims to propose a new family of one-parameter multi-step iterative methods that combine the homotopy perturbation method with a quadrature formula for solving nonlinear equations. The proposed methods are based on a higher-order convergence scheme that allows for faster and more efficient convergence compared to existing methods. It aims also to demonstrate that the efficiency index of the proposed iterative methods can reach up to 43≈1.587 and 84≈1.681, respectively, indicating a high degree of accuracy and efficiency in solving nonlinear equations. To evaluate the effectiveness of the suggested methods, several numerical examples including their performance are provided and compared with existing methods.

Generalized Convergence for Multi-Step Schemes under Weak Conditions

“…This highlights the effectiveness and reliability of our technique for solving complex and large systems of nonlinear equations. The methodology can be applied to other methods [4,5,8,11,16] in order to extend their applicability along the same lines. This is the direction of future research.…”

“…so the first assertion holds in (5), and iterate z m ∈ U(x 0 , b * ). Then, by the third substep of (2), condition (H 2 ), and Formula (2), we have in turn the triangle inequality ∥z (1)…”

“…The computed outcomes for local convergence are detailed in Tables 1 and 2, whereas Tables 3 and 4 present the results for semi-local convergence. Furthermore, we determine the coefficient of convergence (COC) [4,5] using the equations…”

“…Nowadays, nonlinear equation systems are an essential tool in the scientific community for solving challenging real-world issues [1][2][3][4][5]. These equations, characterized by their nonlinearity in variables, find applications in a wide range of disciplines, from physics and biology to economics and engineering [6][7][8][9].…”

Application of homotopy perturbation method to solve a nonlinear mathematical model of depletion of forest resources

“…In 2008, He employed the HPM to solve boundary value problems [20]. In 2007, Javidi and Golbabai used a revised version of the HPM to solve non-linear Fredholm integral equations [21].Recently, HPM with small variations has been applied to study fractal duffing oscillator problems under arbitrary conditions [22], modified HPM for nonlinear oscillators Anjum and He [23], attachment oscillator arising in nanotechnology [24], conservative nonlinear oscillators [25], non-linear oscillator problems in a fractal space [26] and HPM including Aboodh transformation to solve fractional calculus Tao et al [27], vibrating magnetic inverted pendulum Moatimid et al [28], Symmetry-breaking and pull-down motion for the helmholtz-duffing oscillator Niu et al [29], nonlinear fractional Drinfeld-Sokolov-Wilson Equation Nadeem and Alsayaad [30], trajectory analysis of a zero-pitch-angle e-Sail Niccolai et al [31], natural convection between two concentric horizontal circular cylinders Abdulameer and Ali Al-Saif [32], nonlocal initialboundary value problems for parabolic and hyperbolic Al-Hayani and Younis [33], multi-step iterative methods for solving nonlinear equations Saeed et al [34], telegraph equation Moazzzam et al [35], triangular linear diophantine fuzzy system of equations Shams et al [36], condensing coagulation model and Lifshitz-Slyzov equation Arora et al [37], singular nonlinear system of boundary value problems Pathak et al [38], rikitake-yype system Ene and Pop [39], heat and mass transfer with 2D unsteady squeezing viscous flow problem Abdul-Ameer and Ali Al-Saif [40], variable Speed Wind Turbine Control Shalbafian and Ganjefar [41], radial thrust problem Niccolai et al [42], special third grade fluid flow with viscous dissipation effect over a stretching sheet Swain et al [43], and the frequency-amplitude relationship of a nonlinear oscillator with cubic and quintic nonlinearities He et al [44]. The HPM has become a widely-used technique to solve a large variety of problems in different fields and many research papers have been published each year using this method as evidenced by a simple search on Google Scholar.…”