A highly accurate family of stable and convergent numerical solvers based on Daftardar-Gejji and Jafari decomposition technique for systems of nonlinear equations

This study introduces a family of root-solvers for systems of nonlinear equations, leveraging the Daftardar-Gejji and Jafari Decomposition Technique coupled with the midpoint quadrature rule. Despite the existing application of these root solvers to single-variable equations, their extension to systems of nonlinear equations marks a pioneering advancement. Through meticulous derivation, this work not only expands the utility of these root solvers but also presents a comprehensive analysis of their stability and semilocal convergence; two areas of study missing in the existing literature. The convergence of the proposed solvers is rigorously established using Taylor series expansions and the Banach Fixed Point Theorem, providing a solid theoretical foundation for semilocal convergence guarantees. Additionally, a detailed stability analysis further underscores the robustness of these solvers in various computational scenarios. The practical efficacy and applicability of the developed methods are demonstrated through the resolution of five real-world application problems, underscoring their potential in addressing complex nonlinear systems. This research fills a significant gap in the literature by offering a thorough investigation into the stability and convergence of these root solvers when applied to nonlinear systems, setting the stage for further explorations and applications in the field. center dot The proposed methods combine the Daftardar-Gejji and Jafari Decomposition Technique and the midpoint quadrature rule for solving systems of nonlinear equations. center dot The numerical results show the superiority of the proposed methods over some other third- and fourth-order convergent methods from the literature. center dot The proposed methods can be used in various contexts and many real-world applications.