Highly Accurate and Stable Class of Zero‐Finding Numerical Methods Based on Daftardar–Jafari Decomposition Approach for Engineering Applications

This study introduces a class of zero‐finding numerical methods that integrates the Daftardar–Jafari decomposition approach with the midpoint quadrature rule. This hybrid formulation enhances both stability and accuracy by coupling analytical decomposition with a balanced numerical integration scheme. A detailed convergence analysis—via Taylor and semilocal frameworks—demonstrates third‐ and fourth‐order convergence, achieving up to average absolute error across benchmark nonlinear problems. The stability of the proposed solvers is visualized through polynomiography, confirming robust attraction basins compared to classical Newton‐type methods. Extensive simulations on engineering and physical models (including fluid dynamics and boundary value problems) reveal faster convergence and improved efficiency indices. The paper concludes with insights on extending the method to multivariable systems and potential future developments.