Convergence and efficiency of high order parallel schemes for nonlinear engineering models

A novel technique for solving nonlinear equations-a basic problem in science and engineering-is presented in this work. This study tackles the drawbacks of conventional analysis, which frequently rely on Taylor series expansions and higher-order derivative assumptions, while highlighting the importance of high-order convergence techniques in computational mathematics. We expand the theoretical framework by concentrating solely on first-derivative-based local convergence analysis, whereas earlier research used Taylor expansions up to the eighth derivative to analyze an optimal family of eighth-order numerical scheme, even though they only used the first derivative in calculations. In addition, the proposed method is extended to parallel methods capable of finding all nonlinear equation solutions simultaneously with and without derivative evaluations; these developed techniques achieve convergence orders eighth and twelfth, respectively. A Local Convergence Theorem is used to define the domain of the convergence radius, which gives a systematic method for selecting optimal initial guesses and enhancing convergence behavior. Numerical outcomes on benchmark engineering problems reveal that the proposed methods are more efficient and stable as compared exiting method solutions, particularly in terms of residual error, error graph, computational time in seconds, percentage computational efficiency and computational cost. These results assist the development of advanced high-order iterative techniques that improve nonlinear problem-solving accuracy and efficiency while simultaneously ensuring resilience and theoretical soundness.