on efficient iterative schemes for finding all solutions of non-linear engineering problems

Nonlinear equations are important in engineering because they may simulate complicated real-world phenomena such as fluid dynamics, material stress, and electrical circuits, where linear assumptions fail. they allow engineers to more correctly estimate how a system will respond in certain conditions. the main aim of this effort is to develop an efficient higher-order simultaneous computer approach capable of computing all solutions concurrently. the convergence theorem analysis indicates that the scheme has a local convergence order of 10. using a few engineering applications, we show that the order strategy surpasses the current approach in terms of residual error, stability, and consistency.