A convergent multi-step efficient iteration method to solve nonlinear equation systems

This work presents a novel high-order iterative procedure in solving nonlinear set of equations. The developed scheme, based on Jarratt-type methods, achieves tenth-order convergence without requiring the second Fréchet derivative of the function, thereby enhancing computational efficiency. The performance of the iterative scheme is demonstrated through error analysis and applications to nonlinear partial differential equations like the time-dependent Burgers’ and Fisher’s PDEs, highlighting its potential for solving such nonlinear systems.