Further development on Traub’s method for solving system of nonlinear equations and ODE’s

The foremost objective of this work is to propose a eighth and sixteenth order scheme for handling a nonlinear equation. The eighth order method uses three evaluations of the function and one assessment of the first derivative and sixteenth order method uses four evaluations of the function and one appraisal of the first derivative. Kung-Traub conjecture is satisfied, theoretical analysis of the methods are presented and numerical examples are added to confirm the order of convergence. The performance and efficiency of our iteration methods are compared with the equivalent existing methods on some standard academic problems. We tested projectile motion problem, Planck’s radiation law problem as an application. The basins of attraction are also given to demonstrate their dynamical behavior in the complex plane. Further, we attempt to proposed a sixteenth order iterative method for solving system of nonlinear equation with four functional evaluation, namely two F and two F 0 and only one inverse of Jacobian. The theoretical proof of the method is given and numerical examples are included to confirm the convergence order of the presented methods. We apply the new scheme to find solution on 1-D bratu problem. The performance and efficiency of our iteration methods are compared.