Assessment of Numerical Performance of Some Runge-Kutta Methods and New Iteration Method on First Order Differential Problems

This research focuses on the assessment of the numerical performance of some Runge-Kutta methods and New Iteration Method “NIM”  for solving first-order differential problems. The assessment is conducted through extensive numerical experiments and comparative  analyses. Accuracy, efficiency, and stability are among the key factors considered in evaluating the performance of the methods. A range  of first-order differential problems with diverse characteristics and complexity levels is employed to thoroughly examine the methods'  capabilities and limitations. The numerical investigation that is defined in the study as well as the results that are stated in the Tables,  demonstrates that all the approaches produce extremely accurate results. However, the “NIM” was shown to be the most effective of the  three methods used in this study. Conclusively, the “NIM” should be employed to solve first-order nonlinear and linear ordinary  differential equations in place of Runge-Kutta Fourth order method (RK4M) and Butcher Runge-Kutta Fifth order method (BRK5M). In  addition, BRK5M is more applicable and efficient than RK4M when solving first order ordinary differential problems.