Advancements in Solving Higher-Order Ordinary Differential Equations via the Variational Iterative Method.

Abstract

This study presents advancements in solving higher-order ordinary differential equations (ODEs) using the Variational Iterative Method (VIM) and compares its performance with the New Iteration Method (NIM) and Adomian Decomposition Method (ADM). ODEs are critical in modeling the rate of change in various systems over time, and many do not have exact solutions, necessitating the use of numerical methods to obtain approximate results. While several iterative methods exist, a detailed comparison of VIM with other techniques, particularly for higher-order ODEs, is still lacking. This research focuses on understanding the principles and methodology of VIM and applying it to solve higher-order linear and nonlinear ODEs. The study evaluates the accuracy, convergence rate, and computational efficiency of VIM compared to NIM and ADM through the solution of third, fourth, and fifth-order differential problems. The results show that VIM outperforms NIM and ADM, with faster convergence and higher efficiency. Error analysis in Figures 1, 2, and 3 highlights the strengths and limitations of each method, providing valuable insights for researchers and practitioners in selecting the most appropriate technique for solving higher-order ODEs. These findings advance the development of iterative methods in numerical analysis and contribute to progress in the field of differential equations.