Analysis Of Solving Ordinary Differential Equations With A Comparison Of Adam-Bashforth Moulton And Milne-Simpson Methods

Abstract

Ordinary Differential Equations (ODEs) are mathematical models widely used in various fields of science and engineering to represent dynamic phenomena.. This study aims to compare the performance of two multi-step numerical methods, namely the Adams–Bashforth–Moulton (ABM) method and the Milne–Simpson (MS) method, in solving ordinary differential equations. The analysis was carried out by implementing both methods using the Python programming language and comparing their numerical results to the exact solution. Based on the simulation graphs, both methods produced results that closely matched the exact solution, with nearly overlapping curves throughout the time interval from zero to two. However, the absolute error analysis showed that the MS method generated smaller errors and a more stable error growth compared to the ABM method, especially at longer time steps. This indicates that although both methods are accurate, the Milne–Simpson method tends to be more stable over time. This study provides a comprehensive overview of the strengths of each method and can serve as a reference in selecting efficient and accurate numerical methods for solving ordinary differential equations.