Numerical solution of second order ordinary differential equations

Numerical analysis is a modern technology employed by scientists and engineers to tackle complex problems that are challenging to solve through direct methods. In this article, we introduce the Milne-Simpson predictor-corrector technique (MS) and the fourth-order Adam-Bashforth-Moulton predictor-corrector method (ADM) for solving second-order initial value problems (IVPs) of ordinary differential equations.Both methods are highly efficient and particularly suitable for addressing IVPs. We compare the Euler and fourth-order Runge-Kutta methods within the ADM framework to the Milne-Simpson predictor-corrector approach. To validate the accuracy of the numerical solutions, we compare the approximate solutions with the exact solutions and find that they align well. We also observe that initializing the fourth-order ADM method with the fourth-order Runge-Kutta method and using the MS method for approximation yields superior accuracy compared to starting the ADM with the Euler method. Additionally, we evaluate the performance, effectiveness, and computational efficiency of both approaches. Finally, we demonstrate the convergence of these methods and examine the error terms for various step sizes. The numerical experiments are conducted using Matlab 2024.