A Robust Direct Block Hybrid Method for Solving Nonlinear Second-Order Differential Equations

Abstract

This paper introduces a robust Block Hybrid Method (BHM) for the direct numerical integration of second-order nonlinear ordinary differential equations (ODEs), with a particular focus on the generalized Van der Pol and Duffing equations. By avoiding the common practice of reducing the problem to a larger first-order system, our method minimizes computational overhead and potential error accumulation. The one-step, self-starting nature of the block formulation provides solutions at multiple grid points simultaneously, enhancing computational efficiency over traditional step-by-step methods. We first establish a priori bounds for the solutions under general conditions to ensure theoretical soundness. The method's efficacy is then validated against several well-known nonlinear problems, demonstrating its high order of accuracy and stability. The results establish the BHM as a highly competitive and reliable alternative for obtaining accurate solutions to second-order initial value problems in applied science and engineering.