Computing Numerical Scheme for Approximate Solutions of Second-Order Oscillatory Initial Value Problems
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Date
2026
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Faculty of Physical Science, University of Ilorin
Abstract
This study presents a Computational Numerical Scheme (CNS) for the direct solution of second-order oscillatory initial value problems without reducing them to systems of first-order differential equations. The motivation for this approach stems from the need to improve computational efficiency, reduce numerical complexity, and preserve the inherent structure of oscillatory physical models such as Bessel-type equations and mass-spring systems. The CNS was derived using a power series approximation combined with interpolation and collocation techniques, leading to a continuous hybrid linear multistep formulation. The method was analyzed for consistency, order, zero-stability, convergence, and absolute stability, where it was shown to be A-stable. The CNS was applied to three test problems, and its performance was evaluated through comparisons of absolute errors with existing methods in the literature. Numerical results demonstrated that the CNS consistently produced highly accurate approximations with significantly reduced errors, outperforming competing methods. Graphical illustrations further confirmed the superiority and stability of the method across all tested intervals. The findings indicate that the CNS is an efficient, reliable, and accurate approach for solving second-order oscillatory differential equations arising in real-life physical systems.
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Computational Numerical Scheme (CNS); second-order differential equations; oscillatory systems; initial value problems; stability analysis; mass-spring system; Bessel equation; numerical simulation.