A Novel Block Method for Direct Simulation of Higher-Order Oscillatory Differential Equations Using Power Series Polynomials
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Date
2026
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Faculty of Physical Sciences, University of Ilorin
Abstract
This study introduces a novel block method for the direct numerical integration of higher-order oscillatory
differential equations. The method employs power series polynomials as basis function within a collocation and
interpolation framework. The effectiveness of the proposed approach is demonstrated through its application to
second and third-order oscillatory test problems, including the classical mass-spring system. A rigorous theoretical
analysis confirms that the method is consistent, zero-stable, and convergent, achieving a uniform order of five.
Linear stability analysis reveals a substantial region of absolute stability, indicating its suitability for mildly stiff
problems. Numerical results, presented in tables and figures, show that the proposed method achieves significantly
higher accuracy and faster convergence compared to existing techniques. This affirms the reliability and efficacy
of the technique for the direct simulation of higher-order oscillatory differential equations.
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Citation
Novel Block Method, Direct Simulation, Higher-Order Oscillatory Differential Equations, Power Series Polynomials, Stability Analysis