A Novel Block Method for Direct Simulation of Higher-Order Oscillatory Differential Equations Using Power Series Polynomials

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.