Computational Numerical Method for the Direct Solution of Highly Third Order Oscillatory Linear Initial Value Problems
DOI:
https://doi.org/10.62054/ijdm/0302.04Abstract
This paper presents the development and analysis of a Computational Numerical Method (CNM) for the direct solution of highly third order oscillatory linear initial value problems (IVPs). The method was derived using a linear multistep framework based on power series approximation combined with interpolation and collocation techniques. The formulation avoids the reduction of higher-order problems to systems of first-order equations, thereby improving computational efficiency and reducing accumulated errors. The theoretical properties of the method, including order, error constant, consistency, zero-stability, convergence and region of absolute stability, are rigorously analyzed. The results show that the CNM is of uniform order eight, consistent, zero-stable, convergent and A-stable, making it suitable for both stiff and non-stiff differential equations. The effectiveness of the CNM is verified through numerical simulations of several third-order oscillatory initial value problems. The results are presented in tabular form and compared with existing methods in the literature. The comparisons demonstrate that the CNM provides highly accurate approximations and performs competitively or better than the referenced schemes for both stiff and non-stiff problems. The outcomes confirm the reliability, stability and computational efficiency of the CNM approach in solving third-order oscillatory IVPs.
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