BLOCK HYBRID METHOD FOR ACCURATE SOLUTIONS OF HIGHER-ORDER DIFFERENTIAL EQUATIONS
DOI:
https://doi.org/10.33003/fjs-2026-1003-4597Keywords:
higher-order ODEs, One-step block method, hybrid method, numerical stability, interpolation and collocation, initial value problem, absolute stability, convergence analysisAbstract
This study presents a novel one-step block hybrid method for the direct numerical solution of second, third and fourth-order ordinary differential equations without reducing them to equivalent first-order systems. The method is formulated using power series interpolation and collocation techniques, leading to a continuous implicit scheme that is transformed into an explicit block form for efficient computation. Theoretical analysis shows that the method satisfies key properties of numerical algorithms, including consistency, zero-stability, and convergence, with a uniform order of six. The region of absolute stability is established using the Boundary Locus Method, confirming the method’s strong stability characteristics. To validate its performance, the method is applied to several dynamic, linear and oscillatory initial value problems. The numerical results demonstrate excellent agreement between computed and exact solutions, with significantly smaller errors compared to existing methods in the literature. The findings indicate that the proposed method is highly accurate, computationally efficient, and robust, making it a reliable tool for solving complex higher-order differential equations encountered in science and engineering.
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Copyright (c) 2026 Hannatu Samuel, Pius Tumba, Aina Babatunde, Abubakar Ahmad Dauda, Maina Waziri
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