The need for accurate solutions to mathematical models, particularly for linear and nonlinear higher-order initial value problems, is essential across various scientific and engineering fields. Traditional methods often face challenges with stability and precision, especially in non-linear cases, prompting the development of advanced numerical techniques. This study introduces a two-step overlapping adaptive step-size multi-derivative hybrid block method to address these challenges in solving higher-order initial value problems. The method incorporates overlapping elements, using the second-to-last intra-step point from the previous step within each integration block to enhance accuracy. The method uses error estimation and selects an appropriate step-size, ensuring the desired accuracy without wasting computational resources or introducing unnecessary errors. The non-linear initial value problems are efficiently linearized using a modified-Picard iteration. Numerical examples are provided to demonstrate the efficiency and accuracy of the proposed method, and its performance is compared against a similar non-overlapping method as well as other methods reported in the literature.
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Rufai, U. Olamide , Sibanda, P. and Goqo, S. (2026). An overlapping adaptive step-size multi-derivative hybrid block method for higher order initial value problems. Computational Methods for Differential Equations, 14(2), 798-827. doi: 10.22034/cmde.2025.63158.2817
MLA
Rufai, U. Olamide, , Sibanda, P. , and Goqo, S. . "An overlapping adaptive step-size multi-derivative hybrid block method for higher order initial value problems", Computational Methods for Differential Equations, 14, 2, 2026, 798-827. doi: 10.22034/cmde.2025.63158.2817
HARVARD
Rufai, U. Olamide, Sibanda, P., Goqo, S. (2026). 'An overlapping adaptive step-size multi-derivative hybrid block method for higher order initial value problems', Computational Methods for Differential Equations, 14(2), pp. 798-827. doi: 10.22034/cmde.2025.63158.2817
CHICAGO
U. Olamide Rufai , P. Sibanda and S. Goqo, "An overlapping adaptive step-size multi-derivative hybrid block method for higher order initial value problems," Computational Methods for Differential Equations, 14 2 (2026): 798-827, doi: 10.22034/cmde.2025.63158.2817
VANCOUVER
Rufai, U. Olamide, Sibanda, P., Goqo, S. An overlapping adaptive step-size multi-derivative hybrid block method for higher order initial value problems. Computational Methods for Differential Equations, 2026; 14(2): 798-827. doi: 10.22034/cmde.2025.63158.2817
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