The need for accurate solutions to mathematical models, particularly for linear and non-linear 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.
Rufai, U. Olamide , Sibanda, P. and Goqo, S. (2025). AN OVERLAPPING ADAPTIVE STEP-SIZE MULTI-DERIVATIVE HYBRID BLOCK METHOD FOR HIGHER ORDER INITIAL VALUE PROBLEMS. Computational Methods for Differential Equations, (), -. 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, , , 2025, -. doi: 10.22034/cmde.2025.63158.2817
HARVARD
Rufai, U. Olamide, Sibanda, P., Goqo, S. (2025). 'AN OVERLAPPING ADAPTIVE STEP-SIZE MULTI-DERIVATIVE HYBRID BLOCK METHOD FOR HIGHER ORDER INITIAL VALUE PROBLEMS', Computational Methods for Differential Equations, (), pp. -. 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, (2025): -, 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, 2025; (): -. doi: 10.22034/cmde.2025.63158.2817
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