Block Hybrid Numerical Method for the Direct Solutions of Second, Third and Fourth Order Initial Value Problems
DOI:
https://doi.org/10.62054/ijdm/0204.10Abstract
A one-step block hybrid method for the direct numerical solution of second, third and fourth-order ordinary differential equations (ODEs) without reducing them to equivalent systems of first-order equations is developed and implemented in this study. Collocation and interpolation techniques are used in the construction of the method to obtain a continuous implicit scheme that is subsequently converted into an explicit block hybrid form. A thorough analysis is conducted of the scheme's fundamental characteristics, such as order, error constant, consistency, zero-stability, convergence and the region of absolute stability. Comparing the suggested method to existing approaches in the literature, numerical experiments on benchmark problems like the mass-spring system, third-order oscillatory models, and fourth-order oscillatory differential equations show that it is more accurate and stable. The findings indicate a computationally efficient and highly accurate framework for the direct solution of higher-order ODEs, significantly advancing numerical analysis in applied mathematics.
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