Three Step Hybrid Block Method with Two Generalized Off-step Points for Directly Solving Third Order Ordinary Differential Equations

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DOI: https://doi.org/10.28924/2291-8639-23-2025-58
Published: Mar 3, 2025
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Raft Abdelrahim, Abdelgabar Adam Hassan, Hamdy M. Barakat, Mohammad S. Hijazi, Three Step Hybrid Block Method with Two Generalized Off-step Points for Directly Solving Third Order Ordinary Differential Equations, Int. J. Anal. Appl., 23 (2025), 58.

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Raft Abdelrahim, Abdelgabar Adam Hassan, Hamdy M. Barakat, Mohammad S. Hijazi

Abstract

This article presents a hybrid block method with steplength k = 3 and two hybrid points chosen within grid intervals [xn, xn+1] and [xn+1, xn+2]. The block method is applied for the numerical solution of third order ordinary differential equations (ODEs) directly. An improved methodology is introduced where the generalized form to develop any three-step hybrid block method is given for solving third order ODEs irrespective of the value of the hybrid point with respect to the chosen intervals within the grid. The properties of the block method were investigated with the method exhibiting convergence following from satisfying the properties of zero-stability and consistency. To further validate the new hybrid block method, certain numerical examples were considered and the results show improved accuracy in terms of error comparison when compared with previously existing literature.

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