### Applied Engineering

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### Numerical Algorithm for One and Two-Step Hybrid Block Methods for the Solution of First Order Initial Value Problems in Ordinary Differential Equations

In this paper, we developed a numerical Algorithm for one and two-step hybrid block methods for the numerical solution of first order initial value problems in ordinary differential equations using method of collocation and interpolation of Taylor’s series as approximate solution to give a system of non linear equations which was solved to give a hybrid block method. To further justify the usability and effectiveness of this new hybrid block method, the basic properties of the hybrid block scheme was investigated and found to be zero-stable, consistent and convergent. The derived scheme was tested on some numerical examples and was found to give better approximation than the existing methods. The errors displayed after solving some selected initial value problems, revealed that, it is better to increase L (Derivative) rather than the step length k as shown in our numerical results. Also, It was difficult to satisfy the zero-stability for larger k. In addition, the new method converges faster with lesser time of computation, which address the setback associated with other methods in the literature. Finally, the new method has order of accuracy for one-step as order Ten while order Eighteen for two-step.

Block Method, Initial Value Problems, Hybrid, One-Step, Two-Step, Taylor Series

APA Style

Ononogbo Chibuike Benjamin, Airemen Ikhuoria Edward, Ezurike Ugochi Julia. (2022). Numerical Algorithm for One and Two-Step Hybrid Block Methods for the Solution of First Order Initial Value Problems in Ordinary Differential Equations. Applied Engineering, 6(1), 13-23. https://doi.org/10.11648/j.ae.20220601.13

ACS Style

Ononogbo Chibuike Benjamin; Airemen Ikhuoria Edward; Ezurike Ugochi Julia. Numerical Algorithm for One and Two-Step Hybrid Block Methods for the Solution of First Order Initial Value Problems in Ordinary Differential Equations. Appl. Eng. 2022, 6(1), 13-23. doi: 10.11648/j.ae.20220601.13

AMA Style

Ononogbo Chibuike Benjamin, Airemen Ikhuoria Edward, Ezurike Ugochi Julia. Numerical Algorithm for One and Two-Step Hybrid Block Methods for the Solution of First Order Initial Value Problems in Ordinary Differential Equations. Appl Eng. 2022;6(1):13-23. doi: 10.11648/j.ae.20220601.13