Mathematics and Statistics Vol. 11(2), pp. 353 - 372
DOI: 10.13189/ms.2023.110215
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Numerical Solution of Linear and Nonlinear Second Order Initial Value Problems Using Three-Step Generalized Off-Step Hybrid Block Method

Kamarun Hizam Mansor , Oluwaseun Adeyeye , Zurni Omar ^*
School of Quantitative Sciences, Universiti Utara Malaysia, Sintok 06010, Kedah, Malaysia

ABSTRACT

The numerical of second order initial value problems (IVPs) has garnered a lot of attention in literature, with recent studies ensuring to develop new methods with better accuracy than previously existing approaches. This led to the introduction of hybrid block methods which is a class of block methods capable of directly solving second order IVPs without reduction to a system of first order IVPs. Its hybrid characteristic features the addition of off-step points in the derivation of this block method, which has shown remarkable improvement in the accuracy of the block method. This article proposes a new three-step hybrid block method with three generalized off-step points to find the direct solution of second order IVPs. To derive the method, a power series is adopted as an approximate solution and is interpolated at the initial point and one off-step point while its second derivative is collocated at all points in the interval to obtain the main continuous scheme. The analysis of the method shows that the developed method is of order 7, zero-stable, consistent, and hence convergent. The numerical results affirm that the new method performs better than the existing methods it is compared with, in terms of error accuracy when solving the same IVPs of second order ordinary differential equations.

KEYWORDS
Linear, Nonlinear, Second Order, Initial Value Problems, Three-Step, Generalized Off-Step, Hybrid Block Method

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Kamarun Hizam Mansor , Oluwaseun Adeyeye , Zurni Omar , "Numerical Solution of Linear and Nonlinear Second Order Initial Value Problems Using Three-Step Generalized Off-Step Hybrid Block Method," Mathematics and Statistics, Vol. 11, No. 2, pp. 353 - 372, 2023. DOI: 10.13189/ms.2023.110215.

(b). APA Format:
Kamarun Hizam Mansor , Oluwaseun Adeyeye , Zurni Omar (2023). Numerical Solution of Linear and Nonlinear Second Order Initial Value Problems Using Three-Step Generalized Off-Step Hybrid Block Method. Mathematics and Statistics, 11(2), 353 - 372. DOI: 10.13189/ms.2023.110215.