Mathematics and Statistics 11(2): 353-372, 2023 http://www.hrpub.org DOI: 10.13189/ms.2023.110215 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 Received October 20, 2022; Revised January 10, 2023; Accepted February 10, 2023 Cite This Paper in the Following Citation Styles (a): [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): 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. Copyright©2023 by authors, all rights reserved. Authors agree that this article remains permanently open access under the terms of the Creative Commons Attribution License 4.0 International License 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 1. Introduction Consider the following second order initial value problem (IVP) of ordinary differential equations (ODEs)  ′′ = (, ,  ′ ), ()=  0 ,  ′ ()=  1 (1) with ∈ [, ]. Equation (1) could either be linear or nonlinear depending on the properties of the dependent variable, and various numerical methods have been developed to solve either linear, nonlinear, or both. Some of these numerical methods are seen in studies such as [1] where second order Bratu-type IVPs were solved using a sixth order Runge-Kutta method while [2] adopted predictor-corrector method of the same type of IVPs. [3] developed a new class of variable coefficients and two-step semi-hybrid methods to solve problems in the form of Equation (1), [4] enhanced the conventional Numerov method to solve second order IVPs with better accuracy, and [5] solved Equation (1) problems using a novel Lie group based neural network method. In [6], a review of some numerical methods for solving second order IVPs was conducted and the methods under consideration were the third order convergence numerical method, successive approximation method and Adomian decomposition method. Other studies that have considered the numerical solution of linear, nonlinear, or both, for second order IVPs include [7-11]. In their works, they adopted numerical approaches such as the use of Bernoulli polynomials, hybrid linear multistep methods, predictor-corrector