Mathematics and Statistics 8(6): 773-781, 2020 DOI: 10.13189/ms.2020.080618 http://www.hrpub.org Generalised Modified Taylor Series Approach of Developing k -step Block Methods for Solving Second Order Ordinary Differential Equations Oluwaseun Adeyeye * , Zurni Omar Department of Mathematics, School of Quantitative Sciences, University Utara Malaysia, Sintok, Kedah, Malaysia Received October 06, 2020; Revised November 20, 2020; Accepted December 28, 2020 Cite This Paper in the following Citation Styles (a): [1] Oluwaseun Adeyeye, Zurni Omar , ”Generalised Modified Taylor Series Approach of Developing k-step Block Methods for Solving Second Order Ordinary Differential Equations,” Mathematics and Statistics, Vol. 8, No. 6, pp. 773-781, 2020. DOI: 10.13189/ms.2020.080618. (b): Oluwaseun Adeyeye, Zurni Omar , (2020). Generalised Modified Taylor Series Approach of Developing k-step Block Methods for Solving Second Order Ordinary Differential Equations. Mathematics and Statistics, 8(6), 773-781. DOI: 10.13189/ms.2020.080618. Abstract Various algorithms have been proposed for developing block methods where the most adopted approach is the nu- merical integration and collocation approaches. However, there is another conventional approach known as the Taylor series approach, although it was utilised at inception for the development of linear multistep methods for first order differential equa- tions. Thus, this article explores the adoption of this approach through the modification of t he a forementioned conventional Taylor series approach. A new methodology is then presented for developing block methods, which is a more accurate method for solving second order ordinary differential equations, coined as the Modified Taylor Series (MTS) Approach. A further step is taken by presenting a generalised form of the MTS Approach that produces any k-step block method for solving second order ordinary differential equations. The computational complexity of this approach after being generalised to develop k-step block method for second order ordinary differential equations is calculated and the result shows that the generalised algorithm involves less computational burden, and hence is suitable for adoption when developing block methods for solving second order ordinary differential equations. Specifically, an alternate and easy-to-adopt approach to developing k-step block methods for solving second order ODEs with fewer computations has been introduced in this article with the developed block methods being suitable for solving second order differential equations directly. Keywords Generalised, Modified Taylor Series, Block Method, Second Order, Ordinary Differential Equations, k-step 1 Introduction Block methods for the numerical solution of second order ordinary differential equations came to light in a bid to bypass the disadvantages of wastage in computational time of previously existing conventional methods [1, 2]. To develop block methods, there are various approaches, with the most adopted approaches being the numerical integration approach as seen in studies by [3-6] and the interpolation approach as adopted by authors [7-9], amongst many other research work. In the work by [10], these two approaches were displayed and the advantages and shortcomings were discussed. It was stated that although the derivation using integration approach is more complicated in comparison to interpolation approach, this approach is able to generalize the formulation of the integration coefficients while the interpolation approach fails in this regard. [11] however mentioned another approach for developing linear multistep methods which is the derivation by Taylor series. This approach is less rigorous to be adopted as seen in studies by [12, 13]. However the methods developed by [12, 13] were restricted to the discrete schemes subject to boundary conditions, these methods were not further converted to block form and they were not adopted for initial value problems. The advantage of block methods is the ability of the method to simultaneously evaluate the method at different grid points for the equation under consideration. Hence, this study will first give the general overview of how to adopt the Modified Taylor Series (MTS) Approach in obtaining the discrete schemes and the corresponding block methods. Then, the generalised form to develop any k-step block method is presented. The organization of this paper is in the format where Section 2 discusses the general approach for using the MTS approach to derive the two-step and three-step block methods, Section 3 gives the generalised form for developing any k-step