Generalised Modified Taylor Series Approach of Developing k-step Block Methods for Solving Second Order Ordinary Differential Equations

Various algorithms have been proposed for developing block methods where the most adopted approach is the numerical 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 equations. Thus, this article explores the adoption of this approach through the modification of the aforementioned 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.