Hybrid Block Method for Direct Numerical Approximation of Second Order Initial Value Problems Using Taylor Series Expansions

Zurni Omar,Oluwaseun Adeyeye

doi:10.3844/ajassp.2017.309.315

Abstract

In this article, a hybrid block method is utilized for the numerical approximation of second order Initial Value Problems (IVPs). The rigor of reduction to a system of first order initial value problems is bypassed as the hybrid block method directly solves the second order IVPs. Likewise, the methodology utilized also avoids the cumbersome steps involved in the widely adopted interpolation approach for developing hybrid block methods as a simple and easy to implement algorithm using the knowledge from the conventional Taylor series expansions with less cumbersome steps is introduced. To further justify the usability of this hybrid block method, the basic properties which will infer convergence when adopted to solve differential equations are investigated. The hybrid block method validates its superiority over existing methods as seen in the improved accuracy when solving the considered numerical examples.

Highlights

Differential equations of the form:( ) d2y =f dy x, y, y(a) = y, dy (a) y', x ∈ [a,b] (1) dx 2 dx dxAre known as second order initial value problems with initial conditions prescribed at a certain point a
Differential equations of this form play an important role in modeling virtually every physical or biological process because such equations occur in connection with numerous problems that are encountered in various aspects of our everyday life
Block multistep methods have evolved with time for the direct solution of second order initial value problems (Abdelrahim and Omar, 2016; Jator, 2007; Adesanya et al, 2012) and the concept of introducing evaluation at offgrid points birthing the introduction of hybrid block methods for numerical approximation of (1) have been investigated into

Summary

Introduction

Are known as second order initial value problems with initial conditions prescribed at a certain point a Differential equations of this form play an important role in modeling virtually every physical or biological process because such equations occur in connection with numerous problems that are encountered in various aspects of our everyday life. This concept of mathematical modeling involves translating problems from an application area into tractable mathematical formulations whose theoretical and numerical analysis provide insight, answers and guidance useful for the originating application (Abdelrahim and Omar, 2016; Moaddy et al, 2015). Block multistep methods have evolved with time for the direct solution of second order initial value problems (Abdelrahim and Omar, 2016; Jator, 2007; Adesanya et al, 2012) and the concept of introducing evaluation at offgrid points birthing the introduction of hybrid block methods for numerical approximation of (1) have been investigated into

Methods

Results

Conclusion