Accurate Four-Step Hybrid Block Method for Solving Higher-Order Initial Value Problems

Olanegan, O O,Adeyefa, E O

doi:10.21123/bsj.2022.19.4.0787

Abstract

This paper focuses on developing a self-starting numerical approach that can be used for direct integration of higher-order initial value problems of Ordinary Differential Equations. The method is derived from power series approximation with the resulting equations discretized at the selected grid and off-grid points. The method is applied in a block-by-block approach as a numerical integrator of higher-order initial value problems. The basic properties of the block method are investigated to authenticate its performance and then implemented with some tested experiments to validate the accuracy and convergence of the method.

Highlights

The article considers higher-order initial value problems of the form:...(1) Equation (1) applies in applied science and engineering to model problems with linear and nonlinear Initial Value Problems (IVPs)
Our intension in this paper is to propose a four-step hybrid block method that can directly integrate three different higher-order ordinary differential equations to solve initial value problems with better accuracy when compared to existing methods and validated with their basic properties
The Region of Absolute Stability (RAS) of a linear multistep method is the set of points z in the complex plane for which the polynomial ; z satisfies the root condition

Summary

Introduction

The article considers higher-order initial value problems of the form:...(1) Equation (1) applies in applied science and engineering to model problems with linear and nonlinear IVPs. A computational solution to various differential equations creates the need to develop methods to solve such equations. The resulting family of implicit schemes considered a general block form with the incorporation of Taylor series expansion to produce solutions to fourth-order initial value problems at all grid points.

Results

Conclusion