DERIVATION AND IMPLEMENTATION OF A THREE-STEP HYBRID BLOCK ALGORITHM (TSHBA) FOR DIRECT SOLUTION OF LINEAR AND NONLINEAR SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS

O E Abolarin,B G Ogunware,S O Ayinde,A F Adebisi

doi:10.33003/fjs-2020-0404-505

Abstract

The development and application of an implicit hybrid block method for the direct solution of second order ordinary differential equations with given initial conditions is shown in this research. The derivation of the three-step scheme was done through collocation and interpolation of power series approximation to give a continuous linear multistep method. The evaluation of the continuous method at the grid and off grid points formed the discrete block method. The basic properties of the method such as order, error constant, zero stability, consistency and convergence were properly examined. The new block method produced more accurate results when compared with similar works carried out by existing authors on the solution of linear and non-linear second order ordinary differential equations

Highlights

The reduction of second order ordinary differential equations (ODEs) to a system of first order ODEs and solve using any appropriate method for first order ODEs was the principal approach used for solving higher order initial value problems second order ODEs
The results generated by the developed three-step hybrid block method are displayed in tables (1) and (2)
The three-step hybrid scheme gives better result when compared with the method of Abhulimen and Okunuga (2008) for solving the linear second order ODE in problem 2

Summary

INTRODUCTION

The reduction of second order ordinary differential equations (ODEs) to a system of first order ODEs and solve using any appropriate method for first order ODEs was the principal approach used for solving higher order initial value problems second order ODEs. Omar (2016), Omole and Ogunware (2018), Omar and Raft (2016), and many more proposed the block method for the direct solution of second order ordinary differential equations independently. The distinctiveness of the block method is that in each usage, the solution value will be obtained concurrently at several different points and it is found to be cost effective because of the evaluation of few functions involved. The focus of this work is to develop and implement a three-step hybrid block method for the solution of second order. While the evaluation of the first derivative of the continuous scheme at all points yields yn 3 These schemes in equations (7) - (14) are combined together in matrix form and by using the matrix inversion technique, a block method of the following form is produced.

OF RESULTS

CONCLUSION