Hybrid Numerical Method with Block Extension for Direct Solution of Third Order Ordinary Differential Equations

M K Duromola,A L Momoh

doi:10.4236/ajcm.2019.92006

Abstract

This paper focuses on the development of a hybrid method with block extension for direct solution of initial value problems (IVPs) of general third-order ordinary differential equations. Power series was used as the basis function for the solution of the IVP. An approximate solution from the basis function was interpolated at some selected off-grid points while the third derivative of the approximate solution was collocated at all grid and off-grid points to generate a system of linear equations for the determination of the unknown parameters. The derived method was tested for consistency, zero stability, convergence and absolute stability. The method was implemented with five test problems including the Genesio equation to confirm its accuracy and usability. The rate of convergence (ROC) reveals that the method is consistent with the theoretical order of the proposed method. Comparison of the results with some existing methods shows the superiority of the accuracy of the method.

Highlights

IntroductionThe focus of this article is to find an approximate solution on a given interval to third order initial value problems (IVP) of the type
The focus of this article is to find an approximate solution on a given interval to third order initial value problems (IVP) of the type= y′′′( x) f ( x, y ( x), y′( x= )), y ( x) α= a, y′( x) α= b, y′′( x) αc (1)where x ∈[a,b] ⊂ and y ( x), f ( x, y ( x), y′( x), y′′( x)) ∈ n
This paper focuses on the development of a hybrid method with block extension for direct solution of initial value problems (IVPs) of general third-order ordinary differential equations

Summary

Introduction

The focus of this article is to find an approximate solution on a given interval to third order initial value problems (IVP) of the type. In order to remove the difficulties usually encountered by adopting this mode of solution, researchers ([1] [11]-[22]) have proposed direct methods other than Predictor-Corrector methods whose modes of implementation are in block-by-block manner which was first introduced by Milne [23] as a starting step for predictor-corrector. This monumental success has greatly removed the burden of developing predictors and resulted in methods of uniform orders that yielded more accurate results. Conclusion was drawn on the performance of the proposed method when applied to solve the numerical examples

Mathematical Formulation

Analysis of the Proposed Method

Numerical Example

Method

Conclusion