Numerical solution of initial value problems by rational interpolation method using Chebyshev polynomials

J Oboyi,P.T Bukie,S.E Ekoro

doi:10.4314/gjpas.v25i2.8

Abstract

In this research, a modified rational interpolation method for the numerical solution of initial value problem is presented. The proposed method is obtained by fitting the classical rational interpolation formula in Chebyshev polynomials leading to a new stability function and new scheme. Three numerical test problems are presented in other to test the efficiency of the proposed method. The numerical result for each test problem is compared with the exact solution. The approximate solutions are show competitiveness with the exact solutions of the ODEs throughout the solution interval.Keywords and Phrases: Chebyshev polynomial, Rational Interpolation, Minimaxpolynomial, Initial Value Problems and Ordinary Differential Equations (ODEs)

Highlights

Many of the differential equations encountered in practice cannot be solved analytically and recourse must necessarily be made to numerical methods
The need to develop direct methods for solving higher order ordinary differential equation cannot be over emphasized in the theory of initial value problems (Pandey, 2012)
In other to achieve the set objective, we shall be concerned with addressing the problem of improving the accuracy of the rational interpolation method by modifying the existing method of Anetor et al, (2014), (Nwachukwu 2005), (Okosun and Ademuluyi 2007) and others by the introduction of Chebyshev polynomials

Summary

Introduction

Many of the differential equations encountered in practice cannot be solved analytically and recourse must necessarily be made to numerical methods. There is a wide range of methods developed by researchers that can be efficiently implemented with the computer and has become widely applied in engineering, sciences and many fields. Recent research on efficient methods to solve differential equations is the motivation for this work. The need to develop direct methods for solving higher order ordinary differential equation cannot be over emphasized in the theory of initial value problems (Pandey, 2012). Researchers have applied nonstandard finite difference method and obtained competitive results to those obtained with other methods. Our aim is to improve the classical implicit difference methods for systems of first order initial value problems. Though implicit methods are in general more expensive, but they have advantage in terms of stability and convergence (Horner, 1977)

Objectives

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Discussion

Conclusion