A Zero-Stable Optimal Order Method for Direct Solution of Second Order Differential Equations

Kayode Sunday Jacob

doi:10.3844/jmssp.2010.367.371

Abstract

Problem statement: In this study, a numerical method for direct solution of general second order differential equations was considered in order to circumvent the problems of computational burden and computer time wastage associated with method of reduction to system of first order equations. The issue of zero stability of higher order methods is considered in the development of the method. Approach: The method was developed based on collocation and interpolation approach using power series as the basis function to the solution of the problem. The basic properties of the method were considered. A consistent symmetric and zero stable main predictor of order five was also developed for the evaluation of the implicit scheme. The accuracy of the developed method is tested with test problems. Results: The method was zero-stable, consistent and normalized. The order of accuracy was found to be optimal. Both the main method and the predictor were obtained to be normalized and zero-stable. Conclusion/Recommendations: Comparison of the derived method with an existing method of the same order of accuracy showed a higher accuracy of the derived method. In the later research, this accuracy will be improved by developing the main predictor of the same order of accuracy with the main method.

Highlights

Higher order ordinary differential equations of the form: burden and wastage in computer time (Awoyemi and Kayode, 2002)
M = 1, 2, ..., {t, y} ⊂ Rn are of special interest to scientists and engineers. The results of their field-work are most often modeled into equations of type (1) which are conventionally reduced to system of first order equations of the type: F(t, y, y′) = 0, y(t0 ) = y0, t ∈[a, b]
The minimum value of k for the development of any Linear Multistep Methods (LMM) must be equal to the order of the differential equation it is meant to solve

Summary

Introduction

Higher order (linear and non-linear) ordinary differential equations of the form: burden and wastage in computer time (Awoyemi and Kayode, 2002). When the reduced equations are not solvable analytically numerical methods are adopted to approximate the solution. Numerical methods developed for such reduced equations abound in literature (Abhulimen and Otunta, 2006; Ademiluyi et al, 2002; Ademiluyi and Kayode, 2001; Awoyemi and Kayode, 2005; Chan et al, 2004).

Results

Conclusion