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

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.