Continuous Explicit Hybrid Method for Solving Second Order Ordinary Differential Equations

Abstract

This paper presents an explicit hybrid method for direct approximation of second order ordinary differential equations. The approach adopted in this work is by interpolation and collocation of a basis function and its corresponding differential system respectively. Interpolation of the basis function was done at both grid and off-grid points while the differential systems are collocated at selected points. Substitution of the unknown parameters into the basis function and simplification of the resulting equation produced the required continuous, consistent and symmetric explicit hybrid method. Attempts were made to derive starting values of the same order with the methods using Taylor's series expansion to circumvent the inherent disadvantage of starting values of lower order. The methods were applied to solve linear, non-linear, Duffing equation and a system of equation second-order initial value problems directly. Errors in the results obtained were compared with those of the existing implicit methods of the same and even of higher order. The comparison shows that the accuracy of the new method is better than the existing methods.