Construction and Implementation of Hybrid Backward Differentiation Formulas for the Solution of Second Order Differential Equations.

Abstract

In this paper, we propose a family of Hybrid Backward Differentiation Formulas (HBDF) for direct solution of general second order Initial Value Problems (IVPs) The method is derived by the interpolation and collocation of the assumed approximate solution and it’s second derivative respectively, where k is the step number of the methods. The interpolation and collocation procedures lead to a system of (k+1) equations, which are solved to determine the unknown coefficients. The resulting coefficients are used to construct the approximate continuous solution from which the Multiple Finite Difference Methods (MFDMs) are obtained and simultaneously applied to provide the direct solution to IVPs. Two specific methods for k=2 and k=3 are used to illustrate the process. Numerical examples are given to show the efficiency of the method