DIRECT SOLUTION OF SECOND ORDER INITIAL VALUE PROBLEMS BY HYBRID BACKWARD DIFFERENTIATION FORMULAS

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) of the form . The method is derived by the interpolation and collocation of the assumed approximate solution and its 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. A specific methods for k=4 is used to illustrate the process. The methods is shown to be zero stable, consistence and hence convergence. Numerical examples are given to show the efficiency of the method.