A Class of Implicit Six Step Hybrid Backward Differentiation Formulas for the Solution of Second Order Differential Equations

In this paper, we propose a class implicit six step Hybrid Backward Differentiation Formulas (HBDF) for the solution of second order Initial Value Problems (IVPs). The method is derived by the interpolation and collocation of the assumed approximate solution. The single continuous formulation derived is evaluated at grid point of X = Xn+k and its second derivative at X = Xn+j, j = 1,2,.....k - 1 and X = Xn+μ 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. Numerical examples are given to show the efficiency of the method.