Derivation of Five Step Block Hybrid Backward Differential Formulas (HBDF) through the Continuous Multi-Step Collocation for Solving Second Order Differential Equation

Abstract

The study aims to develop the theory of numerical methods used for the numerical solution of second order ordinary differential equations (ODEs). 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. The suggested approach eliminates requirement for a starting value and its speed proved to be up when computations with the block discrete schemes were used. One specific methods for k=5 is used to illustrate the process. The test problem was solved with the proposed numerical method and obtained numerical and analytical solutions were compared.