A TWO-STEP HYBRID BLOCK FALKNER-TYPE METHOD FOR SOLVING GENERAL SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS

Abstract

One distinct family of methods for the numerical approximation of general and special second order ordinary differential equation is the Falkner-type methods which consists of a couple of rational formulas, one to follow the solution and the order to follow the derivative. In this paper, we explore this method by introducing a number of off step points in order to increase the number of function evaluation in the derivation process of a two-step Falkner-type method through the interpolation and collocation technique. The two main Falkner formulas and the additional ones to complete the block procedure are obtained from a continuous formulation. The basic properties of the proposed method were investigated and found to be zero-stable and of order p  9 which implies convergence.The performance of the new method was shown through some numerical examples and found to have higher accuracy than the existing methods considered in the literature.