A FAMILY OF K-STEP TRIGONOMETRICALLY-FITTED BLOCK FALKNER METHODS FOR SOLVING SECOND-ORDER INITIAL-VALUE PROBLEMS WITH OSCILLATING SOLUTIONS

Abstract

A family of K-step Trigonometrically-fitted Block Falkner Methods is considered for the direct solution of second order Oscillatory Initial value problems. As unique to Falkner methods, two main formulas (one for the method and one for the derivative) for each k-step and some additional formulas. This method shall be adapted to general oscillatory second order ordinary differential equations via the multistep collocation technique. The idea employed in this study is the generalized collocation technique based on fitting functions that are combination of trigonometric and algebraic polynomials, which is then implemented in a block mode to get approximations at all the grid points simultaneously. As in other block methods, there is no need of other procedures to provide starting values, and thus the methods are selfstarting (sharing this advantage of Runge-kutta methods). The study of the properties of the proposed adapted block Falkner methods reveals that they are consistent and zero-stable, and thus, convergent. Furthermore, the stability analysis and the algebraic order conditions of the proposed methods are established. As evident from the numerical results, the methods are efficient and accurate when compared with some recent methods in the literature.