4-STEP BLOCK HYBRID METHODS FOR SOLVING SECOND ORDER INITIAL VALUE PROBLEMS

Abstract

This work is based on the derivation of four-step block hybrid methods for the solution of second order initial value problems (IVPs) in block mode. Power series was used as the basis function. Techniques of collocation and interpolation were used in the differential system arising from the basis function and interpolating functions respectively. Continuous linear multistep methods with step number k = 4 were developed by interpolating the basis function at some certain grid points and collocating the differential system at both grid and off-grid points. Unknown parameters in the system of linear equation arising from the collocation and interpolation functions were determined. Consequently the values were substituted into the approximate solution, to generate hybrid methods which were simultaneously applied as numerical integrators by assembling them into block methods. Analysis of the methods were carried out to determine consistency, order, zero stability,convergency, and region of absolute stability. The new methods were tested on some differential equations to determine their efficiency and accuracy, introduction of a hybrid point between points of interpolation validates its superiority over some existing methods in terms of accuracy when solving the considered numerical examples. Thus, numerical solutions obtained yield better results when compared with some existing methods.