Implicit hybrid block methods for solving second, third and fourth orders ordinary differential equations directly

Strong numerical algorithm for simultaneously solving second, third and fourth orders of ordinary differential equations with given initial conditions is considered in this paper. Linear Multi-step collocation approach was applied in the derivation of the new method with the use of power series approximate solution as interpolation polynomial. The numerical integrators that formed the block were derived by evaluating the continuous scheme along with its derivatives at the non-interpolating points within the selected interval of integration. The basic properties of the method were investigated properly. In order to examine the effectiveness of the block, the methods were tested on some second, third and fourth orders ordinary differential equations and the results generated proved its effectiveness over existing methods in terms of accuracy.