Five Steps Block Predictor-Block Corrector Method for the Solution of

Theory has it that increasing the step length improves the accuracy of a method. In order to affirm this we increased the step length of the concept in [1] by one to get k = 5. The technique of collocation and interpolation of the power series approximate solution at some selected grid points is considered so as to generate continuous linear multistep methods with constant step sizes. Two, three and four interpolation points are considered to generate the continuous predictor-corrector methods which are implemented in block method respectively. The proposed methods when tested on some numerical examples performed more efficiently than those of [1]. Interestingly the concept of self starting [2] and that of constant order are reaffirmed in our new methods.


Introduction
In this paper we examine the solution to general second order initial value problem of the form In literature, it has been stated clearly the journey of the development of direct methods to offset the burden of reduction [3]- [6]. Various methods have been proposed by scholars for solving higher order ordinary differential equation (ODE). Notable authors like [1] [7]- [11] have developed direct methods of solving general second order ODE's to cater for the burden inherent in the method of reduction. Now writing computer code is less bur-densome since it no longer requires special ways to incorporate the subroutine to supply the starting values. As a result, this leads to computer time and human effort conservation.
The new methods are continuous in nature with the advantage of possible evaluation at all points within the integration interval. We have taken advantage of the works of [7] [12]- [15] who proposed direct block methods as predictor in the form And also the discrete block formula as corrector in the form =  with the aim to cater for some of the setbacks of predictor-corrector method [16] [17]. The fact that interpolation point cannot exceed the order of the differential equation for block methods is worrisome [9]. Also vital to this paper is the concept of block predictor-corrector method (Milne approach). This method formed a bridge between the predictor-corrector method and block method [4] [10] [13]. In [1] we stated that results generated at an overlapping interval affect the accuracy of the method and the nature of the model cannot be determined at the selected grid points.
In this paper as in [1], we developed a method using the Milne approach but the corrector was implemented at a non overlapping interval. The numerical experiment compared the results generated at different step lengths, when k = 4 and when k = 5 respectively.

Development of the
Solving for the unknown constants j a s ′ using Guassian elimination method and substituting into (4), makes Equation (8) where ( ) Evaluating the first derivatives of (11) at 0,1 t = gives the following ( ) Writing Equations (12) to (16) in block form, the parameters in (3) gives the following In a similar way the results for cases II and III are summarized as:   38625017  2811953  420667  72343  1317556800  658778400 658778400  658778400 1317556800  2956022  2956487  138308  38999  1634  20586825  20586825  20586825  41173650  20586825  10555029  15699717  2674947  48798400

Order of the Block Predictor
From the above, it clearly shows that our methods are consistent.

Zero Stability
A block method is said to be zero stable if 0, h → the root ( ) of the first characteristics polynomials ∑ satisfying 1 R ≤ must have multiplicity equal to unity [9].
Applying this rule, we have that

Discussion
We have considered two non-linear and two linear second order initial value problems in this paper as shown in Table 1 to Table 4. In [1] we compared our method with the existing methods like the block and block predictor-corrector and the results re-affirms the claim of [10] that though block predictor-corrector method takes longer time to implement, it gives better approximation than the block method. In this paper we extended the step length considered in [1] and considered varying the number of interpolation points to observe the effect on the performance of the method.

Conclusion/Recommendation
In this paper we have proposed the varying of the step length from k = 4 [1] to k = 5. Block methods which have the properties of evaluation at all points within the interval of integration are adopted to give independent solutions at non overlapping intervals as predictors to the correctors. The new method k = 5 performed better than that of k = 4. Thus it has been confirmed that varying the step length improves the accuracy of the method. However, increasing the number of interpolation points does not significantly improve the result. We therefore, recommend the block predictor-block corrector method for use in the quest for solutions to second order initial value problems of ordinary differential equations.