One Step, Three Hybrid Block Predictor-Corrector Method for the Solution of

Direct method for the solution of higher order ordinary differential equations has been established in literature to be better than the method of reduction to system of first order ordinary differential equations [15]. Hybrid method has equally been established to have circumvented the Dahlquist barrier theorem, gave better error estimation than the K-step method especially when the problem are stiff and oscillatory [6].


Introduction
This paper considers the numerical solution to the general third order initial value problems of the form , f is continuous and satisfies the uniqueness theorem given by Awoyemi et al. [1].
Direct method for the solution of higher order ordinary differential equations has been established in literature to be better than the method of reduction to system of first order ordinary differential equations [1][2][3][4][5]. Hybrid method has equally been established to have circumvented the Dahlquist barrier theorem, gave better error estimation than the K-step method especially when the problem are stiff and oscillatory [6].
Scholars have proposed different numerical schemes which include the predictor corrector method and block method. It has been reported that predictor corrector method are not efficient because the predictors are in reducing order of accuracy, moreover the cost of developing the separate predictors, human and computer time involved in the execution are too costly [7,8]. Block method was later proposed to cater for some of the setbacks of the predictor corrector method. It should be reminded that Milne in 1953 first developed block method to serve as a predictor to a predictor-corrector algorithm before it was later adopted as a full method [5,[7][8][9][10][11] revisited the Milne approach and they concluded that though the method is expensive than the block method but gives better result than the block method. They tagged Milne's approach as constant order predictor-corrector method.
In this paper, we combine the unique properties of hybrid method and the constant order predictor-corrector method to develop a new numerical scheme for the solution of third order initial value problems of ordinary differential equations. This paper is organised as follows; section two discussed the algorithms in developing both the predictor and the corrector. Section three considers the analysis of the basic properties of both the predictor and the corrector. In section four, we test the efficiency of the derived method on some of numerical examples, and discussion of result and finally we concluded in section five.

Derivation of the block predictor
We consider the approximate solution of the form Where r and s are the number of interpolation and collocation points respectively. ' j a s are the unknown coefficients to be determined. x is the polynomial basis function of degree j.

Order of the method
Let the linear operator associated with the block method be defined as Expanding (11)

Consistency of the method
Consistency of the predictors: A method is said to be consistent, if it has order greater than one. From the above analysis, it is obvious that all our predictors are consistent.

Zero stability
Zero stability of the predictor: A block method is said to be zero stable as

Hence our method is Zero stable
Zero stability of the corrector: A linear multistep method is said to be zero stable if the zero's of the first characteristics polynomial ( ) r ρ Satisfies 1 r ≤ and is simple for 1 r = hence our corrector are zero stable.

Numerical examples
In this section, we test the efficiency of our method on some numerical examples The results are shown in Table 1 -7000 -6000 -5000 -4000 -3000  Table 2 shows clearly that our method gave good result than the existing methods. Problem III is also a stiff problem solved by Awoyemi and Idowu [13], where a hybrid method of order seven implemented in predictor corrector method was proposed. Our new method gave good result as shown in Table 3.

Conclusion
We have developed a one step, three hybrid points method implemented in constant order block predictor corrector method in this paper. The results re-affirmed that hybrid method gave better approximation especially when the step is low than the k-step method as discussed in section one. We have equally established that the new method gave better approximation than the block method and the convectional predictor corrector method which is in reducing order of accuracy. It should be noted that the new method was able to exhaust all interpolation points hence we developed higher order methods without increasing the grid point as discussed by Adesanya et al. [10]. In our future correspondence, we shall investigate the implication when more interpolation points are considered using the same predictor.  The result is shown in Table 2 Problem III: We consider a linear third order initial value problem The result is shown in Table 3 Note: ERBM→

Discussion of Result
We have considered three numerical examples in this paper. Problem I considered a special problem which was solved by Adesanya et al. [12] and Oladode and Yusuf [14] using a k-step block method of order six. It was shown that our new method gave good approximation than the existing methods as shown in Table 1. Problem II is a linear   14. Olabode BT, Yusuf Y (2009) A new block method for special third order ordinary differential equations. J of Mathematics and Statistics 53: 167-170.