Collocation Techniques for Block Methods for the Direct Solution of Higher Order Initial Value Problems of Ordinary Differential Equations

: This paper presents the derivation techniques of block method for solving higher order initial value problems of ordinary differential equations directly. The method was developed via interpolation and collocation of the shifted Legendre polynomials as basis function. The method is capable of providing the numerical solution at several points simultaneously.


Introduction
This work considers solving an ordinary differential equation (ODE) of the th order ( ≥ 2). There are currently two well-known techniques for solving higher order ODEs. The first is to reduce it to a system of first order ordinary differential equations and then solve using predictor corrector or Runge-Kutta method. The second technique is a block method; it is formulated in terms of linear multistep methods. It preserves the traditional advantage of one step methods, of being self-starting and permitting easy change of step length (Lambert, 1973). Their advantage over Runge-Kutta methods lies in the fact that they are less expensive in terms of the number of functions evaluation for a given order. The method generates simultaneous solutions at all grid points as suggested by many researchers such as Anake (2011), Onumanyi and Okunuga (1985), Onumanyi and Yusuph (2002), Onumanyi et al., (1993Onumanyi et al., ( , 1994, Awoyemi (2001Awoyemi ( , 2005 and 1991), Areo et al. (2008), Fatunla (1991Fatunla ( , 1995, Lambert (1991), Awoyemi Yahaya and Badmus (2009) and much recently by Warren and Zill (2013) and Omar and Kuboye (2015). These methods solve higher order initial value problems of ordinary differential equations without going through the process of reduction.
This present method is aimed at developing a general block method for the direct solution of higher order ( ≥ 2) initial value problems of ordinary differential equations, with the block approach, the non-self-starting nature associated with the predictor corrector method has been eliminated. Unlike the approach in predictor corrector method where additional equations were supplied from a different formulation, all our additional equations are obtained from the same continuous formulation.

Derivation of the k-Step Block Method
Consider solving an ordinary differential equation (ODE) of the th order ( ≥ 2) where = .

Numerical Example
Consider a specific case with = 5, = 2 and approximating the solution of (1) by the shifted Legendre polynomials of the form (2). Evaluating (4) at , = (1) and (2) at and > respectively; gives a system of nonlinear equations of the form

h p h p h p h p h p h p h p h p h p h p h p h p h p h p h p h p h p h p h
Solving for ′",# = 0(1)6 in (8) using inverse of a matrix method and then substituting into (6)   x t x t Evaluating (10) at @ = −1,−2, −3 and 5 with its first derivative evaluated at @ = 0, −1, −2, −3, −4 and 5 with @ = ( − ) and the results substituted in (9), to give the following discrete schemes needed for the implementation of the main scheme

Order and Error Constants
Expanding the block solution of (11) in Taylor's series and collecting like terms in powers of ℎ, the following result is obtained; C

Consistency
Following Lambert (1973Lambert ( , 1991, the block method is consistent since it has orders N O = 5,6 > 1

Zero stability
The block solution of the block method (11) is said to be zero stable if the roots U ; = 1, … , W of the first characteristic polynomial X(U), defined by X(U) = YZ@|U\ − ]| satisfies |U | ≤ 1 and every root with |U | = 1 has multiplicity not exceeding two in the limit as ℎ → 0 From the block method (11), we have Substituting, we get U 5) − U L = 0 r z r z r z r z r z The region of absolute stability is shown in Figure 1

Conclusion
In this paper, it has been shown that continuous collocation methods for solving ordinary differential equations can equally be derived through the approach in this study. In this study, a new block method approach which is capable of solving higher order initial value problems of ordinary differential equations is presented, with the block approach; the non self starting nature associated with the predictor corrector method and the Runge Kutta method has been eliminated. Unlike the approach in predictor corrector method where additional equations were supplied from a different formulation, all the required additional equations are obtained from the same continuous formulation. The basic property of the method was investigated and was found to be zero stable, consistent and convergent. The absolute stability region of the block method was also investigated and revelations showed that the newly constructed method is not A-stable as revealed by Figure 1. This method is very simple and effective for a wide-range of ordinary differential equations.