An Accurate Self-Starting Initial Value Solvers for Second Order Ordinary Differential Equations

This paper focuses on the development of continuous and discrete algorithms for the numerical solution of ordinary differential equations. A continuous hybrid two-step method is developed for second order initial value problems with Chebyshev polynomial as basis function through collocation and interpolation techniques. By selection of equally-spaced points for both interpolation and collocation, new efficient, self-starting and zero-stable finite difference method coupled as block method is produced which is generally more accurate when compared with existing methods. The method is analyzed, its order and error constants are determined with investigations made on consistency and stability. Mathematics Subject Clarification: 65L05 Keyword: Hybrid, Collocation, Interpolation, Block Method Introduction The continuous integration algorithms for the numerical solution of the initial value problems (IVPs) have been discussed in the literature. The works of Lie and Norsett (1989) and Onumanyi et al (1994) focused on the construction of continuous multistep methods by employing the multistep collocation approach. The use of finite difference to construct continuous implicit schemes through which 728 E. O. Adeyefa et al. the block formulae are derived has been discussed extensively by some scholars such as Awoyemi et al (2011), Olabode (2009), Adesanya et al (2009) to mention a few. The traditional multistep methods including the hybrid ones can be made continuous through the idea of multistep collocation, Norsett (1989) and Onumanyi et al (1994). To obtain multiple discrete hybrid method, the continuous implicit hybrid method is evaluated at some selected points involving grid and off-grid points along with its first derivative. For the derivation of the block methods, the multiple discrete hybrid formulae obtained are solved simultaneously and the resulting equations constituted a block from which a number of explicit methods will be obtained. Materials and Methods We consider here the derivation of the proposed continuous hybrid two-step block methods. This we do by approximating the analytical solution of 0 0 '' ( , , '), '( ) , ( ) y f x y y y a z y a y    (1) where f is a continuous function, with a Chebyshev polynomial in the form


Introduction
The continuous integration algorithms for the numerical solution of the initial value problems (IVPs) have been discussed in the literature.The works of Lie and Norsett (1989) and Onumanyi et al (1994) focused on the construction of continuous multistep methods by employing the multistep collocation approach.The use of finite difference to construct continuous implicit schemes through which the block formulae are derived has been discussed extensively by some scholars such as Awoyemi et al (2011), Olabode (2009), Adesanya et al (2009) to mention a few.The traditional multistep methods including the hybrid ones can be made continuous through the idea of multistep collocation, Norsett (1989) and Onumanyi et al (1994).To obtain multiple discrete hybrid method, the continuous implicit hybrid method is evaluated at some selected points involving grid and off-grid points along with its first derivative.For the derivation of the block methods, the multiple discrete hybrid formulae obtained are solved simultaneously and the resulting equations constituted a block from which a number of explicit methods will be obtained.

Materials and Methods
We consider here the derivation of the proposed continuous hybrid two-step block methods.This we do by approximating the analytical solution of 00 '' ( , , '), '( ) , ( ) where f is a continuous function, with a Chebyshev polynomial in the form where x[a, b], the aj's are real unknown parameters to be determined and r + s is the sum of the number of collocation and interpolation points.We need to interpolate at at least two points to be able to approximate (2) and, to make this happen, we proceed by selecting two equally spaced offstep points.So, and its second derivative is collocated at xn+i, i = 0, v and 1, so as to obtain a system of seven equations which are solved by Gaussian elimination method.
Analysis of the Method Here, the order, error constant and consistency of the method are discussed.
The explicit schemes (7) The LMM ( 12) is said to be of order p if C0 = C1 = C2 = . . .= Cp = Cp+1 = 0 and Cp+2 ≠ 0 is the error constant, see Lambert (1973).According to this definition, the discrete schemes (7) have order p = (5, 5, 5) T with error constants The discrete schemes derived are all of order greater than one and satisfy the conditions (i) -(iii).

Definition 3
The LMM ( 12) is said to be zero-stable if no root of the first characteristic polynomial has modulus greater than one, and if every root of modulus one has multiplicity not greater than two.
All the roots of the derived schemes have been verified to be less than or equal to 1 and│ z│ = 1, simple.

Numerical Examples
Here, we consider the application of the derived schemes to three test problems for the efficiency and accuracy of the method implemented as block method.
a = x0 < x1 < ... < xn < xn+1 < ... < xN = b of the integration interval [a, b], with a constant step size h, given by h = xn+1 -xn; n = 0, 1, ..., N -1.The second derivative of (2) is given by 12) is said to be consistent if it is of order p ≥ 1 and its first and second characteristic polynomials defined as derived are discrete schemes belonging to the class of LMM of the form

Table of ResultsTable 1 :
The exact solutions, the computed results and the absolute errors from

Table 2 :
The exact solutions, the computed results and the absolute errors from

Table 3 :
The exact solutions, the computed results and the absolute errors from problems 3