Half Step Numerical Method for Solution of Second Order Initial Value Problems

This paper presents a half step numerical method for solving directly general second order initial value problems. The scheme is developed via collocation and interpolation technique invoked on power series polynomial. The proposed method is consistent, zero stable, order four and three. This method can estimate the approximate solution at both step and off step points simultaneously by using variable step size. Numerical results are given to show the efficiency of the proposed scheme over some existing schemes of same and higher order.


Introduction
Numerical Solution of second order Initial Value Problem (IVP) of the form ( ( ) ( ) ) ( ) ( ) ( ) have been treated in a variety of ways including the use of polynomial and non-polynomial via collocation and interpolation approach, where , and is called the step length, where is the solution at and is the initial point, f is a continuous function within the interval of integration, the condition on the function f are such that existence and uniqueness of solution is guaranteed( Wend [1], and prime indicates differentiation with respect to x, while y(x) is the unknown function to be determined.
Adeniran and Ogundare [2], propose a power series polynomial induced one step hybrid numerical scheme with two off grid points for solving directly second order initial value problems, the scheme can estimate the approximate solution at both step and off-step points simultaneously by using variable step size.
Ojo and Okoro [3], use a Bernstein polynomial to develop one step hybrid scheme with one off-grid point via collocation and interpolation techniques for the direct solution of general second order ordinary differential equations. Adeniran and Olanegan [4], develop a continuous two-step method using trigonometric function as basis function with two discrete methods as end products, the performance of the method is demonstrated on some numerical examples to show accuracy and efficiency advantages.
Adeniran and Longe [5], in their paper construct a one-step hybrid numerical scheme with one off grid points for solving directly the general second order initial value problems. The scheme is developed using collocation and interpolation technique invoked on Lucas polynomial. Numerical results of the scheme show some efficiency over some existing schemes of same and higher order. This paper is concerned with the development of half step method using power series polynomial for numerical solution of second order IVP.

Development of the Method
Considering a power series approximate solution in the form: where c and i are number of distinct collocation and interpolation points respectively. Substituting the second derivative of (2) into (1) gives: we consider grid point of step length of half, collocating (3) at points x= , , and , interpolating (2) at x= and give a system of five equations which are solved using Gaussian elimination method to obtained the parameters 's, j=0, 1,…,4. The parameters 's obtained are then substituted back into equation (2) to give a continuous half step method of the form where and are continuous coefficients. The continuous method (4) is used to generate the main method, that is, we evaluate at x = to obtain In order to incorporate the initial condition at (1) into the derived scheme (5), we differentiate (4) with respect to x and evaluate at point at x= , x = and x= to obtain: Combining the schemes derived in equation (5)(6)(7)(8). The block method is employed to simultaneously obtain value for , , and , needed to implement equation (1), thus we have

Analysis of the Method
We analyze the derived method which includes the order and error constant, consistency zero stability, and convergence of the method.

Order and Error Constant
We adopted the method proposed by Fatunla [6] and Lambert [7] to obtain the order of our method as (3, 3, 3,

Consistency
According to Gurjinder, et al. [8], A linear multistep method is said to be Consistent, if it has an order of convergence, say , thus, our derived methods are consistent, since all are of order three and four respectively.

Zero Stability
To obtain the zero stability of the method, we consider the following conditions: i. A block method of linear multistep method is said to be stable if as the roots ( ) of the first , and for those roots with , must have multiplicity equal to unity.(see Fatunla [6] for details.) ii. If the block method be an marix, then, it is zero stable as . For those roots with , the multiplicity must not exceed the order of the differential equation. For the block method, as Taking the determinant of equation (9) and setting it equal to zero, we have ( ) (10) Solving equation (10), gives

Convergence
Theorem 3.1: The necessary and sufficient condition for a linear multistep method to be convergent is for it to be consistent and zero stable.
Thus our block method is convergent since it is zero stable and consistent.

Numerical Implementation of the Scheme
The effectiveness and validity of our newly derived method was tested by applying it to some second order differential equations. All calculations and programs are carried out with the aid of Maple 2016 software.

Conclusion
The half method are derived via multistep collocation technique and implemented in block form. The method are of order 3 and 4, consistent, zero stable and convergent. The method is reliable, efficient display better accuracy than most of the existing ones in literature.