Bernstein polynomial induced two step hybrid numerical scheme for solution of second order initial value problems

This paper presents a two step hybrid numerical scheme with one off-grid point for numerical solution of general second order initial value problems without reducing to two systems of first order. The scheme is developed using collocation and interpolation technique invoked on Bernstein polynomial. The proposed scheme is consistent, zero stable and is of order four(4). The developed scheme can estimate the approximate solutions at both step and off step points simultaneously using variable step size. Numerical results obtained in this paper shows the efficiency of the proposed scheme over some existing methods of same and higher orders.


Introduction
Differential equations are important tools in solving real world problems and many physical phenomena are model into second order differential equations, such models may or may not have exact solutions, thus a need for a numerical solution.
In this paper, we consider a second order initial value problem of the form In order to solve equation (1.1), the conditions stated below need to be imposed y(x 0 ) = y 0 , y (x 0 ) = y 0 (1.2) where a x b, a = x 0 < x 1 < x 2 < · · · , < x N−1 , N = b−a h , N = 0, 1, · · · , N − 1 and h = x n+1 − x n is called the step length. where y 0 is the solution at x 0 and x 0 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.
The numerical solution of equation (1.1) coupled with equation (1.2) is still receiving a lot of attention due to the fact that many physical sciences and engineering problems formulated into mathematical equation result to equation of such type.
In most applications, equation (1.1&1.2) are solved by reducing it to a system of first order ordinary differential equations and appropriate numerical method(such as Runge Kutta mehod, Modified Euler method, e.t.c) could be used to solve the resultant system, this approach has setbacks which had been reported by scholars, among them are Awoyemi et al [2] and Bun and Vasil'yer [3].
Direct method of solving equation (1.1) has been shown to be more efficient and saves computational time rather than method of reduction to system of first order ordinary differential equation (Brown [4]) and this has led to many scholars to attempt to solve equation (1.1) directly without reduction to system of first order equation. Brown [4] proposed a multi-derivative method to solve equation (1.1&1.2) directly. Adeniran and Ogundare [5] propose a one step hybrid numerical scheme with two off grid points for solving directly second over order initial value problems, the scheme can estimate the approximate solution at both step and off step points simultaneously by using variable step size.
Adeniran, Odejide and Ogundare [6] developed a one step hybrid numerical scheme for the direct solution of general second order ordinary differential equations, the scheme was developed using the collocation and interpolation techniques on the power series approximate solution and augmented by the introduction of one offstep point, in order to circumvent Dahlquist zero stability barrier and upgrade the order of consistency of the method. Accuracy of the scheme was tested with numerical examples and the result shows a better performance over the existing schemes.
In recent years, the Bernstein polynomials have gained the attention of many researchers. It has been used to obtained approximate solutions of different differential equations, for example, a method for approximating solutions to differential equations, proposed by Bhatti used Bernstein [7] operational matrix of differentiation. Ojo and Okoro [8] use a Bernstein polynomial to develop one step hybrid scheme with one offgrid point via collocation and interpolation techniques for the direct solution of general second order ordinary differential equations. The paper extend the work of Ojo and Okoro [8] by developing a two step hybrid method for solution of equation 1.1&1.2.

Berstein Polynomial
Aysegul and Nese [9] define Berstein Polynomial of degree m on interval [0, 1] as where the binomial coefficient is there are (m + 1)nth degree of Berstein Polynomial, for mathematically convenience, we usually set B i,m = 0, if i < 0 or i > 0. In general, the approximation of any function y(x) with the first (m + 1) Berstein Polynomial as

Development of the method
We seek numerical approximation of the analytic solution y(x) by assuming an approximate solution of the form where c and i are number of distinct collocation and interpolation points respectively and B k,n (x) is the Bernstein Polynomial derived from the recursive relation Differentiating equation (3.1) twice and substituting into equation (1.1) gives: We consider a grid point of step length two(2) and off step point at x = x n+ 3 2 . Collocating (5) at x = x n , x n+1 , x n+ 3 2 and x n+2 and interpolating (3) at x = x n and x n+ 3 2 give a system of six equations which are solved using Gaussian elimination method to obtained the parameters a j s, j = 0, 1, · · · , 5. The parameter a j s obtained are then substituted back into equation (3.3) to give a continuous two step hybrid method of the form where α and β are continuous coefficients. The continuous method (3.4) is used to generate the main method. That is, we evaluate at x = x n+1 and x = x n+2 In order to incorporate the initial condition at (1.2) in the derived schemes, we differentiate (6) with respect to x and evaluate at point and x n+2 to have: Combining the schemes derived in equation (3.5 − 3.10). The block method is employed to simultaneously obtain value for y n+1 , y n+ 3 2 , y n+2 , y n+1 , y n+ 3 2 and y n+2 needed where all the A i s and B i s are properly chosen r × r matrix coefficient and m = 0, 1, 2, · · · represent the block number, n = mr is the first step number of the mth block and r is the proposed block size. (Chu and Hamilton [10]).
In order to implement equation (3.5) to (3.10), we use a modified block method defined as follows: where λ is the power of the derivative of the continuous method and µ is the order of the problem to be solved; q=r+s. In vector notation, (3.12) can be written as: where the matrices a = (a ij ), e = (e ij ), d = (d ij ) are constant coefficient matrices and Y m = (y n+v i , y n+1 y n+v i , y n+1 ) T , y m = (y n−(r−1) , y n−(r−2) , · · · , y n ), The normalized version of (3.13) is given by (3.14) The modified block formulae (3.13) and (3.14) are employed to simultaneously obtain the values of y n+1 , y n+ 3 2 , y n+2 , y n+1 , y n+

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 [11] and Lambert [12] to obtain the order of our method as

Consistency
According to Gurjinder et al. [13] A linear multistep method is said to be consistent if it has an order of convergence, say p 1. Thus, our derived methods are consistent, since all are of order four.

Zero Stability
To obtain the zero stability of the method, we consider the following conditions: 1. The block (3.12) is said to be stable if as h → 0 the roots r j , j=1(1)k of the first characteristics polynomial ρ(R) = 0, that is ρ(R) = det[ A (i) R k−1 ] = 0, satisfy |R| 1 and for those roots with |R| 1, must have multiplicity equal to unity.( see Fatunla [11] for details).

If (3.12) be an R × R matrix then, it is zero stable if as
For those root with |R j | 1, the multiplicity must not exceed the order of the differential equation. As h → 0, we have

For our method
Taking the determinant of above, we have solving equation (4.2) we obtain λ = 0,, 1.
Since all the two conditions above are satisfied, we conclude that the block method converges.

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
We have proposed a two-step Bernstein polynomial fitted methods for the direct solution of general second order initial value problems. The method process a good accuracy with order 4, consistent and zero stable. The methods are implemented without the need for the development of predictors nor requiring any other method to generate starting values. Implementation of the method with numerical examples showed that the methods can compete favorably most of the existing multistep methods available for approximating similar class of problems.