Derivation of a Six-step Block Method for Direct Solution of Second Order Ordinary Differential Equations

A new six-step block method for solving second order initial value problems of ordinary differential equations is proposed using interpolation and collocation strategies. In developing this method, the power series adopted as an approximate solution is employed as interpolation equation while its second derivative is used as collocation equation. In addition, the stability properties of the developed method are also established. The numerical results reveal that the new method produces better accuracy if compared to existing methods when solving the same problems.


INTRODUCTION
This paper considers the development of numerical method for the direct solution of second order initial value problems of ordinary differential equations (ODEs) of the form (1) where f is continuous in the given interval of integration.Direct solution of higher order ODEs has been found to be more accurate than when they are reduced to their equivalent system of first order ODEs (see [4], [8], [3], [9]).A lot of scholars such as Awoyemi [4][5] and Omar & Suleiman [11][12][13] amongst others have worked on the derivation of direct method for solving higher order ordinary differential equations.
Adesanya et.al. [2] proposed two steps block method for the direct solution of (1).An improved parallel method with a step-length of three was developed by Yayaha [10] for solving (1) directly.Osilagun et.al. [6] increased the step-length by developing four-step implicit method for solving (1) without reduction process.Furthermore, Yahaya and Badmus [1] developed an accurate uniform order six block method for the direct solution of (1) where the step-length of five was considered but the accuracy of the method is low.
In order to improve the accuracy of the existing methods, a new block method of six step-length for direct solution of second order ordinary differential equations is proposed.

DERIVATION OF THE METHOD
Power series approximate solution of the form is considered as an interpolation polynomial where r and s are the number of interpolation and collocation points respectively.Differentiating (2) once and twice gives and collocating equation ( 4) In order to determine the values of j a , Gaussian elimination method is employed.The values of j a are then substituted into the interpolation polynomial (2) which yields a continuous linear multistep method of the form.
where the step-length are given as Derivation of a Six-Step Block Method for Second Order Differential Equations 153 where Evaluating (7) at the non-interpolation points gives the discrete schemes.Similarly, evaluating the first derivative of ( 7) at all the grid points produces the derivative of discrete schemes.Combining the discrete schemes with its derivative in a matrix finite difference equation yields a block method of the form

C
The derivative of the block (8) gives

PROPERTIES OF THE METHOD
This section examines the properties of the new developed block method for solving second order initial value problems of ODEs.

1. Order of the Method
Expanding (8) about the point using Taylor series gives

2. Zero Stability
The block method ( 8) is said to be zero-stable if the roots and the root 1  z has multiplicity not greater than the order of the differential equation.The first characteristic polynomial of the block method ( 8) is given by  Omar & Suleiman [13] also solved the above differential problem with their developed method whereby the maximum errors were also selected.The same problem was also considered by our method and the results are compared with their results as shown in Table II below.Problem 3: Badmus & Yahaya [1] applied their method to approximate the solution of the problem above.The same problem was also solved by our method and the comparison of the numerical results is given in Table III

CONCLUSION
A new six-step block method for the direct solution of second order initial value problems of ODEs has been successfully developed.In Tables I and II above, the new developed method produces better accuracy when compared with results generated by Omar and Suleiman [13].The new method also outperforms the method developed by Badmus and Yahaya [1] in term of accuracy.It is observed that the higher the steplength k, the higher the accuracy of the method.Therefore, in our future work, the value of the step-length k will be increased so that better results will be obtained.
(8)paring the coefficients of the powers of h makes our method(8)to have order