eu ONE-TWELVETH STEP CONTINUOUS BLOCK METHOD FOR THE SOLUTION OF y ′ ′ ′ = f ( x , y , y ′ , y ′ ′ )

Abstract: We consider direct solution to third order ordinary differential equations in this paper. Collocation and interpolation approach was adopted to generate a continuous hybrid multistep method. We adopted the use of power series as a basis function for approximate solution. We evaluated at off-grid points to get a continuous hybrid multistep method. Block method was later adopted to generate the independent solution at selected off-grid points. The properties of the block viz: order, zero stability and region of absolute stability are investigated. Our method was tested on third order ordinary differential equation and found to give better result when compared with existing methods.


Introduction
In this paper, we considered the method of approximate solution of the general second order initial value problem of the form We consider the solution to general third order initial value problem of the form where x 0 is the initial point and f is continuous within the interval of integration and satisfies the existence and uniqueness condition.
Many real life problems in sciences, engineering, biology and social sciences are model of third order ordinary differential equations.Some of these models do not always have theoretical solutions, thus numerical methods are often employed to solve them.Researchers in most cases always use method of reduction of higher order ODEs into system of first order ODEs to solve (1).This technique though quite good, is bedeviled with many problems such tediousness, complexity of the method, waste of time, and the need for large computer storage memories because of too many auxiliary functions, etc. Conventionally, higher order ordinary differential equations are solved directly by the predictorcorrector method where separate predictors are developed to implement the corrector and Taylor series expansion adopted to provide the starting values.Predictor-corrector methods are extensively studied by [1][2][3][4][5][6][7][8][9][10][11][12].These authors proposed linear multistep methods with continuous coefficient, which have advantage of evaluation at all points within the grid over the proposed method in [4] The major setbacks of predictor-corrector method are extensively discussed by [6].Many research works have been grounded on Linear Multistep Method (LMM) and Linear Multistep Hybrid Method (LMHM) where Collocation and Interpolation are done at selected grid and off-grid points.There was then need to interpolate and collocate at grid and off-grid points with evaluation to be done at off-grid points only since this gives higher accuracy and large region of absolute stability.We apply the One-Twelveth Step Method introduced for the direct solution of Third Order Differential Equations with x n being the only grid point.

Mathematical Formulation of the Method
We consider the simple power series as a basis function for approximation: where φ j (x) = x j and xǫ[a, b], a ′ j s are coefficients to be determined and is a polynomial of degree r + s − 1 .We construct a K-step Collocation Method by imposing the following conditions on (2) Substituting (1) into (5) gives We shall consider the grid point of step length Here ] T , and where We hereby present the continuous schemes Writing (12) Expanding the form Y m and F (Y m ) in Taylor series and comparing coefficients of h, we obtained Definition.The linear operator and the associated block method are said to be of order p if C 0 = C 1 = ... = C p = C p+1 = 0, C p+2 = 0 .C p+2 is called the error constant.It implies that the local truncation error is given by T n+k = C p+2 h p+2 y p+2 (x) + O(h p+3 ).

Expanding the block in Taylor series expansion gives
Comparing the coefficients of h, the order of the block is p = 5, with error constant

Consistency
In numerical analysis, it is necessary that the method satisfies the necessary and sufficient conditions.A numerical method is said to be consistent if the following conditions are satisfies 1.The order of the scheme must be greater than or equal to 1 i.e. p ≥ 1.

2.
Where, ρ(r) and σ(r) are the first and second characteristics polynomials of our method.According to [3], the first condition is a sufficient condition for the associated block method to be consistent.Our method is order p ≥ 1. Hence it is consistent

Zero Stability of the Method
The general form of block method is given as Applying ( 14) to (16) gives λ 4 − λ 3 = 0, λ = 0, 0, 0, 1 Since no root has modulus greater than one and |λ| = 1 is simple, the block method is zero stable in the h → 0

Region of Absolute Stability of the Block Method
We express the stability matrix together with the stability function Hence, we express the block method ( 16) in form of The elements of the matrices A, B, U and V are substituted and computing the stability function with Maple software yield, the stability polynomial of the method which is then plotted in MATLAB environment with the Newton-Raphson Method where N = 401 and tol = 10 −11 to produce the required absolute stability region of the method, as shown by the figure below

Implementation of the Method
In this section, we discuss the strategy for the implementation of the method.In addition, the performance of the method is tested on some examples of third order initial value problems in Ordinary Differential Equations.Absolute error of the approximate solution are then compared with the existing methods.In particular, the comparison are made with those proposed by Olabode [16] and Adesanya et al. [2] Discussion of the results of the methods are also done here.

Numerical Experiments
The method is tested on some numerical problems to test the accuracy of the proposed methods and our results are compared with the results obtained using existing methods.The following problems are taken as test problems: Exact solution:

Conclusion
We have proposed a new one-twelveth step hybrid block method for the direct numerical solution of third order initial value problems of ordinary differential equations in this paper.The method is consistent, convergent and zero stable.The method derived efficiently solved third order Initial Value Problems as can be seen in tables 1-4.In terms of accuracy, our method performs better than the existing methods compared with.

Figure 1 :
Figure 1: Region of absolute stability of our method 112 with constant step-size (h), where h = x n+i − x i , i = 0, 1, 2 and off-grid points at x n+ 1 explicitly, gives

Table 1 :
Result of test problem 1

Table 2 :
Result of test problem 2

Table 3 :
Result of test problem 3

Table 4 :
Result of test problem 4