A New Block-Predictor Corrector Algorithm for the Solution of , , , y f x y y y

We consider direct solution to third order ordinary differential equations in this paper. Method of collection and interpolation of the power series approximant of single variable is considered to derive a linear multistep method (LMM) with continuous coefficient. Block method was later adopted to generate the independent solution at selected grid points. The properties of the block viz: order, zero stability and stability region are investigated. Our method was tested on third order ordinary differential equation and found to give better result when compared with existing methods.


Introduction
This paper considers the general third order initial value problems of the form

y f x y y y y x y y x y y x y
Conventionally, higher order ordinary differential equations are solved directly by the predictor-corrector method where separate predictors are developed to implement the correct and Taylor series expansion adopted to provide the starting values.Predictor-corrector methods are extensively studied by [1][2][3][4][5].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 [6] The major setbacks of predictor-corrector method are extensively discussed by [7].
Lately, many authors have adopted block method to solve ordinary differential equations because it addresses some of the setbacks of predictor-corrector method discussed by [6].Among these authors are [8][9][10].
According to [6], the general block formula is given by where is e s s  vector, is r-vector and is vector,  s is the interpolation points and is the collection points.
Given a predictor equation in the form Equation ( 4) is called a self starting block-predictorcorrector method because the prediction equation is gotten directly from the block formula as claimed by [11,12].
In this paper, we propose an order six block method with step length of four using the method proposed by [11] for the solution of third order ordinary differential equation.

Derivation of the Continuous Coefficient
We consider monomial power series as our basis function in the form The third derivative of (5) gives The solution to (1) is soughted on the partition π N : within the integration interval [a,b] with constant step length given as s r j j j f x y x y x y x j j j a x ; , (7) Interpolating (5) at collocating (7) gives a system of equations Solving ( 8) and ( 9) for j a 's and substituting back into (5) gives a LMM with continuous coefficients of the form where j a 's and j  's are given as

Derivation of the Block Method
The general block formula proposed by [6] in the normalized form is given by Evaluating (10) at 1 , 0,4 the first and second derivative at and substituting into (10) gives the coefficient of ( 11

Order of the Method
We define a linear operator on the block (11) to give Expanding in Taylor series , (12) gives The block (11) and associated linear operator are said to have order if p The term 2 p is called the error constant and implies that the local truncation error for the block is given by Hence the block (11)

Zero Stability of the Block
The block ( 11) is said to be zero stable if the roots of the characteristic polynomial satisfies where  is the order of the differential equation, Hence our method is zero stable.

Convergence
A method is said to be convergent if it is zero stable and has order . 1 p  From the theorem above, our method is convergent.

Test Problem
We test our schemes with third order initial value problems: Problem 1.Consider a special third order initial value This problem was solved by [13] using self-starting predictor-corrector method for special third order differential equations where a scheme of order six was proposed.
Problem 2. Consider linear third order initial value problem x This problem was solved by [14] where a method of order six was proposed.They adopted predictor corrector method in their implementation.Our result is shown in Table 1.

Numerical Results
The following notations are used in the table.
XVAL: Value of the independent variable where numerical value is taken; ERC: Exact result at XVAL; NRC: Numerical result of the new result at XVAL; ERR: Magnitude of error of the new result at XVAL.

Discussion
We have proposed a new block method for solving third order initial value problem in this paper.It should be noted that the method performs better when the step-size is chosen within the stability interval.The Tables 1 and 2 had shown our new method is more efficient in terms of accuracy when compared with the self starting predictor  corrector method proposed by [11] and [15].It should be noted that this method performs better when the step size (h) is within the stability interval.