Numerical Solution of Third Order Ordinary Differential Equations Using a Seven-step Block Method

This paper aims to provide a direct solution to third order initial value problems of ordinary differential equations. Multistep collocation approach is adopted in the derivation of the method. The new block method is zero-stable, consistent and convergent. The application of the new method to solving differential equations gives better results when compared with the existing methods.

The numerical solution of higher order ordinary differential equations through the reduction method was majorly used in the past in such a way that the differential equation will be reduced to its equivalent system of first order and suitable numerical methods designed for first order were used to solve the resulting systems (Lambert [1], Brown [2], Jeltsch [3]).
Direct methods of solving higher order ordinary differential equations had been examined by some authors like Awoyemi [4], Mohammed [5] and Omar and Suleiman [6][7][8] .One of the methods of solving higher order ordinary differential equations directly is predictor-corrected method and this is discussed extensively in Awoyemi [4], Adesanya et al. [9], Odekunle et al. [10], Kayode and Adeyeye [11] and Kayode and Obarhua [12].It is observed that the method has a lot of setbacks and this includes:

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Too many functions to be evaluated per step due to the involvement of predictors which always result to computational burden that affects the accuracy of the method.

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The computer program designed to examine the accuracy of the method are always found to be complicated

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A lot of computer time and human effort are involved.
The use of method without predictors was adopted by Omar & Suleiman [6][7][8], Ehigie et al. [13] and Adesanya et al. [14].These scholars independently developed methods for solving higher order ordinary differential equations to proffer solution to the setbacks in predictor-corrector method.The development of block methods for solving second order ordinary differential equations have been carried out by many researchers but few literatures are found on block method for solving (1) directly.Yap, Ismail and Senu [15] developed accurate block hybrid collocation method with order six for solving third order ordinary differential equations.Furthermore, an accurate scheme by block method having an order seven for solving (1) directly was developed by Olabode [16] but the method is of lower accuracy.
This paper proposes a block method with uniform order eight for solving (1) directly.The use of approximated power series as an interpolation equation and its derivative as a collocation equation is adopted in the development of the method.

Derivation of the Method
We consider power series approximate solution of the form as an interpolation equation.Where k=7.The first, second and third derivatives of (2) give We interpolate equation (2 and equation ( 5) is collocated at . The interpolation and collocation equations give where, Solving for the unknown variables aj using Gaussian elimination method which are then substituted into the interpolation equation ( 2) produces a continuous implicit scheme of the form

Numerical solution of third order ordinary differential equations 747
In order to find the discrete schemes and its derivatives, equation ( 7) is evaluated at The first and second derivatives are evaluated at . These discrete schemes and its derivatives at n x are combined in a matrix form to produce a block of the form:    The first and second derivatives of (8) give as defined by Lambert [1].

Zero Stability
The method ( 8) is said to be zero-stable as h 0 and for the root with 1  p z the multiplicity must not exceed the order of differential equation under consideration.This is demonstrated below 0 . Therefore, our method

Conclusion
We have developed a block method of order P=8 for direct solution of general third order ordinary differential equations.The results generated when the new method was applied to third order initial value problems show a better performance over the existing methods.

Table 3 :
[16]arison of the results of the new method with Olabode[16]