IMPLICIT FOUR-STEP APPROACH WITH APPLICATION TO NON-LINEAR THIRD ORDER ORDINARY DIFFERENTIAL EQUATIONS

A unique and efficient implicit four-step approach with application to nonlinear third order ordinary differential equations is considered in this article. In the derivation of this method Collocation and Interpolation techniques were engaged and power series approximate solution was used as the interpolating polynomial. The third derivative of the power series was collocated at the entire grid points, while the interpolation was done at the first three points. Appropriate study of the basic properties of the method was done. The results generated when the new block method was applied on nonlinear third order ordinary differential equations are better in terms of accuracy than the existing methods.


INTRODUCTION
The numerical solution of nonlinear third order initial value problems (IVPs) of ordinary differential equations (ODEs) directly using a unique implicit four-step linear multistep block method is studied in this research. These ODEs which are frequently met in our everyday lives are of the form ( ) = ( , , , ), ( ) = 0 , ( ) = 1 , ( ) = 2 (1) Equation (1) arises in diverse fields of applied mathematics, amongst which are elasticity, fluid mechanics, and quantum mechanics as well as in control system, engineering and physics. The existence and uniqueness of the solution for these equations have been discussed extensively in Adeniran & Omotoye (2016) and Wend (1969). In general, finding the exact solutions of these equations is not easy. For instance, the application problem in fluid mechanics named Fluid flow does not have exact solution, hence it is important to get the numerical solutions [3,7,9]. For a long time, different numerical methods have been developed in order to approximate the solution of equation (1). Among these methods are block method, linear multistep method, hybrid method, Taylor series and Rung-kutta method, see Henrici (1962), Kayode et al., (2018), Adoghe et al., (2016), Adeniran & Omotoye (2016), Abdelrahim et al., (2019), Ukpebor (2019), Ogunware et al., (2018), andYao et al., (2011). This article is motivated to derive a Four-step approach with an application to nonlinear third order ordinary differential equations via power series as the basic function. This work is motivated by the success story of block methods for solving ordinary differential equations directly without reducing it to system of first order ordinary differential equation. The advantages of the method lie in the fact that it is economical, saves time and computationally reliable.

Consistency
Definition 3.1: The Four-step block method (24-27) is said to be consistent if it has an order more than or equal to one i.e.
. Therefore, the method is consistent (Abdelrahim et al., (2019) and Lambert 1973). Zero Stability Definition 3.2: The hybrid block method (24-27) said to be zero stable if the first characteristic polynomial having roots such that , then the multiplicity of must not greater than six as discussed in Wend (1969) and Ogunware et al. (2015).
In order to find the zero-stability of Four-step block method (24-27), we only consider the first characteristic polynomial of the method as follows (37) which implies . Hence the method is zero-stable since .

Convergence
Theorem (3.1): Consistency and zero stability are sufficient condition for linear multistep method to be convergent. Since the method (22-24) are consistent and zero stable, it implies the method is convergent for all point (as reported in Kayode et al., (2018), Adoghe et al., (2016) and Ukpebor (2019)).

Implementation of the Block Methods
In this section, we implement our derived method (24)

Numerical Examples
The method is specifically developed to examine third order nonlinear 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:

DISCUSSION OF RESULTS
In this section, the tables of results will be extensively discussed. Table 1 shows the exact solution, computed solution and the error in the method for problem 1. The comparison of error in new method with another error in the literature is also made. Specifically, Adoghe et al., (2016) who proposed a linear multistep method of order 5. As it could be seen in Table 1 , the four-step block method of order 5 proposed in this work is better in terms of accuracy than that of Adoghe et al (2016). On the other hand, Table 2 shows the computation of an application problem in Fluid Mechanics namely Thin Flow. The problem was solved by Adeniran and Omotoye (2016) using h=0.1. The results show that the proposed method is more accurate when compared with other method in the literature. The method is therefore computationally reliable and recommended for general use.

CONCLUSION
In this article, the derivation of the new block method for solving third order nonlinear ordinary differential equations directly is studied. The method is of order p=5 which shows that it is consistent. The positive aspect of the method over the existing numerical methods is its ability to solve problem with exact solution and without exact solution and performance in terms of accuracy and convergence in the literature. the comparison of errors in the new method with other existing method is shown in Figure 1. The new method gives minimal error and also solves a notable real life problem namely Thin Flow which has application in fluid mechanics.