A CLASS OF SIXTH ORDER HYBRID EXTENDED BLOCK BACKWARD DIFFERENTIATION FORMULAE FOR COMPUTATIONAL SOLUTIONS OF FIRST ORDER DELAY DIFFERENTIAL EQUATIONS

In this paper, we established and carried-out the computational solution of some first order delay differential equations (DDEs) using hybrid extended backward differentiation formulae method in block forms without the application of interpolation techniques in determining the delay term. The discrete schemes were worked-out through the linear multistep collocation technique by matrix inversion approach from the continuous construction of each step number which clearly demonstrated the order and error constants, consistency, zero stability, convergence and region of absolute stability of this method after investigations. The results obtained after the implementation of this method validate that the lower step number integrated with hybrid extended future points performed better than the higher step numbers integrated with hybrid extended future points when compared with the exact solutions and other existing methods.


INTRODUCTION
Delay differential equations have displayed great importance in its applications and diversities in modeling real life situations. Delay differential equations (DDEs) are differential equations which considers time series of past values of dependent variables and derivatives, whereas the evolution of ODEs depends only on the current values of these quantities. Every real life phenomenon involves delay and has been studied by many researchers. For decades, notable scholars have carried-out intensive research activities in the area of computational solutions for delay differential equations which have shown the numerous advantages of DDEs over ODEs for mathematical modeling of real life problems. The applications of DDEs can be seen in many real life situations involving celestial and quantum mechanics, nuclear and theoretical physics, astrophysics, quantum chemistry, molecular dynamics, engineering, medicine and economic dynamics and control. In physical sciences, Tziperman., Stone., Cane., Jarosh.,(1994), delay differential equations was used in the modeling of El Nino temperature oscillations in the Equatorial Pacific to determine the model single-species population growth. In electrical circuits, delays are introduced because it takes time for a signal to travel through a transmission line.
In this research work, we look forward to obtaining the computational solutions of the first order delay differential equations (DDEs) of the form: where () t  is the initial function, is called the delay, () t  − is called the delay argument and 3499 FIRST ORDER DELAY DIFFERENTIAL EQUATIONS delay differential equations without using any interpolation techniques in examining the delay term. Osu, Chibuisi, Okwuchukwu, Olunkwa, and Okore. (2020), implemented third derivative block backward differentiation formulae for numerical solutions of first order delay differential equations without interpolation techniques in investigating the delay argument. Chibuisi, Osu, Amaraihu and Okore. (2020), solved first order delay differential equations using multiple offgrid hybrid block simpson's methods without the application of any interpolation technique in estimating the delay term. Chibuisi, Osu, Edeki and Akinlabi. (2020), solved first DDEs using extended block backward differentiation formulae for efficiency of the numerical solution without the introduction of interpolation techniques in finding the delay argument.
The difficulty in applying these interpolation techniques by Majid et al., (2012) is that the numerical methods to be implemented in solving DDEs should be the same with the interpolating polynomials which is very difficult to carry out; otherwise, the accuracy of the method will not be preserved. It is important that in the evaluation of the delay argument, applying a reliable and coherent formula shall be highly recommended.
In order to prevail over the difficulty caused by using interpolation techniques in evaluating the delay argument; we applied idea of the sequence constructed by Sirisena et al. (2019) which we merge into the first order delay differential equations. Then we implemented hybrid extended block backward differentiation formulae methods to solve some first order delay differential equations containing the evaluated delay argument to improve the performance of the existing extended BBDF been studied by  in terms of efficiency, accuracy, consistency, convergence and region of absolute stability at constant step width a .

Construction of Multistep Collocation Approach
The k-step multistep collocation approach with e collocation points was derived In Sirisena (1997) as; 1 From the matrix equation (5), the columns of give the continuous coefficients of the continuous scheme of (2).

Construction of HEBBDF Method with Integrated One Off-grid Extended Future Point
and One Extended Future Point for 4 k = With the same procedure, we integrated one off-grid extended future points at 9 2 v x x + = and one extended future point at as collocation points, thus the interpolation points, 4 g = and the collocation points 3 e = are considered, therefore, (2) becomes: The matrix A in (5) becomes 22

CONVERGENCE ANALYSIS
Here, the examinations of order, error constant, consistency, zero stability and region of the absolute stability of (9), (11) and (13) are presented.

Region of Absolute Stability
The regions of absolute stability of the numerical methods for DDEs are considered. We considered finding the U -and W -stability by applying (9), (11) and (13) to the DDEs of this form: With the same technique for (11) Applying the same approach for (13), we have and we have, The polynomials of V -and W -stability are constructed by introducing (18), (19) and (20) to (17) and (9), (11) and (13) to (17) as stated below START

NUMERICAL COMPUTATIONS
In this section, some first-order delay differential equations shall be solved using (9), (11) and (13)

RESULTS AND DISCUSSIONS
Here, the solutions of the schemes derived in (9), (11) and (13), shall be examined in solving the two problems above by computing their absolute errors.

Analysis of Results
The analysis of results is obtained by determining absolute differences of the exact solutions and the numerical solutions. The results are presented in the tables 5.1.1 to 5.1.2,

Conclusions
In conclusion, the discrete schemes of (9), (11) and (13), were worked-out from their individual continuous form and were revealed to be convergent, U -and W -stable. Also, it was revealed in 3533 FIRST ORDER DELAY DIFFERENTIAL EQUATIONS tables 5.1.1 to 5.1.2 that the lower step number of HEBBDF method with integrated off-grid extended future points and one extended future point performed better than the higher step numbers of HEBBDF method with integrated off-grid extended future points and one extended future point when compared with the exact solutions. Most Importantly, this method performed better in terms of efficiency, accuracy, consistency, convergence and region of absolute stability at constant step width a when compared with other existing method as presented in table 5

CONFLICT OF INTERESTS
The author(s) declares that there is no conflict of interests.