The Computational Solution of First Order Delay Diﬀerential Equations Using Second Derivative Block Backward Diﬀerentiation Formulae

In this paper, we implemented second derivative block backward diﬀerentiation formulae meth-ods in solving ﬁrst order delay diﬀerential equations without the application of interpolation methods in investigating the delay argument. The delay argument was evaluated using a suitable idea of sequence which we incorporated into some ﬁrst order delay diﬀerential equations before its numerical evaluations. The construction of the continuous expressions of these of block methods was executed through the use of second derivative backward diﬀerentiation formulae method on the bases of linear multistep collocation approach using matrix inversion method to derive the discrete schemes. After the numerical experiments, the new proposed method was observed to be convergent, stable and less time consuming. From the numerical solutions obtained, the scheme for step number k = 4 performed better in terms of accuracy than that of the schemes for step numbers k = 3 and 2 when compared with other existing methods.


Introduction
Research has revealed that most real life situations are more realistic when they are modeled using delay differential equations (DDEs).This is because the unknown function of the delay differential equations (DDEs) does not only depends on the current value but also depends on the past value which is called a delay term.Delay differential equation (DDE) is one of the mathematical models that commonly possess the result in differential equations with time delay.In literature, various types of numerical methods have been developed and implemented in treating the problems of the delay differential equations (DDEs).Most scholars adopted the use of interpolation techniques in the evaluation of the delay term of the delay differential equations in different field of life.These interpolation techniques such as Hermite, Nordsieck, Newton divided difference and Neville's interpolation were applied by [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] in solving delay differential equations numerically have some limitations which affected the accuracy of their method.One of the limitations encountered by these researchers in the use of interpolation techniques to evaluate the delay term of DDEs was studied by [16] that the computational method use in solving DDEs should be at least the same with the order of the interpolating polynomials which is very hard to achieve; otherwise, the accuracy of the method will not be preserved.Therefore, it is required that in the evaluation of the delay term, using an accurate and efficient formula should be considered.In order to overcome the limitation posed by using interpolation techniques in evaluating the delay term, we applied the valid expression of the sequence formulated by [17] and incorporate it into the first order delay differential equations before its numerical evaluations.This approach has been successfully applied by [18][19][20][21] in finding the numerical solution of first order delay differential equations without the application of the interpolation techniques in evaluating the delay term.In this paper, we formulated and applied second derivative block backward differentiation formulae method in solving some first order delay differential equations (DDEs) of this form y (t) = f (t, y(t), y(t − τ )), for t > t0, τ > 0 where e(t) is the initial function, τ is called the delay, (t − τ ) is called the delay argument and y(t − τ ) is the solution of the delay term.The results obtained after the application of the proposed method shall be compared to other existing methods studied by [17,21] to prove its advantage.
2 Construction Techniques

Construction of Second Derivative Backward Differentiation Formulae Method
In [22] the k−step Backward Differentiation Formulae Methods was derived as And its second derivative [22] was expressed as where α b (x), β b (x) and γ b (x) are continuous coefficients of the method defined as where I represents the unit matrix of dimension (u+v)×(u+v) and R and Q are matrices presented as give the continuous coefficients of the continuous scheme (2.2).

Construction of Second Derivative Block Backward Differentiation
Formulae Method for k=2 Here, the number of interpolation points, u = 2 and the number of collocation points v = 2. Therefore, (2.2) becomes The matrix Q in (2.6) becomes The inverse of the matrix R = Q −1 is computed using Maple 18 to obtain the continuous scheme is obtained using (2.6) and evaluating it at x = x a+2 and its derivative at x = x a+1 , the following discrete schemes are obtained

Construction of Second Derivative Block Backward Differentiation Formulae Method for k=3
Here, also the number of interpolation points, u = 3 and the number of collocation points, v = 2. Therefore, (2.2) becomes 12) The matrix Q in (2.6) becomes The inverse of the matrix R = Q −1 is computed using Maple 18 to obtain the continuous scheme is obtained using (2.6) and evaluating it at x = x a+3 and its derivative at x = x a+1 and x = x a+2 , the following discrete schemes are obtained

Construction of Second Derivative Block Backward Differentiation Formulae Method for k=4
Here, also the number of interpolation points, u = 4 and the number of collocation points, v = 2. Therefore, (2.2) becomes The inverse of the matrix R = Q −1 is computed using Maple 18 to obtain the continuous scheme is also obtained using (2.6) and evaluating it at x = x a+4 and its derivative at x = x a+1 , x = x a+2 and x = x a+3 , the following discrete schemes are obtained

Convergence Analysis
Here, the investigations of order, error constant, consistency, zero stability and region of the absolute stability of (2.11), (2.14) and (2.17) will be carried out.

Order and Error Constant
The SDBBDFM (2.2) is said to be of order Ω if C0 = C1 = ... CΩ = 0 and the first non-zero coefficient CΩ+1 = 0 is the error constant as developed by [25].The order and error constant for (2.11) are obtained as follows Following the same approach, (2.14) can be presented as Applying the same approach, (2.17) can be obtained as

Consistency
Since Ω = 1 in (2.11), (2.14) and (2.17) satisfying the condition for consistency of order Ω ≥ 1 as stated by [25], then the discrete schemes are said to be consistent.

Zero Stability
The discrete schemes (2.11), (2.14) and (2.17) are said to be zero stable if the no root of the first characteristic polynomial is greater than 1.

2 ,
the discrete schemes in (2.11) is zero stable.Applying the same technique for (2.14) and presented as follows