Analytical Algorithm for Systems of Neutral Delay Differential Equations

Delay differential equations (DDEs), as well as neutral delay differential equations (NDDEs), are often used as a fundamental tool to model problems arising from various areas of sciences and engineering. However, NDDEs partic-ularly the systems of these equations are special transcendental in nature; it has therefore, become a challenging task or times almost impossible to obtain a convergent approximate analytical solution of such equation. Therefore, this study introduced an analytical method to obtain solution of linear and nonlinear systems of NDDEs. The proposed technique is a combination of Homotopy analysis method (HAM) and natural transform method, and the He’s polynomial is modified to compute the series of nonlinear terms. The presented technique gives solution in a series form which converges to the exact solution or approximate solution. The convergence analysis and the maximum estimated error of the approach are also given. Some illustrative examples are given, and comparison for the accuracy of the results obtained is made with the existing ones as well as the exact solutions. The results reveal the reliability and efficiency of the method in solving systems of NDDEs and can also be used in various types of linear and nonlinear problems.


Introduction
Ordinary differential equations (ODEs) are usually used as a fundamental tool in modelling the problems of the real world. However, in most cases, the mathematical formulation of real-life problems needs to consider both the present and past states of the system behaviour. Different from (ODEs), DDE is a type of differential equation in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous time [1]. Hence, more reliable models of real problems arising from various field of studies such as; biology, population dynamics, chemistry, and physics, control theory to mention but few are now model using DDEs as well as NDDEs [2] [3] [4] [5] [6]. Recently series of methods have been developed to find an approximate analytical solution to different types of DDEs [7]- [12]. However, most of these methods have experienced a series of challenges in finding a convergent approximate analytical solution of NDDEs in particular system of such equations. So, scientist and engineers adopt the use of numerical methods as the best approach to approximate the solutions. Therefore, more analytical approaches are highly needed for solving these equations.
This work aims to develop an analytical technique for solving linear and nonlinear systems of NDDEs from the combination of HAM and Natural transform. The proposed technique improved on the work of Rebenda and Smarda [12] by introducing the concept of NDDE into Natural transform. In addition, the various derivatives for both proportional and constants delay of NDDEs were successfully generated using the Natural transform. This work is also an extension of Efficient Analytical Approach for Nonlinear System of Retarded Delay Differential equations. This approach was developed by Barde and Maan [13] and solutions to different types of nonlinear systems of retarded DDEs were obtained in a series form which converges to exact or approximate solution.
Thus, based on the results of this pervious works, this research focuses to develop a new analytical technique that modifies Efficient Analytical Approach for Nonlinear System of Retarded Delay Differential Equations with aims to obtain approximate analytical solution for both linear and nonlinear system of NDDEs with proportional and constant delays. Using the introduced technique, the He's polynomial is adjusted in order to ease the computational difficulties of nonlinear terms of such equations. Furthermore, the convergence analysis and the maximum estimated error of the technique are also investigated.
Therefore, in this work we were able to develop a new generating function in Equation (29) that provides a convergent analytical solution to various types of linear and nonlinear system of NDDEs in a series form using few numbers of computational terms and minimal error as compared with the previous techniques. Thus, different from some of the existing methods the presented technique provides solution to different form of linear and nonlinear NDDEs without any linearization, perturbation or unnecessary assumptions. In Section 4, some illustrative examples are presented in order to show the reliability and efficiency of our algorithms over the reference methods.

Methods
The idea of this work is come up with analytical approach from the combination of natural transform and HAM for solving systems of linear and nonlinear NDDEs. The He's polynomial is modified to compute the series of nonlinear where + N denotes as natural transform and , s u are transforming variables. Equation (1) can be simplified as [23] ( ) as in the given integral: Theorem 2.1 [24] The generalised natural transform of the function ( ) v t is given as Theorem 2.2 [20] If H τ is the Heaviside function and for any real number 0 τ ≥ we defined Theorem 2.3 [24] Let ( ) ( ) n v t be the nth derivatives of the function ( ) v t then its natural transform is given by Using an induction method, for 1 n = and 2 n = we respectively obtained the natural transform of first and second derivatives of Now suppose the result holds for n, then we have to show its also true for 1 n + . Now from Equation (9) we have which is true for 1 n + and hence the result.
A. Barde Proof Also by induction, for 1 n = and 2 n = we respectively have the natural transform of first and second derivatives of ( ) Now assume Equation (11) is true for n and we have to show for which is true for 1 n + and hence the proof. Another important point to note is that, in both the concept of HAM and natural transform there is no direct approach for the computation of nonlinear terms of the system of NDDEs for both Proportional and constant delays.
Therefore, in this research the He's polynomial will be adjusted for the series calculation of these nonlinear terms.

Analysis of the Result
Consider the following n-order system of NDDEs where Now for simplicity we rewrite Equation (14) in the following form Subject to a given initial conditions. The v γ is defined as N-dimentional vector of the form . The linear terms are decom- Take the natural transform of both sides of Equation (16) to obtain: Note: This research considered two forms of delay functions Case I: Case II: Therefore, by substituting the given initial condition into Equation (17), and simplify using the differential properties of natural transform we respectivily obtained the following for the two types of delay as defined in Case I and Case II.  ; ; e ; 0 lim ; where [ ] 0,1 q ∈ is an embedding parameter, ( ) are functions of variables t and q.
So, by means of HAM we can construct the following Homotopy Equations as where + N denotes as natural transform, Now, from Equation (20) as 0 q = and 1 q = we respectively obtained the following equation.
Thus, as q increases from 0 to 1, the solutions ( ) Define vectors By differentiate Equation (20) m times with respect to q and setting 0 q = and finally divided by ! m we obtain the so called mth-order deformation equation as   Hence, as M → ∞ the following recursive relations of Equations (14) and (15) for the two type of delay as defined respectively in Case I and Case II are obtained , , , The proof for Case II (constant delay) is of the same process with that of Case I.   Barde Hence, for all , m p N ∈ with m p ≥ , by means of Equation (33) and using triangle inequality we seccessively obtained the following , , , Therefore, choosing , p m N > then we obtain the following ( ) This shows that { }  (14) and (15). According to Equation (16) we have    (23) converges to the solutions ( ) i v t of Equations (14) and (15) The Proof is now complete.

Examples and Discussion
The application of the proposed technique will be presented in this section. This involve solving some problems of linear and nonlinear systems of NDDEs with both proportional and constant delays.
Example 4.1 [10] First we seek for a solution of the following 2-dimensional linear system of NDDEs with constant delay By choosing an initial approximations of ( ) The series solutions in Equation (46)  with minimum error as shown in Figure 1 While in most applications only numerical approximations was obtained. For instance, in [10] the numerical approximation of this problem was computed using Implicit Block method with the maximum absolute error of 1.47320 when the tolerance was 1 × 10 −10 in a total number of 25 steps.
Example 4.2 [12] Next we consider a third-order nonlinear system of NDDEs with both proportional and constant delays From Equation (48) define a non-linear operator Now using Equation (29) Using only one iteration of the derived algorithm (first order) in Equation (29) a good approximation of Example 4.2 was successfully obtained. Since this problem has no exact solution therefore, Figure 2 shows the comparison between approximate solution obtained by the proposed technique, Matlab Package DDENSD and the result obtained by Rebenda and Smarda [12] using an algorithm based on the combination of the method of steps and differential transform method (DT).
Threrefore, from Figure 2 we can observed that there is a good correspondence between the first order approximate solution of the proposed tecnique with that of Matlab Package DDENSD and DT. Hence, this shows that the presented method provides reliable results and reduces the computation size as compared with the previous techniques.

Conclusion
This paper presents an efficient analytical approach suitable for solving linear and nonlinear systems of NDDEs with proportional and constant delays via HAM and natural transform. The presented algorithm adjusted the He's polynomial in order to ease the computational difficulties of both proportional and constant delays. Another advantage of this research is that a new algorithm is constructed in Equation (25) which reduces the computational work as compared to other methods, produces a much faster convergent approximate solution and handles more complicated problems (as in the case of second example) in applications than other analytical methods. Therefore, the presented approach is efficient and reliable in solving different form of linear and nonlinear systems of NDDEs which can be also applied to solve various types of linear and nonlinear problems.