1. 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.

2. 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 terms of both proportional and constants delays.

HAM is a powerful technique introduced by Lio [14] for solving different types of linear and nonlinear problems. Details on theory and application of HAM can be found in [1] [15] [16] [17] [18] .

In recent years natural transform is considered as an active topic in research due to its vast application in solving different type of differential and integral equations [19] [20] [21] [22] . This transform was derived from the renowned Fourier integral which converged to either Laplace or Sumudu transforms depending on the values of the transform variables. The basic concepts of natural transform for further use in this research are rendered below.

Definition 2.1 [23] Let $t\in \left(-\infty \mathrm{,}\infty \right)$ , then the natural transform of the function $v\left(t\right)$ is defined by:

$N^{+}\left[v\left(t\right)\right]=V\left(s\mathrm{,}u\right)={\displaystyle {\int }_{-\infty }^{\infty }{\text{e}}^{-st}v\left(ut\right)\text{d}t}\mathrm{<mo>\; ;\; </mo>}s\mathrm{,}u\in \left(\mathrm{0,}\infty \right]\mathrm{.}$ (1)

where $N^{+}$ denotes as natural transform and $s\mathrm{,}u$ are transforming variables.

Equation (1) can be simplified as [23]

$\begin{array}{c}{N}^{+}\left[v\left(t\right)\right]=V\left(s\mathrm{,}u\right)={\displaystyle {\int }_{-\infty }^{\infty }{\text{e}}^{-st}v\left(ut\right)\text{d}t}\mathrm{<mo>\; ;\; </mo>}s\mathrm{,}u\in \left(\mathrm{0,}\infty \right]\\ ={\displaystyle {\int }_{-\infty }^0{\text{e}}^{-st}v\left(ut\right)\text{d}t}\mathrm{<mo>\; ;\; </mo>}s\mathrm{,}u\in \left(-\infty \mathrm{,0}\right)+{\displaystyle {\int }_0^{\infty }{\text{e}}^{-st}v\left(ut\right)\text{d}t}\mathrm{<mo>\; ;\; </mo>}s\mathrm{,}u\in \left(\mathrm{0,}\infty \right)\\ =N^{-}\left[v\left(t\right)\right]+N^{+}\left[v\left(t\right)\right]\\ =N\left[v\left(t\right)H\left(-t\right)\right]+N\left[v\left(t\right)H\left(t\right)\right]\\ =V^{-}\left(s\mathrm{,}u\right)+V^{+}\left(s\mathrm{,}u\right)\end{array}$ (2)

where, $H\left(\mathrm{.}\right)$ is the Heaviside function. Assume the function $v\left(t\right)H\left(t\right)$ is defined on $R^{+}$ and for $t\in R$ then its natural transform can be define over the set

$A=\left\{v\left(t\right)\mathrm{<mo>\; :\; </mo>}\exists M\mathrm{,}{\tau }_1\mathrm{,}{\tau }_2>\mathrm{0,}\left|v\left(t\right)\right|<M{\text{e}}^{\frac{\left|t\right|}{{\tau }_j}}\mathrm{,}t\in {\left(-1\right)}^j\times \left[\mathrm{0,}\infty \right)\mathrm{,}j\in Z^{+}\right\}$

as in the given integral:

$N^{+}\left[v\left(t\right)\right]=V^{+}\left(s\mathrm{,}u\right)={\displaystyle {\int }_0^{\infty }{\text{e}}^{-st}v\left(ut\right)\text{d}t}\mathrm{<mo>\; ;\; </mo>}s\mathrm{,}u\in \left(\mathrm{0,}\infty \right]\mathrm{.}$ (3)

Theorem 2.1 [24] The generalised natural transform of the function $v\left(t\right)$ is given as

$N^{+}\left[v\left(t\right)\right]=V^{+}\left(s\mathrm{,}u\right)={\displaystyle \underset{n=0}{\overset{\infty }{\sum }}}\frac{n\mathrm{<mo>\; !\; </mo>}{a}_n{u}^n}{s^{n+1}}\mathrm{.}$ (4)

Property 2.1 [23] Let a be a non-zero constant and $v\left(at\right)\in A$ then,

$N^{+}\left[v\left(at\right)\right]=\frac{1}{a}V\left(\frac{s}{a}\mathrm{,}u\right)\mathrm{.}$ (5)

Theorem 2.2 [20] If $H_{\tau }$ is the Heaviside function and for any real number $\tau \ge 0$ we defined

$H_{\tau }\left(t\right)=(\begin{array}{l}1,\text{for}t\ge \tau \\ 0\text{for}t<\tau \end{array}$

Then the natural transform of the shifted function $v\left(t-\tau \right)=v\left(t-\tau \right)H_{\tau }\left(t\right)$ is given by

$N^{+}\left[v\left(t-\tau \right)H_{\tau }\left(t\right)\right]={\text{e}}^{\frac{-s\tau }{u}}{N}^{+}\left[v\left(t\right)\right]\mathrm{.}$ (6)

Theorem 2.3 [24] Let $v^{\left(n\right)}\left(t\right)$ be the nth derivatives of the function $v\left(t\right)$ then its natural transform is given by

$N^{+}\left[v^n\left(t\right)\right]=V_n^{+}\left(s\mathrm{,}u\right)=\frac{s^n}{u^n}V\left(s\mathrm{,}u\right)-{\displaystyle \underset{k=1}{\overset{n}{\sum }}}\frac{s^{n-k}}{u^{\left(n-k\right)+1}}{v}^{k-1}\left(0\right)\mathrm{.}$ (7)

Corollary 2.1

Let $v_i^{\left(n\right)}\left(at\right)$ be the nth derivatives of the functions $v_i\left(at\right)$ with respect to t, $\left(i=\mathrm{1,2,}\cdots \mathrm{,}N\right)$ and suppose that $N^{+}\left[v_i\left(at\right)\right]=V_i^{+}\left(as\mathrm{,}u\right)$ then we define the following

$N^{+}\left[v_i^n\left(at\right)\right]=V_{n\mathrm{,}i}^{+}\left(as\mathrm{,}u\right)=\frac{s^n}{{\left(au\right)}^n}{V}_i^{+}\left(as\mathrm{,}u\right)-{\displaystyle \underset{k=1}{\overset{n}{\sum }}}\frac{s^{n-k}}{a{u}^{\left(n-k\right)+1}}{v}_i^{\left(k-1\right)}\left(0\right)$ (8)

Proof

Using an induction method, for $n=1$ and $n=2$ we respectively obtained the natural transform of first and second derivatives of $v_i\left(at\right)$ that is

$\begin{array}{l}{N}^{+}\left[{v^{\prime }}_i\left(at\right)\right]=V_{\mathrm{1,}i}^{+}\left(as\mathrm{,}u\right)=\frac{s}{au}{V}_i^{+}\left(as\mathrm{,}u\right)-\frac{v_i\left(0\right)}{au}\\ N^{+}\left[{v^{″}}_i\left(at\right)\right]=V_{\mathrm{2,}i}^{+}\left(as\mathrm{,}u\right)=\frac{s^2{V}_i^{+}\left(as\mathrm{,}u\right)-s{v}_i\left(0\right)}{{\left(au\right)}^2}-\frac{{v^{\prime }}_i\left(0\right)}{au}\end{array}$ (9)

Now suppose the result holds for n, then we have to show its also true for $n+1$ . Now from Equation (9) we have

$\begin{array}{c}{N}^{+}\left[v_i^{\left(n+1\right)}\left(at\right)\right]=N^{+}\left[{\left(v_i^{\left(n\right)}\left(at\right)\right)}^{\prime }\right]=V_{n+\mathrm{1,}i}^{+}\left(as\mathrm{,}u\right)=\frac{s}{au}{V}_{n\mathrm{,}i}^{+}\left(as\mathrm{,}u\right)-\frac{v_i^{\left(n\right)}\left(0\right)}{au}\\ =\frac{s}{au}\left[\frac{s^n}{{\left(au\right)}^n}{V}_i^{+}\left(as,u\right)-{\displaystyle \underset{k=1}{\overset{n}{\sum }}}\frac{s^{n-k}}{{\left(au\right)}^{\left(n-k\right)+1}}{v}_i^{\left(k-1\right)}\left(0\right)\right]-\frac{v_i^{\left(n\right)}\left(0\right)}{au}\\ =\frac{s^{n+1}}{{\left(au\right)}^{n+1}}{V}_i^{+}\left(as,u\right)-{\displaystyle \underset{k=1}{\overset{n+1}{\sum }}}\frac{s^{\left(n-k\right)+1}}{{\left(au\right)}^{\left(n-k\right)+2}}{v}_i^{\left(k-1\right)}\left(0\right)\end{array}$ (10)

which is true for $n+1$ and hence the result.

Corollary 2.2

Suppose $v_i^{\left(n\right)}\left(t-\tau \right)$ are the nth derivatives of the shifted functions $v_i\left(t-\tau \right)$ with respect to t, then their natural Transforms can be define as

$\begin{array}{c}{N}^{+}\left[v_i^n\left(t-\tau \right)\right]={\text{e}}^{\left(-\frac{s\tau }{u}\right)}{V}_{n\mathrm{,}i}^{+}\left(s\mathrm{,}u\right)\\ =\frac{s^n}{u^n}{\text{e}}^{\left(-\frac{s\tau }{u}\right)}{V}_i^{+}\left(s\mathrm{,}u\right)-{\displaystyle \underset{k=1}{\overset{n}{\sum }}}\frac{s^{n-k}}{{\left(u\right)}^{\left(n-k\right)+1}}\left[\underset{t\to 0}{\lim }{v}_i^{\left(k-1\right)}\left(t-\tau \right)\right]\end{array}$ (11)

Proof

Also by induction, for $n=1$ and $n=2$ we respectively have the natural transform of first and second derivatives of $v_i\left(t-\tau \right)$ , that is

$\begin{array}{l}{N}^{+}\left[{v^{\prime }}_i\left(t-\tau \right)\right]=\frac{s}{u}{\text{e}}^{\frac{-s\tau }{u}}{V}_i^{+}\left(s\mathrm{,}u\right)-\frac{1}{u}\underset{t\to 0}{\lim }{v}_i\left(t-\tau \right)\\ N^{+}\left[{v^{″}}_i\left(t-\tau \right)\right]=\frac{s^2{\text{e}}^{\frac{-s\tau }{u}}{V}_i^{+}\left(s\mathrm{,}u\right)-s\underset{t\to 0}{\lim }{v}_i\left(t-\tau \right)}{u^2}-\frac{\underset{t\to 0}{\lim }{v^{\prime }}_i\left(t-\tau \right)}{u}\end{array}$ (12)

Now assume Equation (11) is true for n and we have to show for $n+1$ . From Equation (12) we have

$\begin{array}{l}{N}^{+}\left[v_i^{\left(n+1\right)}\left(t-\tau \right)\right]=N^{+}\left[{\left(v_i^n\left(t-\tau \right)\right)}^{\prime }\right]=\frac{s}{u}{N}^{+}\left[v_i^{\left(n\right)}\left(t-\tau \right)\right]-\frac{\underset{t\to 0}{\lim }{v}_i^n\left(t-\tau \right)}{u}\\ =\frac{s}{u}\left[\frac{s^n}{u^n}{\text{e}}^{\frac{-s\tau }{u}}{V}_i^{+}\left(s\mathrm{,}u\right)-{\displaystyle \underset{k=1}{\overset{n}{\sum }}}\frac{s^{n-k}}{u^{\left(n-k\right)+1}}\left[\underset{t\to 0}{\lim }{v}_i^{\left(k-1\right)}\left(t-\tau \right)\right]\right]-\frac{\underset{t\to 0}{\lim }{v}_i^{\left(n\right)}\left(t-\tau \right)}{u}\\ =\frac{s^n}{u^n}{\text{e}}^{\frac{-s\tau }{u}}{V}_i^{+}\left(s\mathrm{,}u\right)-{\displaystyle \underset{k=1}{\overset{n+1}{\sum }}}\frac{s^{n-k+1}}{u^{\mathrm{<mo\; stretchy="false">\; (\; </mo>}n-k\mathrm{<mo\; stretchy="false">\; )\; </mo>}+2}}\left[\underset{t\to 0}{\lim }{v}_i^{\left(k-1\right)}\left(t-\tau \right)\right]\end{array}$ (13)

which is true for $n+1$ 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.

3. Analysis of the Result

Consider the following n-order system of NDDEs

$\begin{array}{l}{\left[v_i\left(t\right)+v_i\left(\alpha \left(t\right)\right)\right]}^{\left(n\right)}\\ =F_i\left[t\mathrm{,}{v}_{\gamma }^{\left(p\right)}\left(t\right)\mathrm{,}{v}_{\gamma }^{\left(p\right)}\left({\alpha }_{i\mathrm{,}j}\left(t\right)\right)\right]\mathrm{,}t\in \left[\mathrm{0,}d\right]\mathrm{,}i=\mathrm{1,}\cdots \mathrm{,}N\mathrm{,}J=\mathrm{1,}\cdots \mathrm{,}M\end{array}$ (14)

where

$\begin{array}{l}{v}_{\gamma }^{\left(p\right)}\left(t\right)=\left(v_1^{\left(p\right)},v_2^{\left(p\right)},\cdots ,v_N^{\left(p\right)}\right),\\ v_{\gamma }^p\left({\alpha }_{i,j}\left(t\right)\right)=\left(v_1^{\left(p\right)}\left({\alpha }_{i,j}\left(t\right)\right),v_2^{\left(k\right)}\left({\alpha }_{i,j}\left(t\right)\right),\cdots ,v_N^{\left(p\right)}\left({\alpha }_{i,j}\left(t\right)\right)\right)\end{array}$

for $p=0,1,2,\cdots ,n-1$ and ${\alpha }_{i\mathrm{,}j}\left(t\right)$ are the functions of delay terms such that $\alpha \left(t\right)=\max \left[{\alpha }_{i,j}\left(t\right)\right]$

with the given initial conditions

$v_i^{\left(p\right)}\left(0\right)=v_{i,0}^{\left(p\right)},v_i\left(t\right)={\psi }_i\left(t\right),t<0$ (15)

Now for simplicity we rewrite Equation (14) in the following form

$L_i\left(v_i+v_i\left(\alpha \right)\right)+R_i\left(v_{\gamma }\right)+F\left(v_{\gamma }\right)=g_i\left(t\right)$ (16)

Subject to a given initial conditions. The $v_{\gamma }$ is defined as N-dimentional vector of the form $v_{\gamma }=\left[v_1\left(t\right),v_2\left(t\right),\cdots ,v_N\left(t\right)\right]$ . The linear terms are decomposed into bounded linear operators $L_i+R_i$ (That is there are some positive numbers ${\alpha }_{i\mathrm{,1}}\mathrm{,}{\alpha }_{i\mathrm{,2}}$ such that { $\Vert L_i\left(v_{\gamma }\right)\Vert \le {\alpha }_{i\mathrm{,1}}\Vert v_{\gamma }\Vert$ , $\Vert R_i\left(v_{\gamma }\right)\Vert \le {\alpha }_{i\mathrm{,2}}\Vert v_{\gamma }\Vert$ ) with $L_i$ as the highest order and $R_i$ as remaining of the linear operators, and $F_i$ are continuous functions satisfy the Lipschitz condition with Lipschitz constants ${\mu }_i\in \left[\mathrm{0,}d\right]$ ( $\left|f_i\left(v\right)-f_i\left(u\right)\right|\le {\mu }_i\left|v-u\right|\mathrm{,}\forall t\in \left[\mathrm{0,}d\right]$ }) represent the non-linear terms.

Take the natural transform of both sides of Equation (16) to obtain:

$N^{+}\left[L_i\left(v_i+v_i\left({\alpha }_i\right)\right)\right]+N^{+}\left[R_i\left(v_{\gamma }\right)\right]+N^{+}\left[F_i\left(v_{\gamma }\right)\right]=N^{+}\left[g_i\left(t\right)\right]$ (17)

Note: This research considered two forms of delay functions ${\alpha }_{i\mathrm{,}j}\left(t\right)$ as follows:

Case I: ${\alpha }_{i,j}\left(t\right)=a_{i,j}t$ , where $a_{i\mathrm{,}j}\in \left(\mathrm{0,1}\right)$ (proportional delay).

Case II: ${\alpha }_{i,j}\left(t\right)=t-{\tau }_{i,j}$ , where ${\tau }_{i,j}>0$ are real constants (constant delay) 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.

$\begin{array}{l}{N}^{+}\left[v_i\left(t\right)+\frac{1}{a_i^{\left(n+1\right)}}v\left(a_{i}t\right)\right]-{\displaystyle \underset{k=1}{\overset{n}{\sum }}}\left(1+\frac{1}{a_i^{\left(n+1-k\right)}}\right)\frac{u^{n-k}}{s^k}{v}_i^{k-1}\left(0\right)\\ +\frac{u^n}{s^n}{N}^{+}\left[R_i\left(v_{\gamma }\right)+F_i\left(v_{\gamma }\right)-g_i\left(t\right)\right]=0\\ N^{+}\left[v_i\left(t\right)+{\text{e}}^{\frac{-s{\tau }_i}{u}}{v}_i\left(t\right)\right]-{\displaystyle \underset{k=1}{\overset{n}{\sum }}}\frac{u^{n-k}}{s^k}\left[v_i^{k-1}\left(0\right)+\underset{t\to 0}{\lim }{v}_i^{k-1}\left(t-{\tau }_i\right)\right]\\ +\frac{u^n}{s^n}{N}^{+}\left[R_i\left(v_{\gamma }\right)+F_i\left(v_{\gamma }\right)-g_i\left(t\right)\right]=0\end{array}$ (18)

where $a_i=\max \left[a_{i,j}\right]$ and ${\tau }_i=\max \left[{\tau }_{i,j}\right]$

Now from Equation (18) we can define the following nonlinear operators

$\begin{array}{l}{N}_i\left[{\varphi }_i\left(t\mathrm{<mo>\; ;\; </mo>}q\right)\right]=N^{+}\left[{\varphi }_i\left(t\mathrm{<mo>\; ;\; </mo>}q\right)+\frac{1}{a_i^{\left(n+1\right)}}{\varphi }_i\left(a_{i}t\mathrm{<mo>\; ;\; </mo>}q\right)\right]-{\displaystyle \underset{k=1}{\overset{n}{\sum }}}\left(1+\frac{1}{a_i^{\left(n+1-k\right)}}\right)\frac{u^{n-k}}{s^k}{\varphi }_i^{k-1}\left(9\right)\\ +\frac{u^n}{s^n}{N}^{+}\left[R_i\left({\varphi }_{\gamma }\left(t\mathrm{<mo>\; ;\; </mo>}q\right)\right)+F_i\left({\varphi }_{\gamma }\left(t\mathrm{<mo>\; ;\; </mo>}q\right)\right)-g_i\left(t\right)\right]\\ N_i\left[{\varphi }_i\left(t\mathrm{<mo>\; ;\; </mo>}q\right)\right]=N^{+}\left[{\varphi }_i\left(t\mathrm{<mo>\; ;\; </mo>}q\right)+{\text{e}}^{\frac{-s{\tau }_i}{u}}{\varphi }_i\left(t\mathrm{<mo>\; ;\; </mo>}q\right)\right]-{\displaystyle \underset{k=1}{\overset{n}{\sum }}}\frac{u^{n-k}}{s^k}\left[{\varphi }_i^{k-1}\left(0\right)+\underset{t\to 0}{\lim }{\varphi }_i^{k-1}\left(t\mathrm{<mo>\; ;\; </mo>}q-{\tau }_i\right)\right]\\ +\frac{u^n}{s^n}{N}^{+}\left[R_i\left({\varphi }_{\gamma }\left(t\mathrm{<mo>\; ;\; </mo>}q\right)\right)+F_i\left({\varphi }_{\gamma }\left(t\mathrm{<mo>\; ;\; </mo>}q\right)\right)-g_i\left(t\right)\right]\end{array}$ (19)

where $q\in \left[\mathrm{0,1}\right]$ is an embedding parameter, ${\varphi }_i\left(t\mathrm{<mo>\; ;\; </mo>}q\right)$ are functions of variables t and q.

So, by means of HAM we can construct the following Homotopy Equations as

$\left(1-q\right)N^{+}\left[{\varphi }_i\left(t\mathrm{<mo>\; ;\; </mo>}q\right)-v_{i\mathrm{,0}}\left(t\right)\right]=h_{i}q{H}_i\left(t\right)N_i\left[{\varphi }_{\gamma }\left(t\mathrm{<mo>\; ;\; </mo>}q\right)\right]$ (20)

where $N^{+}$ denotes as natural transform, $v_{i\mathrm{,0}}\left(t\right)$ are initial approximations of $v_i\left(t\right)$ and $h_i\mathrm{,}H\left(t\right)$ are non-zero auxiliary parameters and auxiliary fuctions respectively.

Now, from Equation (20) as $q=0$ and $q=1$ we respectively obtained the following equation.

$\begin{array}{l}{\varphi }_i\left(t,0\right)=v_{i,0}\left(t\right)\\ {\varphi }_i\left(t,1\right)=v_i\left(t\right)\end{array}$ (21)

Thus, as q increases from 0 to 1, the solutions ${\varphi }_i\left(t\mathrm{,}q\right)$ vary from the initial approximations $v_{i\mathrm{,0}}\left(t\right)$ to the exact solutions $v_i\left(t\right)$ . In topology, this type of variation is called deformation and Equation (20) is called zero-order deformation equation.

Therefore, the Taylor series expansion of ${\varphi }_i\left(t\mathrm{,}q\right)$ with respect to q can be obtained as

${\varphi }_i\left(t,q\right)={\varphi }_i\left(t,0\right)+{\displaystyle \underset{m=1}{\overset{\infty }{\sum }}}{v}_{i,m}\left(t\right)q^m$ (22)

where

$v_{im}\left(t\right)=\frac{1}{m!}{\frac{{\partial }^m{\varphi }_i\left(t;q\right)}{\partial q^m}|}_{q=0}$

Suppose that the initial approximations $v_{i\mathrm{,0}}\left(t\right)$ , auxiliary parameters $h_i$ and the auxiliary function $H_i\left(t\right)$ are properly chosen so that the series in Equation (22) converges at $q=1$ , that is

${\varphi }_i\left(t,1\right)=v_{i,0}\left(t\right)+{\displaystyle \underset{m=1}{\overset{\infty }{\sum }}}{v}_{i,m}\left(t\right)$ (23)

Define vectors

$v_{i,n}\left(t\right)=\left[v_{i,0}\left(t\right),v_{i,1}\left(t\right),\cdots ,v_{i,n}\left(t\right)\right]$ (24)

By differentiate Equation (20) m times with respect to q and setting $q=0$ and finally divided by $m\mathrm{<mo>\; !\; </mo>}$ we obtain the so called mth-order deformation equation as

$N^{+}\left[v_{i\mathrm{,}m}\left(t\right)-{\chi }_m{v}_{i\mathrm{,}m-1}\left(t\right)\right]=h_i{H}_i\left(t\right)R_{y_{i}m}\left[v_{i\mathrm{,}m-1}\left(t\right)\right]$ (25)

where

$R_{y_i\mathrm{,}m}\left[v_{i\mathrm{,}m-1}\left(t\right)\right]=\frac{1}{\left(m-1\right)\mathrm{<mo>\; !\; </mo>}}{\frac{{\partial }^{m-1}{N}_i\left[{\varphi }_i\left(t\mathrm{,}q\right)\right]}{\partial q^{m-1}}|}_{q=0}$ (26)

and

${\chi }_m=(\begin{array}{cc}0,& m\le 1\\ 1,& m>1\end{array}$

By taking the inverse natural transform on both sides of Equation (25) we obtained

$v_{i\mathrm{,}m}\left(t\right)={\chi }_m{v}_{i\mathrm{,}m-1}\left(t\right)+h_i{N}^{-}\left[H_i\left(t\right)R_{y_{i}m}\left[v_{i\mathrm{,}m-1}\left(t\right)\right]\right]$ (27)

Therefore, $v_{i\mathrm{,}m}\left(t\right)$ for $m\ge 1$ can be easily obtained from Equation (27), at Mth order we have

$v_i\left(t\right)={\displaystyle \underset{m=0}{\overset{M}{\sum }}}{v}_{i,m}\left(t\right)$ (28)

Hence, as $M\to \infty$ 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

$\begin{array}{l}{v}_{i\mathrm{,}m}\left(t\right)=\left({\chi }_m+h_i\right)v_{i\mathrm{,}m-1}\left(t\right)+h_i\frac{1}{a_i^n}{v}_{i\mathrm{,}m-1}\left(a_{i}t\right)\\ -h_i\left(1-{\chi }_m\right)N^{-}{\displaystyle \underset{k=1}{\overset{n}{\sum }}}\left(1+\frac{1}{a_i^{n-k+1}}\right)\frac{u^{k-1}}{s^k}{v}_i^{k-1}\left(0\right)\\ +h_i{N}^{-}\left\{\frac{u^n}{s^n}{N}^{+}\left[R_i\left(v_{\gamma \mathrm{,}m-1}\left(t\right)\right)+H_{i\mathrm{,}m-1}\left(v_{{\gamma }_1}\mathrm{,}{v}_{{\gamma }_2},\cdots ,v_{{\gamma }_N}\right)-g_i\left(t\right)\right]\right\}\mathrm{,}m\ge 1\\ v_{i\mathrm{,}m}\left(t\right)=\left({\chi }_m+h_i\right)v_{i\mathrm{,}m-1}\left(t\right)+h_i{v}_{i\mathrm{,}m-1}\left(t-{\tau }_i\right)\\ -h_i\left(1-{\chi }_m\right)N^{-}{\displaystyle \underset{k=1}{\overset{n}{\sum }}}\frac{u^{k-1}}{s^k}\left[v_i^{k-1}\left(0\right)+\underset{t\to 0}{\lim }{v}_i^{k-1}\left(t-{\tau }_i\right)\right]\\ +h_i{N}^{-}\left\{\frac{u^n}{s^n}{N}^{+}\left[R_i\left(v_{\gamma \mathrm{,}m-1}\left(t\right)\right)+H_{i\mathrm{,}m-1}\left(v_{{\gamma }_1}\mathrm{,}\cdots ,v_{{\gamma }_N}\right)-g_i\left(t\right)\right]\right\}\mathrm{,}m\ge 1\end{array}$ (29)

Now, the nonlinear operators $F_i\left(v_{\gamma }\right)$ are expanded as series of modified He’s polynomials $H_{i\mathrm{,}m-1}\left(v_{\gamma 1}\mathrm{,}{v}_{\gamma 2},\cdots ,v_{\gamma n}\mathrm{<mo\; stretchy="false">\; )\; </mo>}\right)$ define as

$H_{i,m}\left(v_{\gamma 1},v_{\gamma 2},\cdots ,v_{\gamma N}\right)=\frac{1}{m!}{\frac{{\partial }^m}{\partial q^m}{F}_i\left({\displaystyle \underset{p=0}{\overset{m}{\sum }}}{q}^p{v}_{\gamma ,p}\right)|}_{q=0}$ (30)

where $v_{\gamma \mathrm{,}i}=\left(v_{i\mathrm{,1}}\mathrm{,}{v}_{i\mathrm{,2}}\mathrm{,}\cdots ,v_{i\mathrm{,}N}\right)$ and $v_{\gamma \mathrm{,}p}=\left(v_{\mathrm{1,}p}\mathrm{,}{v}_{\mathrm{2,}p}\mathrm{,}\cdots ,v_{N\mathrm{,}p}\right)$ .

The proof for Case II (constant delay) is of the same process with that of Case I.

Theorem 3.1 Assume $\left(C\left[D\right]\mathrm{,}\Vert .\Vert \right)$ is a Banach Space and let $v_{i\mathrm{,}m}\left(t\right)$ be defined in $\left(C\left[D\right]\mathrm{,}\Vert .\Vert \right)$ , where $\Vert v_{i\mathrm{,}m}\Vert$ are define in form of an operators, that is $v_{i,m}\left(t\right)=A_i\left(v_{i,m-1}\left(t\right)\right)$ such that $A_i\left(v\right)=-N^{-}{R}_{y_i\mathrm{,}m}\left[{\stackrel{\to }{v}}_{i\mathrm{,}m-1}\left(t\right)\right]$ and

$\Vert A_i\left(v\right)-A_i\left(u\right)\Vert \le {\delta }_i\Vert u-v\Vert \mathrm{,}\forall v\mathrm{,}u\in C\left[D\right]$ (31)

where ${\delta }_i=\left({\alpha }_{i\mathrm{,1}}+{\alpha }_{i\mathrm{,2}}+{\mu }_i\right)d$ for some ${\delta }_i\in \left(\mathrm{0,1}\right)$ . Then $A_i$ have unique fixed points in $C\left[D\right]$ . Furthermore, the Homotopy series in Equation (23) converged uniquelly to the solutions $v_i\left(t\right)$ (their respective fixed points in $C\left[D\right]$ ) of Equations (14) and (15).

Proof

Let $\left[C\left(D\right)\mathrm{,}\Vert .\Vert \right]$ be the Banach space of all continuous functions on $C\left(D\right)$ , and from definitions of $L_i\mathrm{,}{R}_i$ and $F_i$ in Equation (16) then we have to show that the $\left\{v_{i\mathrm{,}m}\right\}$ are Cauchy sequences in $\left[C\left(D\right)\mathrm{,}\Vert .\Vert \right]$ . Now from Equation (19) we have

$\begin{array}{c}\Vert v_{i\mathrm{,}m}-v_{i\mathrm{,}p}\Vert =\Vert -N^{-}\left[R_{y_i\mathrm{,}m}\left[v_{i\mathrm{,}m-1}\left(t\right)\right]-R_{y_i\mathrm{,}p}\left[v_{i\mathrm{,}p-1}\left(t\right)\right]\right]\Vert \\ =\Vert N^{-}[N^{+}\left[2\left({\varphi }_i\left(t\mathrm{<mo>\; ;\; </mo>}q\right)+\frac{1}{a_i^{\left(n+1\right)}}{\varphi }_i\left(a_{i}t\mathrm{<mo>\; ;\; </mo>}q\right)\right)\right]\\ -{\displaystyle \underset{k=1}{\overset{n}{\sum }}}\left(1+\frac{1}{a_i^{\left(n+1-k\right)}}\right)\frac{u^{n-k}}{s^k}\left({\varphi }_{i\mathrm{,}m}^{k-1}\left(0\right)-{\varphi }_{i\mathrm{,}p}^{k-1}(0)\right)\end{array}$

$\begin{array}{c}+\frac{u^n}{s^n}{N}^{+}[R_i\left({\varphi }_{i\mathrm{,}m-1}\left(t\mathrm{<mo>\; ;\; </mo>}q\right)-{\varphi }_{i\mathrm{,}p-1}\left(t\mathrm{<mo>\; ;\; </mo>}q\right)\right)\\ \underset{}{\overset{}{{\displaystyle {}_{}{}^{}}}}+F_i\left({\varphi }_{i\mathrm{,}m-1}\left(t\mathrm{<mo>\; ;\; </mo>}q\right)-{\varphi }_{i\mathrm{,}p-1}\left(t\mathrm{<mo>\; ;\; </mo>}q\right)\right)-g_i\left(t\right)]]\Vert \\ \le \Vert N^{-}\left\{N^{+}\left[L_i\left(v_{i\mathrm{,}m-1}-v_{i\mathrm{,}p-1}\right)+R_i\left(v_{i\mathrm{,}m-1}-v_{i\mathrm{,}p-1}\right)+F_i\left(v_{i\mathrm{,}m-1}-v_{i\mathrm{,}p-1}\right)\right]\right\}\Vert \\ \le \left({\alpha }_{i\mathrm{,1}}+{\alpha }_{i\mathrm{,2}}+{\mu }_i\right)\Vert v_{i\mathrm{,}m-1}-v_{i\mathrm{,}p-1}\Vert \le {\delta }_i\Vert v_{i\mathrm{,}m-1}-v_{i\mathrm{,}p-1}\Vert \end{array}$ (32)

For $m=p+1$ , we have

$\Vert v_{i\mathrm{,}p+1}-v_{i\mathrm{,}p}\Vert \le {\delta }_i\Vert v_{i\mathrm{,}p}-v_{i\mathrm{,}p-1}\Vert \le {\delta }_i^2\Vert v_{i\mathrm{,}p-1}-v_{i\mathrm{,}p-2}\Vert \le \cdots \le {\delta }_i^q\Vert v_{i\mathrm{,1}}-v_{i\mathrm{,0}}\Vert$ (33)

Hence, for all $m\mathrm{,}p\in N$ with $m\ge p$ , by means of Equation (33) and using triangle inequality we seccessively obtained the following

$\begin{array}{c}\Vert v_{i,m}-v_{i,p}\Vert \le \Vert v_{i,m}-v_{i,m-1}\Vert +\Vert v_{i,m-1}-v_{i,m-2}\Vert +\cdots +\Vert v_{i,p+1}-v_{i,p}\Vert \\ \le {\delta }_i^{m-1}\Vert v_{i,1}-v_{i,0}\Vert +{\delta }_i^{m-2}\Vert v_{i,1}-v_{i,0}\Vert +\cdots +{\delta }_i^p\Vert v_{i,1}-v_{i,0}\Vert \\ ={\delta }_i^p\Vert v_{i1}-v_{i,0}\Vert {\displaystyle \underset{k=0}{\overset{m-n-1}{\sum }}}{\delta }_i^k\le {\delta }_i^p\Vert v_{i,1}-v_{i,0}\Vert {\displaystyle \underset{k=0}{\overset{\infty }{\sum }}}{\delta }_i^k\\ ={\delta }_i^p\Vert v_{i1}-v_{i,0}\Vert \left(\frac{1}{1-{\delta }_i}\right).\end{array}$ (34)

Since ${\delta }_i<1$ then for arbitrary ${\epsilon }_i$ we can find some large ${\eta }_i\in N$ such that

${\delta }_i^{{\eta }_i}<\frac{{\epsilon }_i\left(1-{\delta }_i\right)}{\Vert v_{i,1}-v_{i,0}\Vert }$

Therefore, choosing $p,m>N$ then we obtain the following

$\Vert v_{i,m}-v_{i,p}\Vert \le {\delta }_i^p\Vert v_{i,1}-v_{i,0}\Vert \left(\frac{1}{1-{\delta }_i}\right)<\frac{{\epsilon }_i\left(1-{\delta }_i\right)}{\Vert v_{i,1}-v_{i,0}\Vert }\Vert v_{i,1}-v_{i,0}\Vert \left(\frac{1}{1-{\delta }_i}\right)={\epsilon }_i$ (35)

This shows that $\left\{v_{i\mathrm{,}m}\right\}$ are Cauchy sequences in $C\left[D\right]$ , and hence the sequences converged. And the Proof is now completed. The proof for Case II (constant delay) is of the same process with that of Case I.

To establish the proof for the uniqueness of these solutions, let $v_i\left(t\right)$ and $u_i\left(t\right)$ be two distinct solutions of Equations (14) and (15). According to Equation (16) we have

$v_i\left(t\right)+v_i\left(\alpha \right)=L_i^{-1}\left[g_i\left(t\right)-R_i\left(v_{\gamma }\right)-F_i\left(v_{\gamma }\right)\right]$ (36)

where $L_i^{-1}$ are the inverse operators defined by ${\displaystyle {\int }_0^t}\left(\mathrm{.}\right)\text{d}t$ . Since $v_i\left(t\right)$ and $u_i\left(t\right)$ are distinct solutions of Equations (14) and (15), so from Equation (36) we obtain the following equation

$\begin{array}{l}\Vert \left(v_i-u_i\right)+\left(v_{i\mathrm{,}\alpha }-u_{i\mathrm{,}\alpha }\right)\Vert \\ \le \Vert \left(v_i-u_i\right)\Vert =\Vert -{\displaystyle {\int }_0^t}\left[R_i\left(v_i-u_i\right)+F_i\left(v_i-u_i\right)\right]\text{d}t\Vert \\ \le {\displaystyle {\int }_0^t}\left[\Vert R_i\left(v_i-u_i\right)\Vert +\Vert F_i\left(v_i\right)-F_i\left(u_i\right)\Vert \right]\text{d}t\\ \le \left({\alpha }_{\mathrm{2,1}}\Vert v_i-u_i\Vert +{\mu }_i\Vert v_i-u_i\Vert \right)d\le {\delta }_i\Vert v_i-u_i\Vert \end{array}$ (37)

According to Equation (37) we have $\left(1-{\delta }_i\right)\Vert v_i-u_i\Vert \le 0$ and since ${\delta }_i\in \left(\mathrm{0,1}\right)$ then $\Vert v_i-u_i\Vert \le 0$ implies that $v_i=u_i$ and hence the proof.

Theorem 3.2 Suppose the Homotopy series in Equation (23) converges to the solutions $v_i\left(t\right)$ of Equations (14) and (15) and let the approximations of $v_i\left(t\right)$ are given by the truncuted series ${\displaystyle {\sum }_{m=0}^M}{v}_{i,m}\left(t\right)$ . Then the maximum absolute error is estimated to be

$\Vert v_i\left(t\right)-{\displaystyle \underset{m=0}{\overset{M}{\sum }}}{v}_{i,m}\left(t\right)\Vert \le \frac{{\delta }_i^n}{1-{\delta }_i}\Vert v_{i,0}\Vert$ (38)

Proof

According to Theorem 3.1 and Equation (34) we have

$\Vert v_{i\mathrm{,}m}-v_{i\mathrm{,}p}\Vert \le \frac{{\delta }_i^n}{1-{\delta }_i}\Vert v_{i\mathrm{,0}}\left(t\right)\Vert$

As $m\to \infty$ then $v_{i\mathrm{,}m}\to v_i\left(t\right)$ , so we have

$\Vert v_i\left(t\right)-v_{i\mathrm{,}p}\Vert \le \frac{{\delta }_i^n}{1-{\delta }_i}\Vert v_{i\mathrm{,0}}\left(t\right)\Vert$ (39)

Since ${\delta }_i\in \left(\mathrm{0,1}\right)$ then $1-{\delta }_i\le 1$ and from Equation (39) we have

$\Vert v_i\left(t\right)-{\displaystyle \underset{m=0}{\overset{M}{\sum }}}{v}_{i,m}\left(t\right)\Vert \le \frac{{\delta }_i^n}{1-{\delta }_i}\Vert v_{i,0}\left(t\right)\Vert$ (40)

The Proof is now complete.

4. 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

$\begin{array}{l}{v^{\prime }}_1\left(t\right)={v^{\prime }}_1\left(t-1\right)+4v_2\left(t\right),0\le t\le 2\\ {v^{\prime }}_2\left(t\right)=v_1\left(t\right)-v_1\left(t-1\right),0\le t\le 2\end{array}$ (41)

$v_1\left(t\right)={\text{e}}^{-2t},v_2\left(t\right)=\frac{1}{2}\left({\text{e}}^{-2\left(t-1\right)}-{\text{e}}^{-2t}\right),t\in \left[-1,0\right].$

Take the natural transform to both sides of Equation (41) and simplify further using Equation (11) to get

$\begin{array}{l}{N}^{+}\left[v_1\left(t\right)-{\text{e}}^{-\frac{s}{u}}{v}_1\left(t\right)\right]-\frac{1}{s}\left[1-{\text{e}}^2\right]-\frac{u}{s}{N}^{+}\left[4v_2\left(t\right)\right]=0\\ N^{+}\left[v_2\left(t\right)\right]-\frac{1}{s}\left[\frac{1}{2}\left({\text{e}}^2-1\right)\right]+\frac{u}{s}{N}^{+}\left[v_1\left(t-1\right)-v_1\left(t\right)\right]=0\end{array}$ (42)

From Equation (42) define a non-linear operator

$\begin{array}{l}N\left[{\varphi }_1\left(t\mathrm{<mo>\; ;\; </mo>}q\right)\right]=N^{+}\left[{\varphi }_1\left(t\mathrm{<mo>\; ;\; </mo>}q\right)-{\text{e}}^{-\frac{s}{u}}{\varphi }_1\left(t\mathrm{<mo>\; ;\; </mo>}q\right)\right]-\frac{1}{s}\left[1-{\text{e}}^2\right]-\frac{u}{s}{N}^{+}\left[4{\varphi }_2\left(t\mathrm{<mo>\; ;\; </mo>}q\right)\right]\\ N\left[{\varphi }_2\left(t\mathrm{<mo>\; ;\; </mo>}q\right)\right]=N^{+}\left[{\varphi }_2\left(t\mathrm{<mo>\; ;\; </mo>}q\right)\right]-\frac{1}{s}\left[\frac{1}{2}\left({\text{e}}^2-1\right)\right]+\frac{u}{s}{N}^{+}\left[{\varphi }_1\left(t-\mathrm{1\; <mo>\; ;\; </mo>}q\right)-{\varphi }_1\left(t\mathrm{<mo>\; ;\; </mo>}q\right)\right]\end{array}$ (43)

Now using Equation (29) the recursive relation of Example 4.1 can be obtained as

$\begin{array}{l}{v}_{\mathrm{1,}m}\left(t\right)=\left({\chi }_m+h_1\right)v_{\mathrm{1,}m-1}\left(t\right)-h_1{v}_1\left(t-1\right)-h_1\left(1-{\chi }_m\right)N^{-}\left[\frac{1}{s}\left(1-{\text{e}}^2\right)\right]\\ -h_1{N}^{-}\left\{\frac{u}{s}{N}^{+}\left[R_1\left(v_{\mathrm{2,}m-1}\left(t\right)\right)\right]\right\}\\ v_{\mathrm{2,}m}\left(t\right)=\left({\chi }_m+h_2\right)v_{\mathrm{2,}m-1}\left(t\right)-h_2\left(1-{\chi }_m\right)N^{-}\left[\frac{1}{2s}\left({\text{e}}^2-1\right)\right]\\ +h_2{N}^{-}\left\{\frac{u}{s}{N}^{+}\left[R_2\left(v_{\mathrm{1,}m-1}\right)\left(t\right)\right]\right\},m\ge 1\end{array}$ (44)