Accurate Solution for the Pantograph Delay Differential Equation via Laplace Transform

Abstract

The Pantograph equation is a fundamental mathematical model in the field of delay differential equations. A special case of the Pantograph equation is well known as the Ambartsumian delay equation which has a particular application in Astrophysics. In this paper, the Laplace transform is successfully applied to solve the Pantograph delay equation. The solution is obtained in a closed series form in terms of exponential functions. This closed form reduces to the corresponding solution in the relevant literature for the Ambartsumian delay equation as a special case. In addition, the convergence of the obtained series is proved theoretically and validated graphically. Furthermore, the accuracy of the numerical results are estimated through several computations of the residual errors. It is shown that such residuals tend to zero, even in a huge domain. The obtained results reveal that the Laplace transform is a powerful approach to solve linear delay differential equations, including the Pantograph model.

1. Introduction

The use of ordinary differential equations (ODEs) and partial differential equations (PDEs) is essential in modeling engineering and physical phenomena. However, the standard forms of ODEs/PDEs are not always effective in modeling the scientific problems with a memory effect. Such phenomena can be better described by means of delay differential equations (DDEs). There are many standard methods to solve linear ODEs of first or higher orders with constant/variable coefficients, e.g., the separation of variables method, variation in parameters method, reduction in order method, and others. Unfortunately, such methods can not be applied in a straightforward manner to solve DDEs for the existence of the delay terms. Actually, each DDE has its own approach to solve.

For linear ODEs, the Laplace transform (LT) is an effective/fundamental method for a solution. In addition, the LT usually leads to exact solutions for mathematical models governed by linear ODEs/PDEs. Regarding the LT, it was widely applied to deal with a considerable amount of engineering/physical problems. For example, Ebaid et al. [1] solved the three-dimensions falling body problem while Aljohani et al. [2] solved the chlorine transport model by means of the LT. Moreover, several classes of 2nd-order boundary value problems were solved by Ebaid et al. [3,4] and Ali et al. [5] using the LT. Moreover, the LT was found to be effective for treating numerous models in fluids in exact forms, see, for example, Refs. [6,7,8,9].

In addition to the LT, there are other effective methods to analyze the ODEs/DEs, such as the Adomian decomposition method (ADM) (see Refs. [10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28]), the homotopy perturbation method (HPM) (see Refs. [29,30,31,32,33,34,35,36,37,38,39,40,41,42,43]), the homotopy analysis method (HAM) (see Refs. [44,45,46,47,48,49,50,51,52]), and the differential transform method (DTM) (see Refs. [53,54,55,56,57]). These methods are effective when the approximate solution is the target. However, the exact solution is rarely obtainable by the just mentioned methods. Furthermore, the HPM and HAM force us to implement auxiliary parameters and they also need to put the equation being solved in an effective canonical form; this leads to a divergent solution if such a canonical form is not appropriate. However, the LT has its own advantage over the above methods. This is simply because the LT usually leads to the exact solution for linear ODEs/PDEs.

The objective of this paper is to extend the application of the LT to analyze one of the fundamental DDEs, called the Pantograph delay differential equation (PDDE), given by [24]

$$y^{\prime }\left(t\right)=ay\left(t\right)+by\left(ct\right),$$

(1)

where $a,b,c$ are real constants such that $b\ne 0$ and $c<1$. Equation (1) is subjected to the initial condition (IC)

$$y\left(0\right)=\lambda ,$$

(2)

where $\lambda$ is a real constant. It will be shown in this paper that the LT is an effective tool to solve the PDDE. The solution will be determined in a closed convergent series form. In addition, the exact solutions of some special cases will be evaluated at particular values of the involved parameters, a, b, and c. On the other hand, the existing results in the literature are to be recovered as special cases of the present analysis. Before launching to the main purpose of this paper, let us introduce the following basic lemma and theorem which are essential in this study.

Lemma 1. 

For $c\in \mathbb{R}-\left\{0\right\}$, the LT of the delay term $y\left(ct\right)$ is

$$\mathcal{L}\left\{y\left(ct\right)\right\}=\frac{1}{c}Y\left(\frac{s}{c}\right),$$

(3)

where $Y\left(s\right)$ is the LT of $y\left(t\right)$.

Proof. 

From the definition of the LT,

$$\mathcal{L}\left\{y\left(t\right)\right\}={\int }_0^{\infty }{e}^{-st}y\left(t\right)dt,$$

(4)

we have

$$\mathcal{L}\left\{y\left(ct\right)\right\}={\int }_0^{\infty }{e}^{-st}y\left(ct\right)dt.$$

(5)

Let $u=ct$ ($c\ne 0$), then

$$\mathcal{L}\left\{y\left(ct\right)\right\}=\frac{1}{c}{\int }_0^{\infty }{e}^{-\frac{s}{c}u}y\left(u\right)du=\frac{1}{c}Y\left(\frac{s}{c}\right),$$

(6)

which completes the proof. □

The following theorem is well-known as the Heaviside’s expansion formula which permits to calculate the inverse LT of the quotient of two polynomials.

Theorem 1 

([58]). If $P\left(s\right)$ and $Q\left(s\right)$ are two polynomials in s such that the degree of $P\left(s\right)$ is less than the degree of $Q\left(s\right)$ and $Q\left(s\right)$ has n distinct zeros ${\alpha }_k,$ $k=1,2,\dots n$, then the inverse LT of $\frac{P\left(s\right)}{Q\left(s\right)}$ is

$${\mathcal{L}}^{-1}\left\{\frac{P\left(s\right)}{Q\left(s\right)}\right\}=\sum _{k=1}^n\frac{P\left({\alpha }_k\right)}{Q^{\prime }\left({\alpha }_k\right)}{e}^{{\alpha }_{k}t}.$$

(7)

2. Application of the LT-ADM

In this section, the LT is applied to solve the PDDE given by Equations (1) and (2). Accordingly, a closed-form solution will be obtained using a simple and straightforward approach. The suggested approach is based on combining the LT and the ADM following clear and simple steps.

2.1. The Transformed Equation

Let us begin with applying the LT on Equation (1); this yields

$$sY\left(s\right)-y\left(0\right)=aY\left(s\right)+\frac{b}{c}Y\left(\frac{s}{c}\right),$$

(8)

which implies

$$\left(s-a\right)Y\left(s\right)=\lambda +\frac{b}{c}Y\left(\frac{s}{c}\right).$$

(9)

A first step for applying the ADM is to put Equation (9) in the canonical form:

$$Y\left(s\right)=\frac{\lambda }{s-a}+\frac{b}{c\left(s-a\right)}Y\left(\frac{s}{c}\right)$$

(10)

The ADM assumes the solution in the form

$$Y\left(s\right)=\sum _{i=0}^{\infty }{Y}_i\left(s\right).$$

(11)

Substituting (11) into (10), we obtain

$$\sum _{i=0}^{\infty }{Y}_i\left(s\right)=\frac{\lambda }{s-a}+\frac{b}{c\left(s-a\right)}\sum _{i=0}^{\infty }{Y}_i\left(\frac{s}{c}\right).$$

(12)

From (12), we have recurrence scheme

$$\left\{\begin{array}{c}{Y}_0\left(s\right)=\frac{\lambda }{s-a},\hfill \\ \\ Y_i\left(s\right)=\frac{b}{c\left(s-a\right)}{Y}_{i-1}\left(\frac{s}{c}\right),i\ge 1.\hfill \end{array}\right.$$

(13)

For $i=1$, we have

$$Y_1\left(s\right)=\frac{b}{c\left(s-a\right)}{Y}_0\left(\frac{s}{c}\right)=\frac{b\lambda }{c\left(s-a\right)\left(\frac{s}{c}-a\right)},$$

(14)

and for $i=2$, we obtain

$$Y_2\left(s\right)=\frac{b}{c\left(s-a\right)}{Y}_1\left(\frac{s}{c}\right)=\frac{b^2\lambda }{c^2\left(s-a\right)\left(\frac{s}{c}-a\right)\left(\frac{s}{c^2}-a\right)}.$$

(15)

Similarly, the case $i=3$ leads to

$$\begin{array}{c}\hfill Y_3\left(s\right)=\frac{b}{c\left(s-a\right)}{Y}_2\left(\frac{s}{c}\right)=\frac{b^3\lambda }{c^3\left(s-a\right)\left(\frac{s}{c}-a\right)\left(\frac{s}{c^2}-a\right)\left(\frac{s}{c^3}-a\right)}.\end{array}$$

(16)

In view of the above results, a unified formula for the general component $Y_i\left(s\right)$ can be expressed as

$$Y_i\left(s\right)=\frac{b^i\lambda }{c^i\left(s-a\right)\left(\frac{s}{c}-a\right)\left(\frac{s}{c^2}-a\right)\cdots \left(\frac{s}{c^i}-a\right)}=\frac{b^i\lambda }{c^i{\prod }_{k=0}^i\left(\frac{s}{c^k}-a\right)}.$$

(17)

Therefore, the solution to Equation (9) is

$$Y\left(s\right)=\sum _{i=0}^{\infty }\frac{b^i\lambda }{c^i{\prod }_{k=0}^i\left(\frac{s}{c^k}-a\right)}=\lambda \sum _{i=0}^{\infty }\frac{b^i}{c^i{\prod }_{k=0}^i\left(\frac{s}{c^k}-a\right)}.$$

(18)

The inverse LT of the expression on the right-hand side of Equation (18) is the subject of the next section. This leads to the solution $y\left(t\right)$ in a closed series form in terms of exponential functions.

2.2. The Closed-Form Solution of the PDDE

Applying the inverse LT on both sides of Equation (18) gives

$$y\left(t\right)=\lambda {\mathcal{L}}^{-1}\left\{\sum _{i=0}^{\infty }\frac{b^i}{c^i{\prod }_{k=0}^i\left(\frac{s}{c^k}-a\right)}\right\}.$$

(19)

Let us define the two polynomials $P_i\left(s\right)$ and $Q_i\left(s\right)$ by

$$P_i\left(s\right)={\left(\frac{b}{c}\right)}^i,Q_i\left(s\right)=\prod _{k=0}^i\left(\frac{s}{c^k}-a\right),$$

(20)

then Equation (19) can be rewritten as

$$y\left(t\right)=\lambda \sum _{i=0}^{\infty }{\mathcal{L}}^{-1}\left\{\frac{P_i\left(s\right)}{Q_i\left(s\right)}\right\}.$$

(21)

It is clear from (20) that the polynomial $Q_i\left(s\right)$ has $(i+1)$ distinct roots (${\alpha }_k$, say), $k=0,1,2,\dots ,i$. Such roots are given by ${\alpha }_k=a{c}^k$. Applying Theorem 1, we obtain

$${\mathcal{L}}^{-1}\left\{\frac{P_i\left(s\right)}{Q_i\left(s\right)}\right\}=\sum _{k=0}^i\frac{P_i\left({\alpha }_k\right)}{Q_i^{\prime }\left({\alpha }_k\right)}{e}^{{\alpha }_{k}t},$$

(22)

i.e,

$${\mathcal{L}}^{-1}\left\{\frac{P_i\left(s\right)}{Q_i\left(s\right)}\right\}=\sum _{k=0}^i\frac{{\left(\frac{b}{c}\right)}^i{e}^{a{c}^{k}t}}{Q_i^{\prime }\left(a{c}^k\right)}.$$

(23)

Inserting (23) into (21), then $y\left(t\right)$ reads

$$y\left(t\right)=\lambda \sum _{i=0}^{\infty }\sum _{k=0}^i{\left(\frac{b}{c}\right)}^i\frac{e^{a{c}^{k}t}}{Q_i^{\prime }\left(a{c}^k\right)}.$$

(24)

Consequently, the solution of the PDDE is given by the closed form (24). The convergence of the series solution (24) will be analyzed in a subsequent section. In addition, this closed form reduces to the corresponding solution in the literature at particular values of the parameters a, b, and c. This issue is addressed in the next section.

3. Special Cases

3.1. Ambartsumian Delay Equation

The Ambartsumian equation is of particular interest in Astrophysics. It is used for studying the surface brightness of the Milky Way. The standard Ambartsumian model is governed by the DDE (see Refs. [21,22]):

$$y^{\prime }\left(t\right)=-y\left(t\right)+\frac{1}{q}y\left(\frac{t}{q}\right),q>1.$$

(25)

Comparing Equation (25) with Equation (1), we find $a=-1$ and $b=c=\frac{1}{q}$. Employing these values in Equation (24), then the solution of the Ambartsumian equation reads

$$y\left(t\right)=\lambda \sum _{i=0}^{\infty }\sum _{k=0}^i\frac{e^{-\frac{t}{q^k}}}{Q_i^{\prime }\left(-\frac{1}{q^k}\right)},Q_i\left(s\right)=\prod _{k=0}^i\left(q^{k}s+1\right),$$

(26)

which is the same result obtained by Bakodah and Ebaid [21].

3.2. Exact Solution at $c=1$

If $c=1$, then Equation (1) takes the form

$$y^{\prime }\left(t\right)=(a+b)y\left(t\right),$$

(27)

which has the exact solution

$$y\left(t\right)=\lambda e^{(a+b)t}.$$

(28)

The expression (28) can be directly derived by finding the inverse LT of the $Y\left(s\right)$ given in Equation (18) when $c=1$. In this case, we find from Equation (18) that

$$\begin{array}{ccc}\hfill y\left(t\right)& =& \lambda {\mathcal{L}}^{-1}\left\{\sum _{i=0}^{\infty }\frac{b^i}{{\prod }_{k=0}^i\left(s-a\right)}\right\}=\lambda \sum _{i=0}^{\infty }{b}^i{\mathcal{L}}^{-1}\left\{\frac{1}{{\left(s-a\right)}^{i+1}}\right\},\hfill \\ & =& \lambda \sum _{i=0}^{\infty }{b}^i{e}^{at}{\mathcal{L}}^{-1}\left\{\frac{1}{s^{i+1}}\right\}=\lambda e^{at}\sum _{i=0}^{\infty }\frac{{\left(bt\right)}^i}{i!}=\lambda e^{\left(a+b\right)t}.\hfill \end{array}$$

(29)

3.3. Exact Solution at $c=-1$

In this case, one can find from Equation (20) that $P_i\left(s\right)={\left(-b\right)}^i$ and $Q_i\left(s\right)={\prod }_{k=0}^i\left({(-1)}^{k}s-a\right)$. It can be shown that such expressions of $P_i\left(s\right)$ and $Q_i\left(s\right)$ lead to the same exact periodic solution reported in Ref. [59]. The derivation of such periodic solution is ignored here just to avoid repetition.

4. Theoretical Results

The n-term approximate solution ${\Phi }_n\left(t\right)$ can be extracted from Equation (24) as

$${\Phi }_n\left(t\right)=\lambda \sum _{i=0}^{n-1}\sum _{k=0}^i{\left(\frac{b}{c}\right)}^i\frac{e^{a{c}^{k}t}}{Q_i^{\prime }\left(a{c}^k\right)}.$$

(30)

This approximation can be rewritten as

$${\Phi }_n\left(t\right)=\lambda \sum _{i=0}^{n-1}{z}_i\left(t\right),$$

(31)

where

$$z_i\left(t\right)=\sum _{k=0}^i{\chi }_{i,k}\left(t\right),{\chi }_{i,k}\left(t\right)={\left(\frac{b}{c}\right)}^i\frac{e^{a{c}^{k}t}}{Q_i^{\prime }\left(a{c}^k\right)}.$$

(32)

Here, we show that the components $z_i\left(t\right)$ enjoy certain properties at $t=0$, given in the following theorem.

Theorem 2. 

At $t=0$, the components $z_i\left(t\right)$ satisfy the properties $z_0\left(0\right)=1$ and $z_i\left(0\right)=0$ $\forall i\ge 1$.

Proof. 

At $t=0$ and for $i=0$, we have from Equation (32) that

$$z_0\left(0\right)={\chi }_{0,0}\left(0\right),{\chi }_{0,0}\left(0\right)=\frac{1}{Q_0^{\prime }\left(a\right)}.$$

(33)

To calculate $Q_0^{\prime }\left(a\right)$, we use Formula (20) for $i=0$ which gives

$$Q_0\left(s\right)=s-a,Q_0^{\prime }\left(s\right)=1,Q_0^{\prime }\left(a\right)=1,$$

(34)

which implies ${\chi }_{0,0}\left(0\right)=1$ and $z_0\left(0\right)=1$. For $i=1$, we have from Equation (32) that

$$\begin{array}{ccc}\hfill z_1\left(0\right)& =& {\chi }_{1,0}\left(0\right)+{\chi }_{1,1}\left(0\right),\hfill \\ & =& \left(\frac{b}{c}\right)\frac{1}{Q_1^{\prime }\left(a\right)}+\left(\frac{b}{c}\right)\frac{1}{Q_1^{\prime }\left(ac\right)}.\hfill \end{array}$$

(35)

The quantities $Q_1^{\prime }\left(a\right)$ and $Q_1^{\prime }\left(ac\right)$ can be evaluated as follows:

$$\begin{array}{ccc}& & Q_1\left(s\right)=(s-a)\left(\frac{s}{c}-a\right)=\frac{s^2}{c}-a\left(1+\frac{1}{c}\right)s+a^2,\hfill \\ & & Q_1^{\prime }\left(s\right)=\frac{2s}{c}-a\left(1+\frac{1}{c}\right),\hfill \\ & & Q_1^{\prime }\left(a\right)=\frac{a}{c}-a,Q_1^{\prime }\left(ac\right)=a-\frac{a}{c}.\hfill \end{array}$$

(36)

Employing (36) in (35), we obtain

$$z_1\left(0\right)=\frac{b}{a-ac}+\frac{b}{ac-a}=0.$$

(37)

Moreover, for $i=2$, we have from Equation (32) that

$$\begin{array}{ccc}\hfill z_2\left(0\right)& =& {\chi }_{2,0}\left(0\right)+{\chi }_{2,1}\left(0\right)+{\chi }_{2,2}\left(0\right),\hfill \\ & =& {\left(\frac{b}{c}\right)}^2\left[\frac{1}{Q_2^{\prime }\left(a\right)}+\frac{1}{Q_2^{\prime }\left(ac\right)}+\frac{1}{Q_2^{\prime }\left(a{c}^2\right)}\right].\hfill \end{array}$$

(38)

Similarly, the magnitudes $Q_2^{\prime }\left(a\right)$, $Q_2^{\prime }\left(ac\right)$, and $Q_2^{\prime }\left(a{c}^2\right)$ are determined from Formula (20) as

$$\begin{array}{ccc}& & Q_2\left(s\right)=(s-a)\left(\frac{s}{c}-a\right)\left(\frac{s}{c^2}-a\right)=\frac{s^3}{c^3}-a\left(\frac{1}{c^3}+\frac{1}{c^2}+\frac{1}{c}\right)s^2+a^2\left(\frac{1}{c^2}+\frac{1}{c}+1\right)s-a^3,\hfill \\ & & Q_2^{\prime }\left(s\right)=\frac{3s^2}{c^3}-2a\left(\frac{1}{c^3}+\frac{1}{c^2}+\frac{1}{c}\right)s+a^2\left(\frac{1}{c^2}+\frac{1}{c}+1\right)s,\hfill \end{array}$$

(39)

and hence

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{Q}_2^{\prime }\left(a\right)=\frac{a^2}{c^3}(c+1){(c-1)}^2,\hfill \\ \\ Q_2^{\prime }\left(ac\right)=-\frac{a^2}{c^2}{(c-1)}^2,\hfill \\ \\ Q_2^{\prime }\left(a{c}^2\right)=\frac{a^2}{c^2}(c+1){(c-1)}^2.\hfill \end{array}\right.\end{array}$$

(40)

From (40), we obtain

$$\begin{array}{ccc}\hfill \frac{1}{Q_2^{\prime }\left(a\right)}+\frac{1}{Q_2^{\prime }\left(ac\right)}+\frac{1}{Q_2^{\prime }\left(a{c}^2\right)}& =& \frac{c^3}{a^2}\left[\frac{1}{(c+1){(c-1)}^2}-\frac{1}{c{(c-1)}^2}+\frac{1}{c(c+1){(c-1)}^2}\right],\hfill \\ & =& \frac{c^3}{a^2}\left[\frac{c-(c+1)-1}{c(c+1){(c-1)}^2}\right]=0.\hfill \end{array}$$

(41)

Inserting this result into Equation (38) yields $z_2\left(0\right)=0$ as required. In fact, one can prove by induction that $z_i\left(0\right)=0$ $\forall i\ge 1$. □

The above theorem is essential for proving that the n-term approximate solution (30) satisfies the IC (2), i.e., ${\Phi }_n\left(0\right)=\lambda$ $\forall n\ge 1$. This point is addressed in the following theorem.

Theorem 3. 

$\forall n\ge 1$, the approximation ${\Phi }_n$ given by Equation (30) satisfies the IC (2).

Proof. 

For $n=1$, we have from Equations (31) and (32) at $t=0$ that

$${\Phi }_1\left(0\right)=\lambda z_0\left(0\right)=\lambda ,$$

(42)

where $z_0=1$ by Theorem 2. For $n\ge 1$, we find

$$\begin{array}{ccc}\hfill {\Phi }_n\left(0\right)& =& \lambda \sum _{i=0}^{n-1}{z}_i\left(0\right)=\lambda \left(z_0\left(0\right)+\sum _{i=1}^{n-1}{z}_i\left(0\right)\right),\hfill \\ & =& \lambda \left(1+\sum _{i=1}^{n-1}{z}_i\left(0\right)\right)=\lambda ,\hfill \end{array}$$

(43)

where $z_i\left(0\right)=0$ $\forall i\ge 1$ (Theorem 2), and this completes the proof. □

5. Convergence Analysis

Theorem 4. 

For $t>0$, the series (24) converges if $\left|c\right|<1$ and $|\frac{b}{a}|<1$.

Proof. 

From Equations (24) and (32), we have

$$y\left(t\right)=\sum _{i=0}^{\infty }{z}_i\left(t\right),z_i\left(t\right)=\sum _{k=0}^i{\chi }_{i,k}\left(t\right),{\chi }_{i,k}\left(t\right)={\left(\frac{b}{c}\right)}^i\frac{e^{a{c}^{k}t}}{Q_i^{\prime }\left(a{c}^k\right)},$$

(44)

and so

$$\begin{array}{ccc}\hfill z_{i+1}\left(t\right)-z_i\left(t\right)& =& \sum _{k=0}^{i+1}{\chi }_{i+1,k}\left(t\right)-\sum _{k=0}^i{\chi }_{i,k}\left(t\right),\hfill \\ & =& {\left(\frac{b}{c}\right)}^{i+1}\sum _{k=0}^{i+1}\frac{e^{a{c}^{k}t}}{Q_{i+1}^{\prime }\left(a{c}^k\right)}-{\left(\frac{b}{c}\right)}^i\sum _{k=0}^i\frac{e^{a{c}^{k}t}}{Q_i^{\prime }\left(a{c}^k\right)}.\hfill \end{array}$$

(45)

In order to facilitate our proof, we rewrite $Q_i\left(s\right)$ in the form

$$Q_i\left(s\right)=\prod _{k=0}^i\frac{1}{c^k}\left(s-a{c}^k\right)=c^{-\frac{1}{2}i\left(i+1\right)}\prod _{k=0}^i\left(s-a{c}^k\right),$$

(46)

or

$$Q_i\left(s\right)=c^{-\frac{1}{2}i(i+1)}{G}_i\left(s\right),G_i\left(s\right)=\prod _{k=0}^i\left(s-a{c}^k\right).$$

(47)

The following quantities can be determined from the definition of $G_i\left(s\right)$ in (47):

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{G}_i\left(a{c}^k\right)=0\forall i\ge k,G_{i+1}\left(s\right)=\prod _{k=0}^{i+1}\left(s-a{c}^k\right)=\left(s-a{c}^{i+1}\right)G_i\left(s\right),\hfill \\ G_{i+1}^{\prime }\left(s\right)=\left(s-a{c}^{i+1}\right)G_i^{\prime }\left(s\right)+G_i\left(s\right),G_{i+1}^{\prime }\left(a{c}^{i+1}\right)=G_i\left(a{c}^{i+1}\right),\hfill \\ G_{i+1}^{\prime }\left(a{c}^k\right)=\left(a{c}^k-a{c}^{i+1}\right)G_i^{\prime }\left(a{c}^k\right)+G_i\left(a{c}^k\right)=a\left(c^k-c^{i+1}\right)G_i^{\prime }\left(a{c}^k\right).\hfill \end{array}\right.\end{array}$$

(48)

Thus, Equation (45) becomes

$$\begin{array}{ccc}\hfill z_{i+1}\left(t\right)-z_i\left(t\right)& =& {\left(\frac{b}{c}\right)}^{i+1}\sum _{k=0}^{i+1}\frac{c^{\frac{1}{2}(i+1)(i+2)}{e}^{a{c}^{k}t}}{G_{i+1}^{\prime }\left(a{c}^k\right)}-{\left(\frac{b}{c}\right)}^i\sum _{k=0}^i\frac{c^{\frac{1}{2}i(i+1)}{e}^{a{c}^{k}t}}{G_i^{\prime }\left(a{c}^k\right)},\hfill \\ & =& {\left(\frac{b}{c}\right)}^{i+1}\left[\sum _{k=0}^i\frac{c^{\frac{1}{2}(i+1)(i+2)}{e}^{a{c}^{k}t}}{G_{i+1}^{\prime }\left(a{c}^k\right)}+\frac{c^{\frac{1}{2}(i+1)(i+2)}{e}^{a{c}^{i+1}t}}{G_{i+1}^{\prime }\left(a{c}^{i+1}\right)}\right]-{\left(\frac{b}{c}\right)}^i\sum _{k=0}^i\frac{c^{\frac{1}{2}i(i+1)}{e}^{a{c}^{k}t}}{G_i^{\prime }\left(a{c}^k\right)},\hfill \\ & =& {\left(\frac{b}{c}\right)}^{i+1}\left[\sum _{k=0}^i\frac{c^{\frac{1}{2}(i+1)(i+2)}{e}^{a{c}^{k}t}}{a\left(c^k-c^{i+1}\right)G_i^{\prime }\left(a{c}^k\right)}+\frac{c^{\frac{1}{2}(i+1)(i+2)}{e}^{a{c}^{i+1}t}}{G_i\left(a{c}^{i+1}\right)}\right]-{\left(\frac{b}{c}\right)}^i\sum _{k=0}^i\frac{c^{\frac{1}{2}i(i+1)}{e}^{a{c}^{k}t}}{G_i^{\prime }\left(a{c}^k\right)},\hfill \\ & =& {\left(\frac{b}{c}\right)}^i\sum _{k=0}^i\left(\frac{\frac{b}{a}{c}^i}{c^k-c^{i+1}}-1\right)\frac{c^{\frac{i}{2}\left(i+1\right)}{e}^{a{c}^{k}t}}{G_i^{\prime }\left(a{c}^k\right)}+\frac{b^{i+1}{c}^{\frac{i}{2}(i+1)}{e}^{a{c}^{(i+1)}t}}{G_i\left(a{c}^{i+1}\right)}.\hfill \end{array}$$

(49)

Moreover, we have

$$\begin{array}{ccc}\hfill G_i\left(a{c}^{i+1}\right)& =& \prod _{k=0}^i\left(a{c}^{i+1}-a{c}^k\right)=\prod _{k=0}^i\left[-a{c}^k\left(1-c^{i-k+1}\right)\right],\hfill \\ & =& \prod _{k=0}^i\left(-a{c}^k\right)\prod _{k=0}^i\left(1-c^{i-k+1}\right),\hfill \\ & =& {(-a)}^{i+1}{c}^{\frac{i}{2}(i+1)}\prod _{j=0}^i\left(1-c^{j+1}\right)={(-a)}^{i+1}{c}^{\frac{i}{2}(i+1)}{(c:c)}_{i+1},\hfill \end{array}$$

(50)

where the quantum calculus notation ${(\alpha :\beta )}_i={\prod }_{j=0}^{i-1}\left(1-\alpha {\beta }^j\right)$ is used when $\alpha =\beta =c$, see Refs. [60,61]. Let

$${\Omega }_{i,k}=\frac{\frac{b}{a}{c}^i}{c^k-c^{i+1}}-1=\frac{\left(c+\frac{b}{a}\right)c^{i-k}-1}{1-c^{i-k+1}},$$

(51)

then

$$z_{i+1}\left(t\right)-z_i\left(t\right)=b^i{c}^{\frac{i}{2}(i-1)}\sum _{k=0}^i\frac{{\Omega }_{i,k}}{G_i^{\prime }\left(a{c}^k\right)}{e}^{a{c}^{k}t}+\frac{{\left(-\frac{b}{a}\right)}^{i+1}{e}^{a{c}^{i+1}t}}{{(c:c)}_{i+1}}.$$

(52)

Hence,

$$\left|z_{i+1}\left(t\right)-z_i\left(t\right)\right|\le \left|b^i\right|{\left|c\right|}^{\frac{1}{2}i(i-1)}\sum _{k=0}^i\left|\frac{{\Omega }_{i,k}}{G_i^{\prime }\left(a{c}^k\right)}\right|e^{a{c}^{k}t}+\frac{{\left|\frac{b}{a}\right|}^{i+1}{e}^{a{c}^{i+1}t}}{\left|{(c:c)}_{i+1}\right|}.$$

(53)

It should be noted that the quantities ${\Omega }_{i,k}$, $G_i^{\prime }\left(a{c}^k\right)$, and ${(c:c)}_{i+1}$ are bounded as $i\to \infty$ if $\left|c\right|<1$, where for ${lim}_{i\to \infty }\left|{\Omega }_{i,k}\right|\to 1$ and $0<{(c:c)}_{\infty }<1$. Accordingly, the first term on the right-hand side of Equation (53) vanishes when $\left|c\right|<1$, while the second term vanishes if $|\frac{b}{a}|<1$; this agrees with Ref. [62] and completes the proof. □

6. Results and Validation

This section explores and validates the accuracy of the present approach. In the first part, we focus on showing the convergence of the obtained approximate solution. Such a target is achieved through several plots of a sequence of the approximations $\left\{{\Phi }_n\right\}$, $n\ge 1$. The considered values of the proportional delay parameter c and the involved constants a and b meet the conditions of the convergence theorem. For declaration, the values are chosen so that they satisfy the requirements $|\frac{b}{a}|<1$ and $\left|c\right|<1$.

In this regard, the approximations ${\Phi }_3$, ${\Phi }_5$, ${\Phi }_7$, and ${\Phi }_9$ are depicted in Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8 at different values of the constants a and b and the proportional delay parameter c. It can be observed that the convergence of these approximations is verified, even for a few terms of the current series solution. In addition, it is shown in these figures that the approximations ${\Phi }_n$ converge rapidly to a specific function, and the nature of such a function depends mainly on the signs of a, b, and c. Moreover, the behavior of the ${\Phi }_n$ follows the exponential functions which increase or decrease according to the signs of the included parameters.

In the second part of this discussion, we focus on the error analysis. The numerical calculations are performed to estimate the residual errors $R{E}_n\left(t\right)$ defined by

$$R{E}_n\left(t\right)=|{\Phi }_n^{\prime }\left(t\right)-a{\Phi }_n\left(t\right)-b{\Phi }_n\left(ct\right)|,n\ge 1.$$

(54)

For this purpose, the numerical calculations of the residuals $R{E}_n\left(t\right)$, for $n=7,8,9$, are extracted and displayed through Figure 9. The results show an acceptable error and the advantage of the proposed approach is clear, even in a huge domain. This conclusion is confirmed in Figure 10; however, the accuracy enjoys a better performance than the corresponding accuracy in Figure 9. The reason behind that is the increase in the number of terms, where $R{E}_n\left(t\right)$ is plotted in Figure 10 at $n=10,11,12$. Of course, one can increase the number of terms as needed to achieve the desired accuracy. In view of the above discussion, one may trust the effectiveness and the efficiency of the LT to treat the PDDE.

7. Conclusions

In this paper, the PDDE was solved by means of a hybrid approach. The suggested technique was mainly based on combining the LT and the ADM. The analytic approximations were successfully conducted and theoretically proved for convergence. The results in the literature were recovered as special cases of our analysis. The performed numerical calculations confirmed the theoretical theorem of convergence. Furthermore, the calculated residuals reveal the accuracy of the obtained results and also confirm the effectiveness/efficiency of our approach to solve the PDDE. The capability of the present analysis to solve extended versions of the PDDE may deserve further future works.