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]
where
are real constants such that
and
. Equation (
1) is subjected to the initial condition (IC)
where
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 , the LT of the delay term iswhere is the LT of . Proof. From the definition of the LT,
we have
Let
(
), then
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 and are two polynomials in s such that the degree of is less than the degree of and has n distinct zeros , then the inverse LT of is 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
which implies
A first step for applying the ADM is to put Equation (
9) in the canonical form:
The ADM assumes the solution in the form
Substituting (11) into (10), we obtain
From (12), we have recurrence scheme
For
, we have
and for
, we obtain
Similarly, the case
leads to
In view of the above results, a unified formula for the general component
can be expressed as
Therefore, the solution to Equation (
9) is
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
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
Let us define the two polynomials
and
by
then Equation (
19) can be rewritten as
It is clear from (20) that the polynomial
has
distinct roots (
, say),
. Such roots are given by
. Applying Theorem 1, we obtain
i.e,
Inserting (23) into (21), then
reads
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]):
Comparing Equation (
25) with Equation (
1), we find
and
. Employing these values in Equation (
24), then the solution of the Ambartsumian equation reads
which is the same result obtained by Bakodah and Ebaid [
21].
3.2. Exact Solution at
If
, then Equation (
1) takes the form
which has the exact solution
The expression (28) can be directly derived by finding the inverse LT of the
given in Equation (
18) when
. In this case, we find from Equation (
18) that
3.3. Exact Solution at
In this case, one can find from Equation (
20) that
and
. It can be shown that such expressions of
and
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
can be extracted from Equation (
24) as
This approximation can be rewritten as
where
Here, we show that the components enjoy certain properties at , given in the following theorem.
Theorem 2. At , the components satisfy the properties and .
Proof. At
and for
, we have from Equation (
32) that
To calculate
, we use Formula (20) for
which gives
which implies
and
. For
, we have from Equation (
32) that
The quantities
and
can be evaluated as follows:
Employing (36) in (35), we obtain
Moreover, for
, we have from Equation (
32) that
Similarly, the magnitudes
,
, and
are determined from Formula (20) as
and hence
Inserting this result into Equation (
38) yields
as required. In fact, one can prove by induction that
. □
The above theorem is essential for proving that the n-term approximate solution (30) satisfies the IC (2), i.e., . This point is addressed in the following theorem.
Theorem 3. , the approximation given by Equation (30) satisfies the IC (2). Proof. For
, we have from Equations (31) and (32) at
that
where
by Theorem 2. For
, we find
where
(Theorem 2), and this completes the proof. □
5. Convergence Analysis
Theorem 4. For , the series (24) converges if and .
Proof. From Equations (24) and (32), we have
and so
In order to facilitate our proof, we rewrite
in the form
or
The following quantities can be determined from the definition of
in (47):
Thus, Equation (
45) becomes
Moreover, we have
where the quantum calculus notation
is used when
, see Refs. [
60,
61]. Let
then
It should be noted that the quantities
,
, and
are bounded as
if
, where for
and
. Accordingly, the first term on the right-hand side of Equation (
53) vanishes when
, while the second term vanishes if
; 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 , . 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 and .
In this regard, the approximations
,
,
, and
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
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
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
defined by
For this purpose, the numerical calculations of the residuals
, for
, 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
is plotted in
Figure 10 at
. 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.
Author Contributions
Methodology, H.K.A.-J.; Validation, R.A. and H.K.A.-J.; Formal analysis, R.A. and H.K.A.-J.; Investigation, R.A. and H.K.A.-J.; Resources, R.A.; Writing—review & editing, H.K.A.-J. All authors have read and agreed to the published version of the manuscript.
Funding
This research received no external funding.
Conflicts of Interest
The authors declare no conflict of interest.
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Figure 1.
Plots of the approximations for at , , and .
Figure 1.
Plots of the approximations for at , , and .
Figure 2.
Plots of the approximations for at , , and .
Figure 2.
Plots of the approximations for at , , and .
Figure 3.
Plots of the approximations for at , , and .
Figure 3.
Plots of the approximations for at , , and .
Figure 4.
Plots of the approximations for at , , and .
Figure 4.
Plots of the approximations for at , , and .
Figure 5.
Plots of the approximations for at , , and .
Figure 5.
Plots of the approximations for at , , and .
Figure 6.
Plots of the approximations for at , , and .
Figure 6.
Plots of the approximations for at , , and .
Figure 7.
Plots of the approximations for at , , and .
Figure 7.
Plots of the approximations for at , , and .
Figure 8.
Plots of the approximations for at , , and .
Figure 8.
Plots of the approximations for at , , and .
Figure 9.
Variation in the residual for at , , and .
Figure 9.
Variation in the residual for at , , and .
Figure 10.
Variation in the residual for at , , and .
Figure 10.
Variation in the residual for at , , and .
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