Long time behavior of higher-order delay differential equation with vanishing proportional delay and its convergence analysis using spectral method

1.   Introduction

In nature there are so many physical phenomena where the state of the system not only depends on the current state but also depends on the history of the function. In this case it is more natural to model such type of phenomena using the delay differential equations (DDEs), where the state of the function depends on the current state as well as on the history of the function. Among the many available delay differential system, pantograph type delay differential equation is more commonly used in mathematical modeling of chemical and pharmaceutical kinetics, control problems and ships aircraft where they are used in navigational control electronic systems. It is called pantograph type equation because it was first used to investigate that how an electric current is obtained by the electric locomotive of a pantograph. Consider the kth-order functional differential equation of pantograph type of the form:

subject to

For $k\ge 2,$ spectral method will be based on the integrated form of given equation. The functions ${\mathrm{\lambda }}_{l}\left(x\right)$ and ${\mu }_{l}\left(x\right)$ are given analytical functions on $I: = \left[0, T\right]$ and ${\alpha }_{l}\in \left(0, 1\right), l = 0, 1$ is a fixed constant known as proportional delay. For simplicity, we employed and analyzed spectral method for the second order functional differential equation, that is $k = 2.$ Eq (1) will take the form:

subject to

Equation (3) plays an important role in modeling of many physical phenomena like, for example in electrodynamics and in nonlinear dynamical systems. In practice it is arises in the problems when ${\mathrm{\lambda }}_{l}\left(x\right)$ and ${\mu }_{l}\left(x\right)$ are real constant and ${\alpha }_{l}\left(x\right)$ is real valued function. The case ${\mathrm{\lambda }}_{l}\left(x\right) = 0,$ is also important in the number theory in the context of partitioning ^[12,14]. Numerical solution of first order pantograph type delay differential equation has been studied extensively in numerous papers such as ^{[1,2,3,4,5,9,10,11,12,13,14,15,17,18,19,20,21,22,23,24,25,26,27,28]}. A spectral collocation method is applied to integro-delay differential equation with proportional delay in ^[6,29,30]. A comprehensive list of references for the solution of DDEs can be found in ^[16]. A very limited work is available regarding the approximate solution of higher-order pantograph equation, especially using the higher-order schemes. To this end, we will use spectral discretization based on Legendre spectral collocation method to solve Eq (3) subject to Eq (4) numerically in order to get an accurate solution for a very few numbers of collocation points, as spectral methods are well known for their exponential rate of convergence.

The rest of the paper is organized as follows. In Section 2, we discuss the spectral method for the approximate solution of Eq (3). Section 3 includes some useful lemmas and convergence analysis of our proposed scheme in ${L}_{\mathrm{\infty }}$ norm. In Section 4, we perform some numerical test to confirm the spectral accuracy and Section 5 consist of conclusion.

2.   Spectral collocation method

In order to fully use the properties of orthogonal polynomials for ease of convergence analysis, spectral methods will be employed on the standard interval $\left[-1, 1\right]$. For this reason, we use the following transformation

Using this transformation, Eqs (3) and (4) will take the form:

subject to

Where

with

where ${u}_{-1} = {y}_{0}, {u}_{-1}' = {\left(\frac{T}{2}\right)y}_{1}.$

For any given positive integer $N,$ we denote the collocation points by ${\left\{{t}_{j}\right\}}_{j = 0}^{N}$, which is the set of $\left(N+1\right),$ Legendre Gauss points corresponding to weights ${\omega }_{i}$. Let ${P}_{N}$ denote the space of all polynomials of degree not exceeding $N$. For any $v\in C\left[-1, 1\right]$, we define the Lagrange interpolating polynomial

where ${\left\{{F(t}_{j})\right\}}_{j = 0}^{N}$ is the Lagrange interpolation polynomial associated with the Legendre collocation points ${\left\{{t}_{j}\right\}}_{j = 0}^{N}$. Spectral method will be employed to the integrated form. For this reason, integrate Eq (5), and using ${u}_{-1} = {y}_{0}, {u}_{-1}' = {\left(\frac{T}{2}\right)y}_{1},$ we get,

We assume that Eqs (8) and (9) holds at collocation points ${\left\{{t}_{j}\right\}}_{j = 0}^{N}$, to get

for $0\le j\le N.$ In order to compute all these integral terms efficiently for the higher-order accuracy, we transform the interval of integral from $\left[-1, {t}_{j}\right]$ to $\left[-1, 1\right]$, as we have a very little information available for both $u\left(s\right)$ and ${u}'\left(s\right)$, we get

where, we use $s = \frac{1+{t}_{j}}{2}\varphi +\frac{{t}_{j}-1}{2}\approx s\left({t}_{j}, \varphi \right)$.

Now using the $\left(N+1\right)$ Gauss quadrature rule relative to the Legendre weight to approximate the integral term, we get

Let ${u}_{j}'$ $\approx u\text{'}\left({t}_{j}\right), {u}_{j}\approx u\left({t}_{j}\right)$, 0 $\le j\le N.$ The set ${\left\{{\varphi }_{k}\right\}}_{k = 0}^{N}$ coincide with the collocation points ${\left\{{t}_{j}\right\}}_{j = 0}^{N}.$

We expand ${u}', u$ and $G$ using Lagrange interpolating polynomials, that is

The Legendre spectral collocation method is to seek ${\left\{{u}_{j}'\right\}}_{j = 0}^{N}, {\left\{{u}_{j}\right\}}_{j = 0}^{N}, {\left\{{G}_{j}\right\}}_{j = 0}^{N},$ holds at colocation points, then the spectral approximation to Eqs (3) and (4) is given by:

To compute ${F}_{p}\left(s\right)$ efficiently, we express it in terms of the Legendre functions of the form ^[8].

where ${f}_{pm}$ is called the discrete polynomial coefficients of ${F}_{p}$. The inverse relation is

and ${\gamma }_{N} = {\left(N+{}^{1}\!\!\diagup\!\!{}_{2}\;\right)}^{-1}$ for the Gauss and Gauss-Radau formulas.

3.   Convergence analysis

To prove the convergence analysis of our method we first introduce the following useful lemmas.

3.1.   Lemma 1

Assume that a $\left(N+1\right)-$ point Gauss, or Gauss-Radau, or Gauss-Lobatto quadrature formula relative to the Legendre weight is used to integrate the product $u\varphi ,$ where $u\in {H}^{m}\left(I\right)$ with $I: = \left(-1, 1\right)$ for some $m\ge 1$ and $\varphi \in {p}_{N}.$ Then there exist a constant $C$ independent of $N$ such that ^[7].

where

3.2.   Lemma 2

Assume that $u\in {H}^{m}\left(I\right)$ and denote ${I}_{N}u$ the interpolation polynomial associated with the $\left(N+1\right)-$ point Gauss, or Gauss-Radau, or Gauss-Lobatto points ${\left\{{x}_{k}\right\}}_{k = 0}^{N}.$ Then

Proof. The estimation in Eq (21) is given on p. 289 of ^[7]. The following estimate is also given in ^[7].

using the above estimate and the inequality

one obtains the estimation given in Eq (22).

3.3.   Lemma 3

Assume that ${F}_{j}\left(t\right)$ be the $j-th$ Lagrange interpolation polynomial with the $\left(N+1\right)-$ point Gauss, or Gauss-Radau, or Gauss-Lobatto points ${\left\{{t}_{k}\right\}}_{k = 0}^{N}.$ Then

3.4.   Lemma 4 (Gronwall inequality)

Let $T > 0 \ and \ {C}_{1}, {C}_{2}\ge 0.$ If a non-negative integrable function $E\left(t\right)$ satisfies

where $0 < \alpha < 1$ is a constant and $G\left(t\right)$ is a nonnegative function, then

Proof. It follows from Eq (23) and a simple change of variable that

since $0 < \alpha < 1$ and where $G\left(s\right)\ge 0,$ we have

which is a standard Gronwall inequality. This leads to the estimate given in Eq (26).

3.5.   Theorem 1

Consider the pantograph Eqs (3) and (4) and its spectral approximations Eqs (17) and (18). If the functions ${\mathrm{\lambda }}_{l}\left(t\right)$ and ${\mu }_{l}\left(t\right)$ are smooth (which implies that the solution of Eqs (3) and (4) is also smooth), then

where $U$ is the polynomial of degree $N$ associated with the spectral approximation Eqs (17) and (18) and $C$ is a constant independent of $N$.

Proof. Following the notation in Lemma 1, the numerical scheme given in Eqs (17) and (18), can be written as

In order to use Lemma 1, we write Eqs (28) and (29) as

where

Using the estimation given in Lemma 2, we get

Multiplying ${F}_{j}\left(t\right)$ on both sides of Eqs (30) and (31) summing up from $0$ to $N$ yield

Similarly, multiplying ${F}_{j}\left(t\right)$ on both sides of Eqs (10) and (11) summing up from $0$ to $N$ yield

It follows from Eqs (31)–(34) that

where

Consequently,

where ${f}_{1}\left(t\right) = {u}'\left(t\right)-{I}_{N}{U}'\left(t\right),$ ${f}_{3}\left(t\right) = u\left(t\right)-{I}_{N}u\left(t\right)$

Using Lemma 2,

Using Lemma 2, with $m = 1,$

It follows from the Gronwall inequality presented in Lemma 4, to get

Our next concern is the estimation of ${‖{J}_{1}‖}_{{L}^{\infty }\left(I\right)} \ and \ {‖{J}_{2}‖}_{{L}^{\infty }\left(I\right)}.$ First

The above two estimates, together with Eq (35), yields:

which leads to the result of Theorem 1.

4.   Numerical examples

In order to confirm the theoretical results, we perform some numerical test to illustrate the accuracy and efficiency of the proposed scheme.

4.1.   Example 1

Consider the following constructed example ^[1]

The exact solution is given by $y\left(x\right) = {x}^{2}.$ The maximum point-wise error between numerical solution and exact solution for different value of $N$ is shown in Figure 1, Table 1.

4.2.   Example 2

Consider the following nonlinear second-order equation of pantograph type

The exact solution is given by $y\left(x\right) = {e}^{-2x}.$ The maximum point-wise error between numerical solution and exact solution for with respect to different $N$ is shown in Figure 2, Table 2.

4.3.   Example 3

Consider the following second-order equation

The exact solution is given by $y\left(x\right) = \frac{5-cos2x}{2}.$ The error behavior relative to $N$ is displayed in Figure 3, Table 3.

5.   Conclusions

A spectral method was introduced for the numerical solution of higher-order delay differential equation of pantograph equation to achieve the high order accuracy. A detail analysis of the proposed scheme is provided in ${L}_{\infty }$ norm. By solving some numerical examples, it is shown that the error between the exact and numerical solution decays exponentially, which further authenticates our theoretical results.

Acknowledgments

The author acknowledges the Deanship of Scientific Research at King Faisal University for the financial support under Nasher Track (Grant No. 206105).

Conflict of interest

The author declares that they have no conflicts of interest to report regarding the present study.