#  Existence and Uniqueness Theorems for Differential Equations with
Proportional Delay

Prajakta Rajmanea∗<EMAIL_ADDRESS>Jayvant Patadeb
<EMAIL_ADDRESS>M. T. Gophanea<EMAIL_ADDRESS>aDepartment of Mathematics, Shivaji University, Kolhapur - 416004, India.
bDepartment of Mathematics, Jaysingpur College, Jaysingpur - 416101, India.

###### Abstract

The differential equation (DE) with proportional delay is a particular case of
the time-dependent delay differential equation (DDE). In this paper, we solve
non-linear DEs with proportional delay using the successive approximation
method (SAM). We prove the existence, uniqueness of theorems, and stability
for DEs with proportional delay using SAM. We derive convergence results for
these equations by using the Lipschitz condition. We generalize these results
to the fractional differential equations (FDEs) and system of FDEs containing
Caputo fractional derivative. Further, we obtain the series solution of the
pantograph equation and Ambartsumian equation in the form of a power series
which are convergent for all reals. Finally, we illustrate the efficacy of the
SAM by example. The results obtained by SAM are compared with exact solutions
and other iterative methods. It is observed that SAM is simpler compared to
other methods and the solutions obtained using SAM are consistent with the
exact solution.

###### keywords:

Successive approximation method; Lipschitz condition; Caputo derivative;
Existence-uniqueness; proportional delay; pantograph equation; Ambartsumian
equation.

###### MSC:

[2020] 26A33; 34A08; 34K06; 34K20.

††journal: Journal of LaTeX Templates

##  1 Introduction

The delay differential equations (DDE) contain the state variable term at a
past time $t-\tau$. The inclusion of the delay $\tau$ makes the DDE an
infinite dimensional dynamical system. Even if it is very difficult to analyze
and solve such equations, this branch is popular among the applied scientists
due to the applications in various fields.

On the other hand, if the order of the derivative in a differential equation
is any arbitrary number (instead of a positive integer) then the equation is
called as the fractional differential equation (FDE). Even though there are
several inequivalent definitions of fractional derivative operator, one can
select the derivative which is appropriate for the model under consideration.
This flexibility is a key feature behind the popularity of fractional
calculus.

Daftardar-Gejji and coworkers proposed numerical schemes [1, 2] for solving
fractional order delay differential equations (FDDE). Modified Laguerre
wavelets method [3], spectral collocation method [4], fractional-order
fibonacci-hybrid functions [5] are few other methods for solving FDDEs.
Stability analysis of FDDEs is proposed in [6, 7, 8, 9]. Applications of FDDE
are presented in [10, 11, 12, 13].

In general, the delay $\tau$ in the DDE ${x^{\prime}(t)}=f(t,x(t),x(t-\tau))$
is not constant. The analysis becomes more difficult when $\tau$ depends on
time or state. The proportional delay differential equation
${x^{\prime}(t)}=f(t,x(t),x(qt))$ or a pantograph equation is a particular
case of time-dependent DDE with $\tau(t)=(1-q)t$. These equations are proposed
by Ockendon and Tayler in the seminal work [14] to model the motion of an
overhead trolley wire. Few other applications of these equations are discussed
in [15, 16]. The Daftardar-Gejji and Jafari method (DJM) is applied in [17] to
find analytical solutions of pantograph equation. Further, the authors
presented the various relations of the solution series with the existing
special functions. Patade and Bhalekar proposed the power series solution
Ambartsumian equation [18] by using DJM. The analytical solution of pantograph
equation are discussed in [19]. In this paper, we generalize the results in
[20] on the existence-uniqueness of ordinary differential equations (ODE) to
differential equations with proportional delay, FDEs proportional delay and
system of FDEs with proportional delay. We use the successive approximation
method (SAM) to prove our results.

The paper is organized as follows. In Section 2, we give definitions and
notations of fractional derivatives and integrals. The differential equations
with proportional delay are described in Section 3. Successive approximate
solutions and existence theorem are discussed in Section 4. The stability
analysis is presented in Section 5. The series solution of the pantograph
equation and Ambartsumian equation are described in Section 6. The
generalization of these results to FDEs and the system of FDEs are derived in
Section 7 and Section 8. Section 9 deals with illustrative example and the
conclusions are summarized in Section 10.

##  2 Preliminaries and Notations

######  Definition 2.1.

[21] The Riemann-Liouville fractional integral of order $\alpha>0$ of $f\in
C[0,\infty)$ is defined as

$I^{\alpha}f(t)=\frac{1}{\Gamma(\alpha)}\int_{0}^{t}(t-\zeta)^{\alpha-1}f(\zeta)d\zeta,\quad
t>0.$ (2.1)

######  Definition 2.2.

[21] The (left sided) Caputo fractional derivative of $f,f\in
C_{-1}^{m},m\in\mathbb{N}\cup\\{0\\}$, is defined as:

$\displaystyle D^{\alpha}f(t)$ $\displaystyle=$
$\displaystyle\frac{d^{m}}{dt^{m}}f(t),\quad\alpha=m$ (2.2) $\displaystyle=$
$\displaystyle I^{m-\alpha}\frac{d^{m}}{dt^{m}}f(t),\quad{m-1}<\alpha<m,\quad
m\in\mathbb{N}.$

Note that for $0\leq m-1<\alpha\leq m$ and $\beta>-1$

$\displaystyle I^{\alpha}x^{\beta}$ $\displaystyle=$
$\displaystyle\frac{\Gamma{(\beta+1)}}{\Gamma{(\beta+\alpha+1)}}x^{\beta+\alpha},$
$\displaystyle\left(I^{\alpha}D^{\alpha}f\right)(t)$ $\displaystyle=$
$\displaystyle f(t)-\sum_{k=0}^{m-1}f^{(k)}(0)\frac{t^{k}}{k!}.$ (2.3)

######  Definition 2.3.

[21] The Mittag-Leffler function is defined as

$E_{\alpha}(t)=\sum_{n=0}^{\infty}\frac{t^{n}}{\Gamma{(\alpha
n+1)}},\quad\alpha>0.$ (2.4)

######  Definition 2.4.

[21] The multi-parameter Mittag-Leffler function is defined as:

$E_{(\alpha_{1},\cdots,\alpha_{n}),\beta}(z_{1},z_{2},\cdots,z_{n})=\sum^{\infty}_{k=0}\mathop{\sum_{l_{1}+\cdots+l_{n}=k}}_{l_{j}\geq
0}(k;l_{1},\cdots,l_{n})\left[\frac{\displaystyle\prod_{j=1}^{n}z^{l_{j}}_{j}}{\Gamma(\beta+\displaystyle\sum^{n}_{j=1}\alpha_{j}l_{j})}\right].$

where, $\left(k;l_{1},l_{2},\cdots,l_{n}\right)$ is the multinomial
coefficient defined as

$\left(k;l_{1},l_{2},\cdots,l_{n}\right)=\frac{k!}{l_{1}!l_{2}!\cdots
l_{n}!}.$ (2.5)

##  3 Differential Equations with Proportional delay

Consider the differential equations with proportional delay

$y^{\prime}(t)=f(t,{y}(t),{y}(qt)),y(0)=y_{0},0<q<1,$ (3.1)

where $f$ is a continuous function defined on some rectangle

$R=\\{|t|\leq a,|y(t)-y_{0}|\leq b,|y(qt)-y_{0}|\leq b,a>0,\,b>0\\}.$

######  Theorem 1.

A function $\phi$ is a solution of the IVP (3.1) on an interval $I$ if and
only if it is a solution of the integral equation

$y(t)=y_{0}+\int_{0}^{t}f(x,{y}(x),{y}(qx))dx\quad\text{on}\quad I$ (3.2)

###### Proof.

Let $\phi$ is a solution of the IVP (3.1) on an interval $I$. Then

$\phi^{\prime}(t)=f(t,{\phi}(t),{\phi}(qt)),\phi(0)=y_{0},0<q<1$ (3.3)

The equivalent integral equation (3.3) is

$\phi(t)=\phi(0)+\int_{0}^{t}f(x,{\phi}(x),{\phi}(qx))dx.$ (3.4)

and $\phi(0)=y_{0}$. Thus $\phi$ is a solution of the IVP (3.2).
Conversely, suppose equation (3.4) hold. Differentiate equation (3.4) w.r.t.
$t$, we get

$\phi^{\prime}(t)=f(t,{\phi}(t),{\phi}(qt)),0<q<1\quad\forall t\in I$

From equation (3.2) $\phi(0)=y_{0}$.
Hence $\phi$ is a solution of the IVP (3.1). ∎

##  4 Successive Approximate Solution for Differential Equations with
Proportional Delay

Let $\phi_{0}(t)=y_{0}$ be the first approximate solution of the IVP (3.1).
Then

$\displaystyle\phi_{1}(t)$ $\displaystyle=$ $\displaystyle
y_{0}+\int_{0}^{t}f(x,{\phi_{0}}(x),{\phi_{0}}(qx))dx.$
$\displaystyle\phi_{2}(t)$ $\displaystyle=$ $\displaystyle
y_{0}+\int_{0}^{t}f(x,{\phi_{1}}(x),{\phi_{1}}(qx))dx.$

Continuing in this way, we obtain

$\phi_{k+1}(t)=y_{0}+\int_{0}^{t}f(x,{\phi_{k}}(x),{\phi_{k}}(qx))dx.\quad
k=0,1,2,\cdots.$ (4.1)

######  Theorem 2.

Let f is continuous and $|f|\leq M$ on $R$. The successive approximation (4.1)
exist and continuous on the interval $I=[-\zeta,\zeta]$, where
$\zeta=\text{min}\left\\{a,\frac{b}{M}\right\\}$. If $t\in I$ then
$\left(t,y(t),y(qt)\right)\in R$ and $|\phi_{k}(t)-y_{0}|\leq M|t|$,
$|\phi_{k}(qt)-y_{0}|\leq M|t|$.

###### Proof.

We prove the result by mathematical induction.
(i) Clearly $\phi(0)=y_{0}$ is continuous on $I$. Thus, theorem is true for
$k=0$.
(ii) For $k=1$, we have

$\displaystyle\phi_{1}(t)$ $\displaystyle=$ $\displaystyle
y_{0}+\int_{0}^{t}f(x,{\phi_{0}}(x),{\phi_{0}}(qx))dx.$
$\displaystyle\phi_{1}(t)$ $\displaystyle=$ $\displaystyle
y_{0}+\int_{0}^{t}f(x,y_{0},y_{0})dx.$

Since $f$ is continuous and hence, $\phi_{1}(t)$ exist.

$\displaystyle|\phi_{1}(t)-y_{0}|$ $\displaystyle=$
$\displaystyle|\int_{0}^{t}f(x,{\phi_{0}}(x),{\phi_{0}}(qx))dx|.$
$\displaystyle\leq$
$\displaystyle\int_{0}^{t}|f(x,{\phi_{0}}(x),{\phi_{0}}(qx))|dx.$
$\displaystyle\leq$ $\displaystyle M|t|$ $\displaystyle\leq$ $\displaystyle
b,\quad t\in I$ $\displaystyle\text{and}\quad|\phi_{1}(qt)-y_{0}|$
$\displaystyle\leq$ $\displaystyle M|qt|$ $\displaystyle\leq$ $\displaystyle
M|t|,\quad 0<q<1$ $\displaystyle\leq$ $\displaystyle b,\quad t\in I$

Thus, for $t\in I$, $\left(t,y(t),y(qt)\right)\in R$ and
$|\phi_{1}(t)-y_{0}|\leq M|t|$, $|\phi_{1}(qt)-y_{0}|\leq M|t|$.
The theorem is true for $k=1$
(iii) Assume that theorem is true for $k=n$.
i.e. For $t\in I$, $\left(t,y(t),y(qt)\right)\in R$ and
$|\phi_{n}(t)-y_{0}|\leq M|t|$, $|\phi_{n}(qt)-y_{0}|\leq M|t|$.
(iv) To prove the theorem for $k=n+1$.
If $t\in I$, then

$\displaystyle\phi_{n+1}(t)$ $\displaystyle=$ $\displaystyle
y_{0}+\int_{0}^{t}f(x,{\phi_{n}}(x),{\phi_{n}}(qx))dx.$

Since $f$ is continuous and hence, $\phi_{n+1}(t)$ exist on $I$.

$\displaystyle|\phi_{n+1}(t)-y_{0}|$ $\displaystyle\leq$ $\displaystyle M|t|$
$\displaystyle\leq$ $\displaystyle b,\quad t\in I$
$\displaystyle\text{and}\quad|\phi_{n+1}(qt)-y_{0}|$ $\displaystyle\leq$
$\displaystyle M|qt|$ $\displaystyle\leq$ $\displaystyle M|t|,\quad 0<q<1$
$\displaystyle\leq$ $\displaystyle b,\quad t\in I$

Thus, if $t\in I$, $\left(t,y(t),y(qt)\right)\in R$ and
$|\phi_{n+1}(t)-y_{0}|\leq M|t|$, $|\phi_{n+1}(qt)-y_{0}|\leq M|t|$.
Hence by mathematical induction, the result is true for all positive integer
$n$. ∎

######  Theorem 3.

(Existence Theorem) Let f is continuous and $|f|\leq M$ on the rectangle

$R=\\{|t|\leq a,|y(t)-y_{0}|\leq b,|y(qt)-y_{0}|\leq b,a>0,\,b>0\\}.$

Suppose $f$ satisfies Lipschitz condition in second and third variable with
Lipschitz constants $L_{1}$ and $L_{2}$ such that

$|f(t,{y_{1}}(t),{y_{1}}(qt))-f(t,{y_{2}}(t),{y_{2}}(qt))|\leq
L_{1}|y_{1}(t)-y_{2}(t)|+L_{1}|y_{1}(qt)-y_{2}(qt)|.$

Then the successive approximations (4.1) converges on the interval
$I=[-\zeta,\zeta]$, where $\zeta=\text{min}\left\\{a,\frac{b}{M}\right\\}$ to
a solution $\phi$ of the IVP (3.1) on $I$.

###### Proof.

We have

$\phi_{k}(t)=\phi_{0}(t)+\sum_{n=1}^{k}[\phi_{n}(t)-\phi_{n-1}(t)].$

To prove the sequence $\\{\phi_{k}\\}$ converges, it is enough to prove the
series

$\phi_{0}(t)+\sum_{n=1}^{\infty}[\phi_{n}(t)-\phi_{n-1}(t)]$ (4.2)

is convergent.
By theorem (2) the function $\phi_{k}$ all exist and continuous on $I$.
Also, $|\phi_{1}(t)-\phi_{0}(t)|\leq M|t|$ and
$|\phi_{1}(qt)-\phi_{0}(qt)|\leq M|t|$ for $t\in I$.
Now,

$\displaystyle\phi_{2}(t)-\phi_{1}(t)$ $\displaystyle=$
$\displaystyle\int_{0}^{t}[f(x,{\phi_{1}}(x),{\phi_{1}}(qx))-f(x,{\phi_{0}}(x),{\phi_{0}}(qx))]dx$
$\displaystyle\therefore|\phi_{2}(t)-\phi_{1}(t)|$ $\displaystyle\leq$
$\displaystyle\int_{0}^{t}|f(x,{\phi_{1}}(x),{\phi_{1}}(qx))-f(x,{\phi_{0}}(x),{\phi_{0}}(qx))|dx$
$\displaystyle\leq$
$\displaystyle\int_{0}^{t}[L_{1}|{\phi_{1}}(x)-{\phi_{0}}(x)|+L_{2}|{\phi_{1}}(qx)-{\phi_{0}}(qx)|]dx$
$\displaystyle\leq$ $\displaystyle M(L_{1}+L_{2})\frac{|t|^{2}}{2}.$

We shall prove by mathematical induction

$|\phi_{n}(t)-\phi_{n-1}(t)|\leq M(L_{1}+L_{2})^{n-1}\frac{|t|^{n}}{n!}$ (4.3)

We have prove that equation (4.3) true for $n=1,2.$
Assume that (4.3) true for $n=m.$
We have

$\displaystyle\phi_{m+1}(t)-\phi_{m}(t)$ $\displaystyle=$
$\displaystyle\int_{0}^{t}[f(x,{\phi_{m}}(x),{\phi_{m}}(qx))-f(x,{\phi_{m-1}}(x),{\phi_{m-1}}(qx))]dx$
$\displaystyle\therefore|\phi_{m+1}(t)-\phi_{m}(t)|$ $\displaystyle\leq$
$\displaystyle\int_{0}^{t}|f(x,{\phi_{m}}(x),{\phi_{m}}(qx))-f(x,{\phi_{m-1}}(x),{\phi_{m-1}}(qx)|dx$
$\displaystyle\leq$
$\displaystyle\int_{0}^{t}[L_{1}|{\phi_{m}}(x)-\phi_{m-1}(x)|+L_{2}|{\phi_{m}}(qx)-{\phi_{m-1}}(qx)|]dx$
$\displaystyle\leq$ $\displaystyle
M(L_{1}+L_{2})^{m}\frac{|t|^{m+1}}{(m+1)!}.$

Thus, the result is true for $n=m+1$.
Hence, by the mathematical induction result is true for all $n=1,2,\cdots.$
Therefore, the infinite series (4.3) is absolutely convergent on $I$. This
shows that the $n^{th}$ term of the series
$|\phi_{0}(t)|+\sum_{n=1}^{\infty}|\phi_{n}(t)-\phi_{n-1}(t)|$ is less than
$\frac{M}{(L_{1}+L_{2})}$ times the $n^{th}$ term of the power series
$e^{(L_{1}+L_{2})|t|}$. Hence The series (4.3) is convergent. ∎

##  5 Stability Analysis

The differential equations with proportional delay

$y^{\prime}(t)=f\left(t,y(t),y(qt)\right),$ (5.1)

is a special case of the time-dependent delay differential equation (DDE)

$y^{\prime}(t)=f\left(t,y(t),y\left(t-\tau(t)\right)\right)\quad\textrm{with}\quad\tau(t)=(1-q)t,$

######  Definition 5.1.

[22] Consider the DDE,

$y^{\prime}(t)=f(y(t),y(t-\tau(t))),$ (5.2)

where $f:R\times R\rightarrow R$. The flow $\phi_{t}(t_{0})$ is a solution
$y(t)$ of (5.2) with initial condition $y(t)=t_{0},\,t\leq 0$. The point
$y^{*}$ is called equilibrium solution of (5.2) if $f(y^{*},y^{*})=0$.
(a) If, for any $\epsilon>0$, there exist $\delta>0$ such that
$|t_{0}-y^{*}|<\delta\Rightarrow|\phi_{t}(t_{0})-y^{*}|<\epsilon,$ then the
system (5.2) is stable (in the Lyapunov sense) at the equilibrium $y^{*}$.
(b) If the system (5.2) is stable at $y^{*}$ and
moreover,$\lim\limits_{t\rightarrow\infty}|\phi_{t}(t_{0})-y^{*}|=0$ then the
system (5.2) is said to be asymptotically stable at $y^{*}$.

The following results are similar to those in [22]

######  Theorem 4.

Suppose that the equilibrium solution $y^{*}$ of the equation

$y^{\prime}=f(y(t),y(t-\tau^{*})),\quad\tau^{*}=\tau(t_{0})$ (5.3)

is stable and
$\|f(y(t),y(t-\tau(t)))-f(y(t),y(t-\tau(t_{1})))\|<\epsilon_{1}|t-t_{1}|$, for
some $\epsilon_{1}>0$ and $t,t_{1}\in[t_{0},t_{0}+c),$ c is a positive
constant, then there exists $\bar{t}>0$ such that the equilibrium solution
$y^{*}$ of Eq. (5.2) is stable on finite time interval $[t_{0},\bar{t})$.

######  Corollary 5.

If the real parts of all roots of $\lambda-a-be^{-\lambda\tau^{*}}=0$ are
negative, where $a=\partial_{1}f,b=\partial_{2}f$ evaluated at equilibrium.
Then there exist $\epsilon_{c},\bar{t}(>t_{0})$, such that when
$\epsilon_{1}<\epsilon_{c}$, the solution $y^{*}=0$ of Eq. (5.2) is stable on
finite time interval $[t_{0},\bar{t})$.

##  6 Series Solution of Pantograph Equation

A pantograph is a device used in electric trains to collect current from
overloaded lines. The pantograph equation was formulated by Ockendon and
Taylor in 1971 and originates in electrodynamics.
Consider the pantograph equation,

$y^{\prime}(t)=ay(t)+by(qt),\quad y(0)=1,$ (6.1)

where $0<q<1$, $a,b\in{R}$.

Integrating (6.1), we get

$y(t)=1+\int_{0}^{t}\left(ay(x)+by(qx)\right)dt$ (6.2)

Suppose $\phi_{k}(t)$ be the $k^{th}$ approximate solution, where the initial
approximate solution is taken as

$\phi_{0}(t)=1.$ (6.3)

For $k\geq 1$, the recurrent formula as below:

$\phi_{k}(t)=1+\int_{0}^{t}\left(a\phi_{k-1}(x)+b\phi_{k-1}(qx)\right)dx.$
(6.4)

From the recurrent formula, we have

$\displaystyle\phi_{1}(t)$ $\displaystyle=$ $\displaystyle
1+\int_{0}^{t}\left(a\phi_{0}(x)+by_{0}(qx)\right)dx$ $\displaystyle=$
$\displaystyle 1+(a+b)\frac{t}{1!},$ $\displaystyle\phi_{2}(t)$
$\displaystyle=$ $\displaystyle
1+\int_{0}^{t}\left(a\phi_{1}(x)+b\phi_{1}(qx)\right)dt$ $\displaystyle=$
$\displaystyle 1+(a+b)\frac{t}{1!}+(a+b)(a+bq)\frac{t^{2}}{2!},$
$\displaystyle\phi_{3}(t)$ $\displaystyle=$ $\displaystyle
1+\int_{0}^{t}\left(a\phi_{2}(t)+b\phi_{2}(qt)\right)dt$ $\displaystyle=$
$\displaystyle
1+(a+b)\frac{t}{1!}+(a+b)(a+bq)\frac{t^{2}}{2!}(a+b)(a+bq)(a+bq^{2})\frac{t^{3}}{3!},$
$\displaystyle\vdots$ $\displaystyle\phi_{k}(x)$ $\displaystyle=$
$\displaystyle 1+\frac{t^{k}}{k!}\prod_{j=0}^{k-1}\left(a+bq^{j}\right),\quad
k=1,2,3\cdots.$ $\displaystyle\text{As}\quad
k\rightarrow\infty,\quad\phi_{k}(t)\rightarrow y(t)$ $\displaystyle y(t)$
$\displaystyle=$ $\displaystyle
1+\sum_{m=1}^{\infty}\frac{t^{m}}{m!}\prod_{j=0}^{m-1}\left(a+bq^{j}\right).$

If we define $\prod_{j=0}^{m-1}\left(a+bq^{j}\right)=1$, for $m=0$, then

$y(t)=\sum_{m=0}^{\infty}\frac{t^{m}}{m!}\prod_{j=0}^{m-1}\left(a+bq^{j}\right).$
(6.5)

######  Theorem 6.

For $0<q<1$, the power series (6.5) is convergent for $t\in{R}$.

######  Corollary 7.

The power series (6.5) is absolutely convergent for all $t$ and hence it is
uniformly convergent on any compact interval on ${R}$.

######  Theorem 8.

If $0<q<1$, $a,b\geq 0$, then

$e^{at}\leq
y(t)=\sum_{m=0}^{\infty}\frac{t^{m}}{m!}\prod_{j=0}^{m-1}\left(a+bq^{j}\right)\leq
e^{(a+b+c)t},\quad 0\leq t<\infty.$

######  Theorem 9.

If $(a+b)<0$ then zero solution of (6.1) is asymptotically stable.

###### Proof.

$\displaystyle\text{Define}\qquad u(t)$ $\displaystyle=$
$\displaystyle\max_{0\leq x\leq t}y^{2}(t)$
$\displaystyle\therefore\frac{1}{2}u^{\prime}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\frac{d}{dt}(y^{2}(t))$ $\displaystyle=$
$\displaystyle y(t)y^{\prime}(t)$ $\displaystyle=$ $\displaystyle
y(t)(ay(t)+by(qt))$ $\displaystyle=$ $\displaystyle ay^{2}(t)+by(t)y(qt)$
$\displaystyle\leq$ $\displaystyle(a+b)u(t)$ $\displaystyle\Rightarrow u(t)$
$\displaystyle\leq$ $\displaystyle u(0)e^{2(a+b)t}$
$\displaystyle\therefore\lim\limits_{x\rightarrow\infty}y(t)$ $\displaystyle=$
$\displaystyle 0,\quad\textrm{if}\quad(a+b)<0.$

∎

###  6.1 Series Solution of Ambartsumian Equation

In [23] Ambartsumian derived a delay differential equation describing the
fluctuations of the surface brightness in a milky way. The equation is
described as:

$y^{\prime}(t)=-y(t)+\frac{1}{q}y\left(\frac{t}{q}\right)$ (6.6)

where $q>1$ and is constant for the given model.
The Eq.(6.6) with initial condition $y(0)=\lambda$ can be written equivalently
as

$y(t)=\lambda+\int_{0}^{t}\left(\frac{1}{q}y\left(\frac{x}{q}\right)-y(x)\right)dx.$
(6.7)

Suppose $\phi_{k}(t)$ be the $k^{th}$ approximate solution, where the initial
approximate solution is taken as

$\phi_{0}(t)=\lambda.$ (6.8)

For $k\geq 1$, the recurrent formula as below:

$\phi_{k}(t)=\lambda+\int_{0}^{t}\left(\frac{1}{q}\phi_{k-1}\left(\frac{x}{q}\right)-\phi_{k-1}(x)\right)dx.$
(6.9)

From the recurrent formula, we have

$\displaystyle\phi_{1}(t)$ $\displaystyle=$
$\displaystyle\lambda+\int_{0}^{t}\left(\frac{1}{q}\phi_{0}\left(\frac{x}{q}\right)-\phi_{0}(x)\right)dx$
$\displaystyle=$
$\displaystyle\lambda+\int_{0}^{t}\left(\frac{\lambda}{q}-\lambda\right)dx$
$\displaystyle=$
$\displaystyle\lambda+\left(\frac{\lambda}{q}-\lambda\right)\frac{t}{1!}$
$\displaystyle=$
$\displaystyle\left(1+\left(\frac{1}{q}-1\right)\frac{t}{1!}\right)\lambda,$
$\displaystyle\phi_{2}(t)$ $\displaystyle=$
$\displaystyle\lambda+\int_{0}^{t}\left(\frac{1}{q}\phi_{1}\left(\frac{x}{q}\right)-\phi_{1}(x)\right)dx$
$\displaystyle=$
$\displaystyle\left(1+\left(\frac{1}{q}-1\right)\frac{t}{1!}+\left(\frac{1}{q}-1\right)\left(\frac{1}{q^{2}}-1\right)\frac{t^{2}}{2!}\right)\lambda,$
$\displaystyle\vdots$ $\displaystyle\phi_{k}(t)$ $\displaystyle=$
$\displaystyle\left(1+\sum_{m=1}^{k}\frac{t^{m}}{m!}\prod_{j=1}^{m}\left(\frac{1}{q^{j}}-1\right)\right)\lambda.$
$\displaystyle\text{As}\quad k\rightarrow\infty,\quad\phi_{k}(t)\rightarrow
y(t)$ $\displaystyle y(t)$ $\displaystyle=$
$\displaystyle\left(1+\sum_{m=1}^{\infty}\frac{t^{m}}{m!}\prod_{j=1}^{m}\left(\frac{1}{q^{j}}-1\right)\right)\lambda.$

If we define $\prod_{j=1}^{m}\left(\frac{1}{q^{j}}-1\right)=1$, for $m=0$,
then

$y(t)=\left(\sum_{m=0}^{\infty}\frac{t^{m}}{m!}\prod_{j=1}^{m}\left(\frac{1}{q^{j}}-1\right)\right)\lambda.$
(6.10)

######  Theorem 10.

For $q>1$, the power series (6.10) is convergent for $t\in{R}$.

######  Corollary 11.

The power series (6.10) is absolutely convergent for all $t$ and hence it is
uniformly convergent on any compact interval on ${R}$.

######  Theorem 12.

The zero solution of (6.6) is asymptotically stable.

##  7 Fractional order differential equations with proportional delay

Consider the initial value problem (IVP)

$\displaystyle D^{\alpha}y(t)$ $\displaystyle=$ $\displaystyle
f(t,{y}(t),{y}(qt)),0<\alpha\leq 1,0<q<1$ $\displaystyle y(0)$
$\displaystyle=$ $\displaystyle y_{0},$ (7.1)

where $D^{\alpha}$ denotes Caputo fractional derivative and $f$ is a
continuous function defined on the rectangle

$R=\\{|t|\leq a,|y(t)-y_{0}|\leq b,|y(qt)-y_{0}|\leq b,a>0,\,b>0\\}.$

######  Theorem 13.

A function $\phi$ is a solution of the IVP (7.1) on an interval $I$ if and
only if it is a solution of the integral equation

$y(t)=y_{0}+\int_{0}^{t}\frac{(t-x)^{\alpha-1}}{\Gamma(\alpha)}f(x,{y}(x),{y}(qx))dx\quad\text{on}\quad
I.$ (7.2)

######  Theorem 14.

Let f is continuous and $|f|\leq M$ on $R$. The successive approximation

$\displaystyle\phi_{k+1}(t)$ $\displaystyle=$ $\displaystyle y_{0}$
$\displaystyle\phi_{k+1}(t)$ $\displaystyle=$ $\displaystyle
y_{0}+\int_{0}^{t}\frac{(t-x)^{\alpha-1}}{\Gamma(\alpha)}f(x,{\phi_{k}}(x),{\phi_{k}}(qx))dx.\quad
k=0,1,2,\cdots.$ (7.3)

exist and continuous on the interval $I=[-\zeta,\zeta]$, where
$\zeta=\text{min}\left\\{a,(\frac{\Gamma(\alpha+1)b}{M})^{\frac{1}{\alpha}}\right\\}$.
If $t\in I$ then $\left(t,y(t),y(qt)\right)\in R$ and $|\phi_{k}(t)-y_{0}|\leq
M\frac{|t|^{\alpha}}{\Gamma(\alpha+1)}$, $|\phi_{k}(qt)-y_{0}|\leq
M\frac{|t|^{\alpha}}{\Gamma(\alpha+1)}$.

######  Theorem 15.

(Existence Theorem) Let f is continuous and $|f|\leq M$ on the rectangle

$R=\\{|t|\leq a,|y(t)-y_{0}|\leq b,|y(qt)-y_{0}|\leq b,a>0,\,b>0\\}.$

Suppose $f$ satisfies Lipschitz condition in second and third variable with
Lipschitz constants $L_{1}$ and $L_{2}$ such that

$|f(t,{y_{1}}(t),{y_{1}}(qt))-f(t,{y_{2}}(t),{y_{2}}(qt))|\leq
L_{1}|y_{1}(t)-y_{2}(t)|+L_{1}|y_{1}(qt)-y_{2}(qt)|.$

Then the successive approximations (7.3) converges on the interval
$I=[-\zeta,\zeta]$, where
$\zeta=\text{min}\left\\{a,(\frac{\Gamma(\alpha+1)b}{M})^{\frac{1}{\alpha}}\right\\}$
to a solution $\phi$ of the IVP (7.1) on $I$.

###  7.1 Series Solution of Fractional Order Pantograph Equation

Consider the fractional order pantograph equation as :

$D^{\alpha}y(t)=ay(t)+by(qt),\quad y(0)=1,$ (7.4)

where $0<\alpha\leq 1$, $0<q<1$, $a,b\in{R}$.

The solution of (7.4) using successive approximation is

$y(t)=\sum_{m=0}^{\infty}\frac{t^{\alpha m}}{\Gamma{(\alpha
m+1)}}\prod_{j=0}^{m-1}\left(a+bq^{\alpha j}\right).$ (7.5)

######  Theorem 16.

If $0<q<1$, then the power series (7.5) is convergent for all finite values of
$t$.

######  Theorem 17.

If $0<q<1$, $a,b\geq 0$, then

$E_{\alpha}{(at^{\alpha})}\leq y(t)=\sum_{m=0}^{\infty}\frac{t^{\alpha
m}}{\Gamma{(\alpha m+1)}}\prod_{j=0}^{m-1}\left(a+bq^{\alpha j}\right)\leq
E_{\alpha}{((a+b)t^{\alpha})},\quad 0\leq t<\infty.$

###  7.2 Series Solution of Fractional Order Ambartsumian Equation

Consider the fractional order Ambartsumian equation as:

$D^{\alpha}y(t)=-y(t)+\frac{1}{q}y\left(\frac{t}{q}\right),\quad y(0)=1$ (7.6)

where $q>1$ and is constant for the given model.

The solution of (7.6) using successive approximation is

$y(t)=\sum_{m=0}^{\infty}\frac{t^{\alpha m}}{\Gamma{(\alpha
m+1)}}\prod_{j=0}^{m-1}\left(\frac{1}{q^{1+\alpha j}}-1\right).$ (7.7)

######  Theorem 18.

If $q>1$, then the power series (7.7) is convergent for all finite values of
$t$.

##  8 System of fractional order differential equations with proportional
delay

Consider the initial value problem (IVP)

$\displaystyle D^{\alpha_{i}}y_{i}(t)$ $\displaystyle=$ $\displaystyle
f_{i}(t,\bar{y}(t),\bar{y}(qt)),0<\alpha_{i}\leq 1,0<q<1$ $\displaystyle
y_{i}(0)$ $\displaystyle=$ ${}^{i}y_{0},\quad 1\leq i\leq n,$ (8.1)

where $D^{\alpha_{i}}$ denotes Caputo fractional derivative,
$\bar{y}(t)=(y_{1}(t),y_{2}(t)\cdots,y_{n}(t))$,
$\bar{y}(qt)=(y_{1}(qt),y_{2}(qt)\cdots,y_{n}(qt))$ and
$f=(f_{1},f_{2}\cdots,f_{n})$ is a continuous function defined on the
rectangle

$R=\\{|t|\leq a,|y_{i}(t)-^{i}y_{0}|\leq b_{i},|y_{i}(qt)-^{i}y_{0}|\leq
b_{i},a>0,\,b_{i}>0,1\leq i\leq n\\}.$

######  Theorem 19.

A function $\bar{\phi}$ is a solution of the IVP (8.1) on an interval $I$ if
and only if it is a solution of the integral equation

$y_{i}(t)=^{i}y_{0}+\int_{0}^{t}\frac{(t-x)^{\alpha_{i}-1}}{\Gamma(\alpha_{i})}f(x,\bar{y}(x),\bar{y}(qx))dx\quad\text{on}\quad
I,$ (8.2)

where
$\bar{\phi}_{m}=\left({}^{1}\phi_{m},^{2}\phi_{m},\cdots,^{n}\phi_{m}\right)$

######  Theorem 20.

Let $||f||=M$ on rectangle R. The successive approximation

${}^{i}\phi_{0}(t)$ $\displaystyle=$ ${}^{i}y_{0},\quad i=0,1,2,\cdots.$
${}^{i}\phi_{k+1}(t)$ $\displaystyle=$ $\displaystyle
y_{0}+\int_{0}^{t}\frac{(t-x)^{\alpha_{i}-1}}{\Gamma(\alpha_{i})}f(x,{\bar{\phi}_{k}}(x),{\bar{\phi}_{k}}(qx))dx.\quad
k=0,1,2,\cdots.$ (8.3)

exist and continuous on the interval $I=[-\zeta,\zeta]$, where

$\zeta=\text{min}\left\\{a,\left(\frac{\Gamma(\alpha_{1}+1)b_{1}}{M}\right)^{\frac{1}{\alpha_{1}}},\cdots,\left(\frac{\Gamma(\alpha_{n}+1)b_{n}}{M}\right)^{\frac{1}{\alpha_{n}}}\right\\}.$

If $t$ is in interval $I$ then $\left(t,\bar{y}_{m}(t),\bar{y}_{m}(qt)\right)$
is in rectangle R and $||\bar{y}_{m}(t)-\bar{y}(0)||\leq
M\sum_{i=1}^{m}\frac{|t|^{\alpha_{i}}}{\Gamma(\alpha_{i}+1)}$,
$||\bar{y}_{m}(qt)-\bar{y}(0)||\leq
M\sum_{i=1}^{m}\frac{|t|^{\alpha_{i}}}{\Gamma(\alpha_{i}+1)}\,\forall m$.

######  Theorem 21.

Let $f$ be a continuous function defined on the rectangle

$R=\\{|t|\leq a,|y_{i}(t)-^{i}y_{0}|\leq b_{i},|y_{i}(qt)-^{i}y_{0}|\leq
b_{i},a>0,\,b_{i}>0,1\leq i\leq n\\}.$

Suppose $f$ satisfies Lipschitz condition in second and third variable with
Lipschitz constants $L_{1}$ and $L_{2}$ such that
$|f(t,\bar{y}(t),\bar{y}(qt))-f(t,\bar{y}(t),\bar{y}(qt))|\leq
L_{1}|\bar{y}_{1}(t)-\bar{y}_{2}(t)|+L_{1}|\bar{y}_{1}(qt)-\bar{y}_{2}(qt)|.$
Then the successive approximations (8.3) converges on the interval
$I=[-\zeta,\zeta]$, where
$\zeta=\text{min}\left\\{a,\left(\frac{\Gamma(\alpha_{1}+1)b_{1}}{M}\right)^{\frac{1}{\alpha_{1}}},\cdots,\left(\frac{\Gamma(\alpha_{n}+1)b_{n}}{M}\right)^{\frac{1}{\alpha_{n}}}\right\\}$
to a solution of the $\phi$ of the IVP (8.1) on $I$.

###  8.1 System of Fractional Order Pantograph Equation

Consider the system of fractional order pantograph equation

$D^{\alpha}y(t)=Ay(t)+By(qt),\quad y(0)=y_{0},\quad 0<\alpha\leq 1$ (8.4)

where $0<q<1$, $A=\left(a_{ij}\right)_{n\times n}$,
$B=\left(b_{ij}\right)_{n\times n}$ and $y=[y_{1},y_{2},\cdots,y_{n}]^{T}$ The
solution of (6.4) using successive approximation is

$\displaystyle y(t)$ $\displaystyle=$
$\displaystyle\left[\sum_{k=0}^{\infty}\prod_{j=1}^{k}(A+Bq^{-(k-j)\alpha})\frac{t^{k\alpha}}{\Gamma(k\alpha+1)}\right]\lambda.$
(8.5)

######  Theorem 22.

For $0<q<1$, the power series (8.5) is convergent for $t\in{R}$.

###  8.2  System of Fractional Order Ambartsumian Equations

In this section, we generalize the Ambartsumian equation (3.1) to the system
of fractional order Ambartsumian equations [24] as:

$D^{\alpha}y(t)=-Iy(t)+By\left(\frac{t}{q}\right),\quad y(0)=\lambda,\quad
0<\alpha\leq 1,$ (8.6)

where $D^{\alpha}$ denotes Caputo fractional derivative, $I$ is the identity
matrix of order $n$, $1<q$,
$y=$$\begin{bmatrix}y_{1}\\\ y_{2}\\\ \vdots\\\ y_{n}\end{bmatrix}$,
$\lambda=$ $\begin{bmatrix}\lambda_{1}\\\ \lambda_{2}\\\ \vdots\\\
\lambda_{n}\end{bmatrix}$ and $B=$$\begin{bmatrix}\frac{1}{q}&0&0&\cdots&0\\\
0&\frac{1}{q}&0&\cdots&0\\\ \vdots&\vdots&\vdots&\ddots&0\\\
0&0&0&\cdots&\frac{1}{q}\end{bmatrix}_{n\times n}.$

Applying SAM to the initial value problem (8.6), we have

$y(t)=y(0)-IJ^{\alpha}y(t)+BJ^{\alpha}y\left(\frac{t}{q}\right).$ (8.7)

Suppose $\phi_{k}(t)$ be the $k$th approximate solution, where the initial
approximate solution is taken as

$\phi_{0}(t)=\lambda.$ (8.8)

For $k\geq 1$, the recurrent formula as below:

$\phi_{k}(t)=\lambda-
IJ^{\alpha}\phi_{k-1}(t)+BJ^{\alpha}\phi_{k-1}\left(\frac{t}{q}\right).$ (8.9)

From the recurrent formula, we have

$\displaystyle\phi_{1}(t)$ $\displaystyle=$ $\displaystyle\lambda-
IJ^{\alpha}\phi_{0}(t)+BJ^{\alpha}\phi_{0}\left(\frac{t}{q}\right)$
$\displaystyle=$ $\displaystyle\lambda-I\frac{\lambda
t^{\alpha}}{\Gamma(\alpha+1)}+B\frac{\lambda t^{\alpha}}{\Gamma(\alpha+1)}$
$\displaystyle=$
$\displaystyle\left(I+(-I+B)\frac{t^{\alpha}}{\Gamma(\alpha+1)}\right)\lambda,$
$\displaystyle\phi_{2}(t)$ $\displaystyle=$ $\displaystyle\lambda-
IJ^{\alpha}\phi_{1}(t)+BJ^{\alpha}\phi_{1}\left(\frac{t}{q}\right)$
$\displaystyle=$ $\displaystyle\lambda-
IJ^{\alpha}\left[\left(I+(-I+B)\frac{t^{\alpha}}{\Gamma(\alpha+1)}\right)\lambda\right]+BJ^{\alpha}\left(I+(-I+B)\frac{q^{-\alpha}t^{\alpha}}{\Gamma(\alpha+1)}\right)\lambda$
$\displaystyle=$ $\displaystyle\lambda-I\left[\frac{\lambda
t^{\alpha}}{\Gamma(\alpha+1)}+(-I+B)\frac{\lambda
t^{2\alpha}}{\Gamma(2\alpha+1)}\right]+B\left[\frac{\lambda
t^{\alpha}}{\Gamma(\alpha+1)}+(-I+B)\frac{\lambda
q^{-\alpha}t^{2\alpha}}{\Gamma(2\alpha+1)}\right]$ $\displaystyle=$
$\displaystyle\left[I+(-I+B)\frac{t^{\alpha}}{\Gamma(\alpha+1)}+(-I+Bq^{-\alpha})(-I+B)\frac{t^{2\alpha}}{\Gamma(2\alpha+1)}\right]\lambda,$

$\displaystyle\phi_{3}(t)$ $\displaystyle=$
$\displaystyle\left[I+(-I+B)\frac{t^{\alpha}}{\Gamma(\alpha+1)}+(-I+Bq^{-\alpha})(-I+B)\frac{t^{2\alpha}}{\Gamma(2\alpha+1)}\right.$
$\displaystyle\left.+(-I+Bq^{-2\alpha})(-I+Bq^{-\alpha})(-I+B)\frac{t^{3\alpha}}{\Gamma(3\alpha+1)}\right]\lambda,$
$\displaystyle\cdots,$ $\displaystyle\phi_{k}(t)$ $\displaystyle=$
$\displaystyle\left[I+\sum_{m=1}^{k}\prod_{j=1}^{m}(-I+Bq^{-(m-j)\alpha})\frac{t^{m\alpha}}{\Gamma(m\alpha+1)}\right]\lambda$

As $k\rightarrow\infty$,$\quad\phi_{k}(t)\rightarrow y(t)$

$\displaystyle y(t)$ $\displaystyle=$
$\displaystyle\left[I+\sum_{k=1}^{\infty}\prod_{j=1}^{k}(-I+Bq^{-(k-j)\alpha})\frac{t^{k\alpha}}{\Gamma(k\alpha+1)}\right]\lambda.$

If we set $\prod_{j=1}^{k}(-I+Bq^{(k-j)\alpha})=I$, for $k=0$, then

$\displaystyle y(t)$ $\displaystyle=$
$\displaystyle\left[\sum_{k=0}^{\infty}\prod_{j=1}^{k}(-I+Bq^{-(k-j)\alpha})\frac{t^{k\alpha}}{\Gamma(k\alpha+1)}\right]\lambda.$
(8.10)

######  Theorem 23.

For $q>1$, the power series

$\displaystyle y(t)$ $\displaystyle=$
$\displaystyle\left[\sum_{k=0}^{\infty}\prod_{j=1}^{k}(-I+Bq^{-(k-j)\alpha})\frac{t^{k\alpha}}{\Gamma(k\alpha+1)}\right]\lambda$

is convergent for $t\in{R}$.

Proof:  Result follows immediately by ratio test [25].

##  9 Illustrative Examples

Example 1. Consider the non-linear differential equations with proportional
delay [26, 27, 28, 29]

$\frac{dy(t)}{dt}=1-2y^{2}\left(\frac{t}{2}\right),\quad y(0)=0.$ (9.1)

The corresponding integral equation is

$y(t)=\int_{0}^{t}\left(1-2u^{2}\left(\frac{t}{2}\right)\right)dx.$ (9.2)

By using successive approximation method (4.1), we obtain

$\displaystyle\phi_{0}(t)$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle\phi_{1}(t)$ $\displaystyle=$ $\displaystyle t,$
$\displaystyle\phi_{2}(t)$ $\displaystyle=$ $\displaystyle t-\frac{t^{3}}{6},$
$\displaystyle\phi_{3}(t)$ $\displaystyle=$ $\displaystyle
t-\frac{t^{3}}{6}+\frac{t^{5}}{120}-\frac{t^{7}}{8064},$
$\displaystyle\phi_{4}(t)$ $\displaystyle=$ $\displaystyle
t-\frac{t^{3}}{6}+\frac{t^{5}}{120}-\frac{t^{7}}{5040}+\frac{61t^{9}}{23224320}-\frac{67t^{11}}{3406233600}+\frac{t^{13}}{12881756160}-\frac{t^{15}}{7990652436480},$
$\displaystyle\phi_{5}(t)$ $\displaystyle=$ $\displaystyle
t-\frac{t^{3}}{6}+\frac{t^{5}}{120}-\frac{t^{7}}{5040}+\frac{61t^{9}}{23224320}-\cdots-\frac{t^{31}}{1062664199886151693758358595882188800},$

and so on.
The exact solution of Eq.(9.1) is $y(t)=\sin t$.
The 5-term solutions of Eq.(9.1) using Adomian decomposition method (ADM)
[26], variational iteration method (VIM) [27], homotopy analysis method (HAM)
[28], optimal homotopy asymptotic method (OHAM) [29] are same and is given by

$\displaystyle y(t)$ $\displaystyle=$ $\displaystyle
t-\frac{t^{3}}{6}+\frac{t^{5}}{120}-\frac{t^{7}}{5040}+\frac{t^{9}}{362880}-\frac{t^{11}}{39916800}+\frac{t^{13}}{6227020800}$
(9.3)
$\displaystyle-\frac{t^{15}}{1307674368000}+\frac{t^{17}}{355687428096000}.$

The 4-term OHAM solution [29] of Eq.(9.1) is

$\displaystyle y(t)$ $\displaystyle=$ $\displaystyle
t-0.166665t^{3}+0.00832857t^{5}-0.000192105t^{7}.$ (9.4)

We compare $5^{th}$ approximation solution (SAM) and 5-term solutions ( ADM,
VIM, HAM) with exact solution in Fig.(1) and $4^{th}$ approximation solution
(SAM) with 4-term solution (OHAM) in Fig.(2). The absolute errors in
computation are shown in Figs.(3)-(4). It can be observed that SAM solution is
better than the solution obtained by using other methods.

---
Fig.1:Comparison of SAM, ADM/VIM/HAM solutions with exact solution of
Eq.(9.1).

---
Fig.2:Comparison of SAM, OHAM solutions with exact solution of Eq.(9.1).

Remark: The ADM/VIM/HAM and OHAM solutions given in [26, 27, 28, 29] are
considered only for the interval $[0,1]$ . Here we have successfully extended
the solution using SAM in intervals $[0,8]$.

---
Fig.3:Comparison of absolute errors in SAM and ADM/VIM/HAM solutions.

---
Fig.4:Comparison of absolute errors in SAM and OHAM solutions.

##  10 Conclusions

In this paper, we solved non-linear differential equations with proportional
delay using the successive approximation method (SAM). The existence,
uniqueness, and stability theorems for differential equations with
proportional delay are presented. The convergence results are derived by using
the Lipschitz condition. A generalization of fractional order and a system of
fractional order cases are also presented. The series solution of the
pantograph equation and Ambersumian equation is obtained using SAM. Finally,
we illustrated the effectiveness of SAM through an example.

##  11 References

## References

  * [1] SACHIN Bhalekar and VARSHA Daftardar-Gejji.  A predictor-corrector scheme for solving nonlinear delay differential equations of fractional order.  Journal of Fractional Calculus and Applications, 1(5):1–9, 2011\.
  * [2] Varsha Daftardar-Gejji, Yogita Sukale, and Sachin Bhalekar.  Solving fractional delay differential equations: a new approach.  Fractional Calculus and Applied Analysis, 18(2):400–418, 2015.
  * [3] Muhammad Asad Iqbal, Umer Saeed, and Syed Tauseef Mohyud-Din.  Modified laguerre wavelets method for delay differential equations of fractional-order.  Egyptian journal of basic and applied sciences, 2(1):50–54, 2015\.
  * [4] Khalid K Ali, Mohammad A Abd El Salam, and Emad M Mohamed.  Chebyshev operational matrix for solving fractional order delay-differential equations using spectral collocation method.  Arab Journal of Basic and Applied Sciences, 26(1):342–353, 2019\.
  * [5] Sedigheh Sabermahani, Yadollah Ordokhani, and Sohrab-Ali Yousefi.  Fractional-order fibonacci-hybrid functions approach for solving fractional delay differential equations.  Engineering with Computers, 36(2):795–806, 2020.
  * [6] Sachin B Bhalekar.  Stability analysis of a class of fractional delay differential equations.  Pramana, 81(2):215–224, 2013.
  * [7] Sachin Bhalekar.  Stability and bifurcation analysis of a generalized scalar delay differential equation.  Chaos: An Interdisciplinary Journal of Nonlinear Science, 26(8):084306, 2016.
  * [8] Eva Kaslik and Seenith Sivasundaram.  Analytical and numerical methods for the stability analysis of linear fractional delay differential equations.  Journal of Computational and Applied Mathematics, 236(16):4027–4041, 2012.
  * [9] Sachin Bhalekar.  Analysis of 2-term fractional-order delay differential equations.  In Fractional Calculus and Fractional Differential Equations, pages 59–75. Springer, 2019.
  * [10] Sachin Bhalekar, Varsha Daftardar-Gejji, Dumitru Baleanu, and Richard Magin.  Fractional bloch equation with delay.  Computers & Mathematics with Applications, 61(5):1355–1365, 2011\.
  * [11] FA Rihan and G Velmurugan.  Dynamics of fractional-order delay differential model for tumor-immune system.  Chaos, Solitons & Fractals, 132:109592, 2020.
  * [12] V Preethi Latha, Fathalla A Rihan, R Rakkiyappan, and G Velmurugan.  A fractional-order delay differential model for ebola infection and cd8+ t-cells response: stability analysis and hopf bifurcation.  International Journal of Biomathematics, 10(08):1750111, 2017.
  * [13] Fathalla A Rihan.  Fractional-order delay differential equations with predator-prey systems.  In Delay Differential Equations and Applications to Biology, pages 211–232. Springer, 2021.
  * [14] John Richard Ockendon and Alan B Tayler.  The dynamics of a current collection system for an electric locomotive.  Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 322(1551):447–468, 1971.
  * [15] Martin Buhmann and Arieh Iserles.  Stability of the discretized pantograph differential equation.  Mathematics of Computation, 60(202):575–589, 1993.
  * [16] Sachin Bhalekar and Jayvant Patade.  Series solution of the pantograph equation and its properties.  Fractal and Fractional, 1(1):16, 2017.
  * [17] Sachin Bhalekar and Jayvant Patade.  Analytical solutions of nonlinear equations with proportional delays.  Appl. Comput. Math, 15(3):331–345, 2016.
  * [18] Jayvant Patade and Sachin Bhalekar.  On analytical solution of ambartsumian equation.  National Academy Science Letters, 40:291–293, 2017.
  * [19] Jayvant Patade and Sachin Bhalekar.  Analytical solution of pantograph equation with incommensurate delay.  Physical Sciences Reviews, 2(9), 2017.
  * [20] Earl A Coddington.  An introduction to ordinary differential equations.  Courier Corporation, 2012.
  * [21] Anatoliĭ Kilbas.  Theory and applications of fractional differential equations elsevier, 2006.
  * [22] Weihua Deng, Yujiang Wu, and Changpin Li.  Stability analysis of differential equations with time-dependent delay.  International Journal of Bifurcation and Chaos, 16(02):465–472, 2006.
  * [23] VA Ambartsumian.  On the fluctuation of the brightness of the milky way.  In Dokl. Akad. Nauk SSSR, volume 44, pages 244–247, 1944.
  * [24] Jayvant Patade.  Series solution of system of fractional order ambartsumian equations: Application in astronomy.  arXiv preprint arXiv:2008.04904, 2020.
  * [25] Tom M Apostol and CM Ablow.  Mathematical analysis.  Physics Today, 11(7):32, 1958.
  * [26] David J Evans and KR Raslan.  The adomian decomposition method for solving delay differential equation.  International Journal of Computer Mathematics, 82(1):49–54, 2005\.
  * [27] Yulita Molliq Rangkuti and MSM Noorani.  The exact solution of delay differential equations using coupling variational iteration with taylor series and small term.  Bulletin of Mathematics, 4(01):1–15, 2012.
  * [28] AK Alomari, Mohd Salmi Md Noorani, and R Nazar.  Solution of delay differential equation by means of homotopy analysis method.  Acta Applicandae Mathematicae, 108:395–412, 2009.
  * [29] N Ratib Anakira, AK Alomari, and Ishak Hashim.  Optimal homotopy asymptotic method for solving delay differential equations.  Mathematical Problems in Engineering, 2013, 2013.