# Application of Natural Transform Method to Fractional Pantograph Delay Differential Equations

Authors: M. Valizadeh; Y. Mahmoudi; F. Dastmalchi Saei
Journal: Journal of Mathematics
 (2019)

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## Abstract

In this paper, a new method based on combination of the natural transform method (NTM), Adomian decomposition method (ADM), and coefficient perturbation method (CPM) which is called “perturbed decomposition natural transform method” (PDNTM) is implemented for solving fractional pantograph delay differential equations with nonconstant coefficients. The fractional derivative is regarded in Caputo sense. Numerical evaluations are included to demonstrate the validity and applicability of this technique.


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## Body

## 1. Introduction

Delay differential equations (DDEs) occur when the rate of change of process is specified by its state at a certain past state known as its history. These types of differential equations occur in traffic models, control systems, population dynamics, and many natural phenomena [1–3]. The expression of delay in mathematical modeling of real-life phenomena leads to a precise and acceptable description of the dynamics of the model. DDEs are complex in nature because the history of the system over the delayed interval is given as an initial condition. Due to this reason, these equations are difficult to solve analytically; hence, a numerical method is required. Pantograph delay equations are one of the well-known functional differential equations with proportional delay and often appear in many scientific models such as mechanics and electrodynamics [4, 5]. In 1851, it was the first time that a device named “pantograph” was used in construction of the electric locomotive after which this name was originated from that time. Pantograph was modeled mathematically in 1971 (see [6] for more details). Thereafter, pantograph delay equation was studied by many authors and solved by several numerical methods. The most important of them are collocation method [7], spline method [8], Runge–Kutta method [9], homotopy perturbation method [10], and Adomian decomposition method [11]. In this paper, we intend to apply the natural transform method to solve fractional delay differential equations with variable coefficients. To solve differential and integral equations, several integral transforms such as Fourier, Laplace, and Sumudu are used [12–16]. The natural transform method is a new integral transform was introduced by Khan and Khan [17], and its properties were described. Belgacem used the natural transform method to solve Bessel’s differential equation, Maxwell’s equation, and nonlinear Klein–Gordon equations (see [18, 19] and its references). A table for some natural transformation properties on certain functions was presented by Khan and Khan in [17]. Moreover, this method is applied to find the solution of diffusion equations [20]. Finally, they proved that the natural transform method converges to the Laplace and Sumudu transforms.In this paper, a novel technique is applied to find an approximate solution for the fractional pantograph delay differential equation of the following form:(1)D∗0αxt=ft+∑i=1naitxqit,0<qi≤1,x0=c,where 0<α<1, ft and ait are the given analytic functions, and D∗0α denotes the Caputo fractional differential operator. The proposed method is a combination of natural transform method (NTM), Adomian decomposition method (ADM) [21], and coefficient perturbation method (CPM) [22], which is called “perturbed decomposition natural transform method.” Unlike existing analytical and numerical methods, this method gives an approximate and/or exact solution for problem (1) by a simple calculation and in an elegant way.The structure of this paper is as follows: In Section2, a brief description of the fractional calculus, natural transform method, and Adomian decomposition method are provided. The existence of unique solution for problem (1) is investigated in Section 3. In Section 4, we introduce our new method (called briefly, PDNTM) for obtaining the approximate solution for problem (1). Numerical examples illustrate efficiency and applicability of the proposed method in Section 5, and conclusions are drawn in Section 6.


## 2. Preliminaries

In this section, we recall some preliminary results which will be needed throughout the paper.


### 2.1. Fractional Calculus

There are some kinds of definitions for fractional derivatives and integrals, such as Riemann–Liouville integral [23], which are described as follows.Forf∈L1a,b, the Riemann–Liouville fractional integral of order α is defined as(2)Jaαfx=1Γα∫axx−tα−1ftdt,α≥0.Forα=0, set Ja0≔I, the identity operator. Let fx=x−aβ for some β>−1 and α>0. Then,(3)Jaαfx=Γβ+1Γα+β+1x−aα+β.Letm=α. The operator Daα defined by(4)Daαfx=DmJam−αfx,is called the Riemann–Liouville fractional differential operator of order α. For α=0, set Da0≔I, the identity operator. The Caputo differential operator of order n is defined by(5)D∗aαfx=Daαfx−Tm−1fx;a,whenever Daαfx−Tm−1fx;a exists, where Tm−1fx;a denotes the Taylor polynomial of degree m−1 for the function f around the point a. In the case m=0, define Tm−1fx;a≔0. Under the above conditions, it is easy to show that(6)D∗aαfx=Jam−αDmfx.More details can be known by referring to [23, 24].


### 2.2. Natural Transform Method (NTM)

In this section, we review some properties of the atural transform method that will be used in later sections. For the functionft defined in 0,∞, piecewise function continues in every finite interval 0≤t≤K and f∈A where(7)A=ft:∃M,τ1,τ2>0,ft<Met/τj,t∈−1j×0,∞.The natural transform of ft is defined as follows:(8)Ru,s=Nf=∫0∞e−stfutdt.With the above conditions, the right-hand side integral in (39) is convergent. By taking u=1 in (39), the Laplace transform of ft,(9)Fs=Lft=∫0∞e−stftdt,s>0,and for s=1, the Sumudu transform of ft,(10)Su=Sft=∫0∞e−tfutdt,t>0,will be obtained. The inverse natural transform of Ru,s is defined in [19] by(11)ft=N−1Ru,s=12πi∫c−i∞c+i∞est/uRu,sds.We recall some useful properties of the natural transform method from [19] in Table 1.Table 1
Natural transforms of some functions.
Functional form
Natural transform form
(1)
af(t) + bg(t)
a
N
f
t
+
b
N
g
t
(2)
1
1
/
s
(3)
t
α
Γ
α
+
1
u
α
/
s
α
+
1
(4)
∑
i
=
0
∞
a
i
t
i
α
∑
i
=
0
∞
Γ
i
α
+
1
u
i
α
/
s
i
α
+
1
(5)
D
∗
0
α
f
t
s
α
/
u
α
R
s
,
u
−
∑
k
=
0
n
−
1
s
α
−
k
+
1
/
u
α
−
k
f
k
0
,
n
−
1
<
α
≤
n


### 2.3. Adomian Decomposition Method (ADM)

Consider the general form of a differential equation as(12)Lyt+Syt+Ryt=gt,where Ly is the linear invertible part, Ny corresponds to the nonlinear term, and Ry is the remaining part. Solving Ly from (12), we have(13)Lyt=gt−Nyt−Ryt.AsL is invertible,(14)yt=y0+L−1gt−L−1Nyt−L−1Ryt.The Adomian decomposition method (ADM) [21] is a powerful tool for solving both linear and nonlinear functional equations. This method decomposes the solution yx into a rapidly convergent series of solution components and then decomposes the analytic nonlinearity Ny into the series of the Adomian polynomials:(15)yt=∑i=0∞yit,Nyt=∑i=0∞Ait,where Ai=Aiy0t,y1t,…,yit are the well-known Adomian polynomials as follows:(16)An=1n!dndtnN∑i=0nλiyitλ=0,n=0,1,2,….Substituting (15) in (14), we have(17)∑i=0∞yit=y0+L−1gt−L−1N∑i=0∞Ait−L−1R∑i=0∞yit.Consequently, we write(18)y0t=y0+L−1gt,y1t=−L−1Ny0t−L−1Ry0t,y2t=−L−1Ny1t−L−1Ry1t,⋮The convergence analysis of the Adomian decomposition method was studied by Cherruault in [25, 26].


## 2.1. Fractional Calculus

There are some kinds of definitions for fractional derivatives and integrals, such as Riemann–Liouville integral [23], which are described as follows.Forf∈L1a,b, the Riemann–Liouville fractional integral of order α is defined as(2)Jaαfx=1Γα∫axx−tα−1ftdt,α≥0.Forα=0, set Ja0≔I, the identity operator. Let fx=x−aβ for some β>−1 and α>0. Then,(3)Jaαfx=Γβ+1Γα+β+1x−aα+β.Letm=α. The operator Daα defined by(4)Daαfx=DmJam−αfx,is called the Riemann–Liouville fractional differential operator of order α. For α=0, set Da0≔I, the identity operator. The Caputo differential operator of order n is defined by(5)D∗aαfx=Daαfx−Tm−1fx;a,whenever Daαfx−Tm−1fx;a exists, where Tm−1fx;a denotes the Taylor polynomial of degree m−1 for the function f around the point a. In the case m=0, define Tm−1fx;a≔0. Under the above conditions, it is easy to show that(6)D∗aαfx=Jam−αDmfx.More details can be known by referring to [23, 24].


## 2.2. Natural Transform Method (NTM)

In this section, we review some properties of the atural transform method that will be used in later sections. For the functionft defined in 0,∞, piecewise function continues in every finite interval 0≤t≤K and f∈A where(7)A=ft:∃M,τ1,τ2>0,ft<Met/τj,t∈−1j×0,∞.The natural transform of ft is defined as follows:(8)Ru,s=Nf=∫0∞e−stfutdt.With the above conditions, the right-hand side integral in (39) is convergent. By taking u=1 in (39), the Laplace transform of ft,(9)Fs=Lft=∫0∞e−stftdt,s>0,and for s=1, the Sumudu transform of ft,(10)Su=Sft=∫0∞e−tfutdt,t>0,will be obtained. The inverse natural transform of Ru,s is defined in [19] by(11)ft=N−1Ru,s=12πi∫c−i∞c+i∞est/uRu,sds.We recall some useful properties of the natural transform method from [19] in Table 1.Table 1
Natural transforms of some functions.
Functional form
Natural transform form
(1)
af(t) + bg(t)
a
N
f
t
+
b
N
g
t
(2)
1
1
/
s
(3)
t
α
Γ
α
+
1
u
α
/
s
α
+
1
(4)
∑
i
=
0
∞
a
i
t
i
α
∑
i
=
0
∞
Γ
i
α
+
1
u
i
α
/
s
i
α
+
1
(5)
D
∗
0
α
f
t
s
α
/
u
α
R
s
,
u
−
∑
k
=
0
n
−
1
s
α
−
k
+
1
/
u
α
−
k
f
k
0
,
n
−
1
<
α
≤
n


## 2.3. Adomian Decomposition Method (ADM)

Consider the general form of a differential equation as(12)Lyt+Syt+Ryt=gt,where Ly is the linear invertible part, Ny corresponds to the nonlinear term, and Ry is the remaining part. Solving Ly from (12), we have(13)Lyt=gt−Nyt−Ryt.AsL is invertible,(14)yt=y0+L−1gt−L−1Nyt−L−1Ryt.The Adomian decomposition method (ADM) [21] is a powerful tool for solving both linear and nonlinear functional equations. This method decomposes the solution yx into a rapidly convergent series of solution components and then decomposes the analytic nonlinearity Ny into the series of the Adomian polynomials:(15)yt=∑i=0∞yit,Nyt=∑i=0∞Ait,where Ai=Aiy0t,y1t,…,yit are the well-known Adomian polynomials as follows:(16)An=1n!dndtnN∑i=0nλiyitλ=0,n=0,1,2,….Substituting (15) in (14), we have(17)∑i=0∞yit=y0+L−1gt−L−1N∑i=0∞Ait−L−1R∑i=0∞yit.Consequently, we write(18)y0t=y0+L−1gt,y1t=−L−1Ny0t−L−1Ry0t,y2t=−L−1Ny1t−L−1Ry1t,⋮The convergence analysis of the Adomian decomposition method was studied by Cherruault in [25, 26].


## 3. Existence and Uniqueness of Solution

In this section, we prove existence and uniqueness of the solution for the fractional delay differential equation (1). This study is based on the Banach contraction principle.Theorem 1.
Assumeft and ait are the given analytic functions in (1) and Mi≔max0≤t≤Tait. If(19)∑i=1nTαMiΓα+1<1,then problem (1) has a unique solution xt∈C10,T.Proof.
The equivalent integral equation representation of problem (1) is of the following form (see [24]):(20)xt=c+J0αft+∑i=1nJ0αaitxqit.
Based on the Banach contraction principle, it is enough to prove that the operatorA defined by(21)Axt≔c+J0αft+∑i=1nJ0αaitxqit,has a unique fixed point. It is clear that the operator A maps C0,T into itself and that(22)Axt−Ax˜t≤∑i=1nJ0αaitxqit−x˜qit≤∑i=1nMiJ0αxqit−x˜qit≤∑i=1nTαMiΓα+1xt−x^t∞,which implies, under our assumption, that A is a contraction mapping. Then, the Banach contraction principle implies that A has a unique fixed point xt.


## 4. Main Results

Let us approximate the coefficientsft and ait,i=1,…,n, in equation (1) by polynomials of fractional order α as follows:(23)ft≃fρt≔∑l=0ρfltlα,ait≃ai,σit≔∑l=0σial,itlα,i=1,…,n.Then, problem (1) can be written in perturbed form as(24)D∗0αx^t=∑l=0ρfltlα+∑i=1n∑l=0σial,itlαx^qit,x^0=x0=c,which is called “perturbed problem” with the exact solution x^t that is an approximation for the exact solution xt of the main problem (1). This is a first step of our new method that converts the main problem (1) to new one with polynomial coefficients. This strategy will simplify the calculation in the implementation of the proposed method. Similar ideas have been used in [27, 28] for solving boundary value problem and eigenvalue problems.Theorem 2.
Letn=1 and(25)f−fρ∞≤ϵ,a1−a1,σ1∞≤ϵ,for some constant ϵ in (23). Then, we have xt−x^t∞=Oϵ, i.e.,(26)xt−x^t∞⟶0,ϵ⟶0.Proof.
It can be easily proved thatxt∈C0,T is a solution of equation (1) if and only if xt∈C0,T is a solution for the integral equation(27)xt=c+J0αft+J0αatxqt,where J0α is the Riemann–Liouville fractional differential operator of order α (see [24]). Consequently, for perturbed problem (24), we have(28)x^t=c+J0αfρt+J0αaσtx^qt.
Subtracting equation (28) from equation (27) and some manipulation yield(29)xt−x^t=J0αft−fρt+J0αatxqt−aσtx^qt=J0αft−fρt+J0αat−aσtx^qt+J0αatxqt−x^qt.
By taking⋅∞ from both sides of equation (29), we obtain(30)xt−x^t≤J0αft−fρt∞+J0αat−aσtx^qt∞+J0αatxqt−x^qt∞.
From (25), we have(31)J0αft−fρt∞≤1Γα∫0tt−sα−1fs−fρs∞ds≤ϵΓα∫0Tt−sα−1ds=ϵTαΓα+1,J0αat−aσtx^qt∞≤x^qt∞Γα∫0tt−sα−1as−aρs∞ds≤ϵx^qt∞Γα∫0Tt−sα−1ds=ϵTαx^qt∞Γα+1,J0αatxqt−x^qt∞≤at∞xs−x^s∞qΓα∫0qtt−sqds≤at∞xs−x^s∞qΓα×qTαα=Tαat∞Γα+1xt−x^t∞.
Then, from (30) and (31), we get(32)xt−x^t∞≤ϵTαΓα+1+ϵTαx^qt∞Γα+1+Tαat∞Γα+1xt−x^t∞.
Consequently,(33)xt−x^t∞1−Tαat∞Γα+1≤ϵTα1+x^qt∞Γα+1.
After some manipulation on (33), we obtain(34)xt−x^t∞≤ϵTα1+x^qt∞Γα+1−Tαat∞.
From the upper boundϵTα1+x^qt∞/Γα+1−Tαat∞ in (34), it is seen that if ϵ⟶0, then xt−x^t∞⟶0.


### 4.1. Method of Solution

Taking the natural transform of both sides of equation (24) yields(35)ND∗0αx^t=N∑l=0ρfltlα+N∑i=1n∑l=0σial,itlαx^qit.Using the properties of the natural transform method in Table1, we get(36)sαuαRu,s−sα−1uαx^0=∑l=0ρflΓlα+1ulαslα+1+∑i=1n∑l=0σial,iNtlαx^qit.Consequently,(37)Ru,s=x^0s+∑l=0ρflΓlα+1ul+1αsl+1α+1+∑i=1n∑l=0σial,iuαsαNtlαx^qit.By taking the inverse natural transform of (37), we obtain(38)x^t=x^0+∑l=0ρflΓlα+1tl+1αΓl+1α+1+∑i=1n∑l=0σial,iN−1uαsαNtlαx^qit.Now, based on the Adomian decomposition method, we construct the solutionx^t in (38) by an infinite convergent series of functions as follows:(39)x^t=∑k=0∞x^kt.Substituting (39) in (38) yields(40)∑k=0∞x^kt=x^0+∑l=0ρflΓlα+1tl+1αΓl+1α+1+∑i=1n∑l=0σial,iN−1uαsαNtlα∑k=0∞x^kqit,in which the components x^kt will be obtained by the following recurrence relation:(41)x^k+1t=∑i=1n∑l=0σial,iN−1uαsαNtlαx^kqit,k=0,1,2,…,such that(42)x^0t=c+∑l=0ρflΓlα+1tl+1αΓl+1α+1.TheM+1-term approximation of the solution x^t is(43)ΦMt=∑k=0Mx^kt,so that(44)x^t=limM⟶∞ΦMt.Therefore, the solutionx^t comes in the form of a convergent series with easily computed components. From Theorem 2 and (23), we know that x^t⟶xt as ρ,σi⟶∞.Remark 1.
As is clear, in recurrence relation (41), the terms Ntlαx^kqit and N−1uα/sαNtlαx^kqit can be calculated easily using properties (3) in Table 1, since the term tlαx^kqit is a fractional-order polynomial for i=1,…,n and k=0,1,2,…. This is a main advantage in our proposed method for calculating the components x^kt in (43) and making our new method easy to be used and writing a computer program for it.
We summarize the steps of our proposed method (called PDNTM method) as an implementation algorithm as follows (Algorithm1).Algorithm 1:
Input Choose values for parameters ρ, σi in (23), and M in (44)
Step 1. Find the approximation of functions ft and ait in (23)
Step 2. Compute the components x^kt from recurrence relation (41) for k=0,…,M
Output Construct the approximate solution ΦMt=∑k=0Mx^kt


## 4.1. Method of Solution

Taking the natural transform of both sides of equation (24) yields(35)ND∗0αx^t=N∑l=0ρfltlα+N∑i=1n∑l=0σial,itlαx^qit.Using the properties of the natural transform method in Table1, we get(36)sαuαRu,s−sα−1uαx^0=∑l=0ρflΓlα+1ulαslα+1+∑i=1n∑l=0σial,iNtlαx^qit.Consequently,(37)Ru,s=x^0s+∑l=0ρflΓlα+1ul+1αsl+1α+1+∑i=1n∑l=0σial,iuαsαNtlαx^qit.By taking the inverse natural transform of (37), we obtain(38)x^t=x^0+∑l=0ρflΓlα+1tl+1αΓl+1α+1+∑i=1n∑l=0σial,iN−1uαsαNtlαx^qit.Now, based on the Adomian decomposition method, we construct the solutionx^t in (38) by an infinite convergent series of functions as follows:(39)x^t=∑k=0∞x^kt.Substituting (39) in (38) yields(40)∑k=0∞x^kt=x^0+∑l=0ρflΓlα+1tl+1αΓl+1α+1+∑i=1n∑l=0σial,iN−1uαsαNtlα∑k=0∞x^kqit,in which the components x^kt will be obtained by the following recurrence relation:(41)x^k+1t=∑i=1n∑l=0σial,iN−1uαsαNtlαx^kqit,k=0,1,2,…,such that(42)x^0t=c+∑l=0ρflΓlα+1tl+1αΓl+1α+1.TheM+1-term approximation of the solution x^t is(43)ΦMt=∑k=0Mx^kt,so that(44)x^t=limM⟶∞ΦMt.Therefore, the solutionx^t comes in the form of a convergent series with easily computed components. From Theorem 2 and (23), we know that x^t⟶xt as ρ,σi⟶∞.Remark 1.
As is clear, in recurrence relation (41), the terms Ntlαx^kqit and N−1uα/sαNtlαx^kqit can be calculated easily using properties (3) in Table 1, since the term tlαx^kqit is a fractional-order polynomial for i=1,…,n and k=0,1,2,…. This is a main advantage in our proposed method for calculating the components x^kt in (43) and making our new method easy to be used and writing a computer program for it.
We summarize the steps of our proposed method (called PDNTM method) as an implementation algorithm as follows (Algorithm1).Algorithm 1:
Input Choose values for parameters ρ, σi in (23), and M in (44)
Step 1. Find the approximation of functions ft and ait in (23)
Step 2. Compute the components x^kt from recurrence relation (41) for k=0,…,M
Output Construct the approximate solution ΦMt=∑k=0Mx^kt


## 5. Numerical Results

In this section, we present some examples to support the accuracy and simplicity of the proposed method. These results in this section indicate the applicability of the method and verification of the theoretical results in the previous section.Example 1.
We consider the following initial value problem:(45)D∗0αxt=1+2xt2−xt,x0=0.
The exact solution forα=1 is xt=t. According to the proposed method, by taking natural transform to both sides of equation (45), we obtain(46)sαuαRs,u−sα−1uαx0=N1+N2xt2−xt.
From initial conditionx0=0, we obtain(47)Rs,u=uαsα+1+uαsαN2xt2−xt.
Applying the inverse of natural transform in (47) yields(48)xt=tαΓα+1+N−1uαsαN2xt2−xt.
Now, based on (41), we obtain(49)∑n=0∞xnt=tαΓα+1+N−1uαsαN2∑n=0∞xnt2−∑n=0∞xnt,which gives(50)x0t=tαΓα+1,xk+1t=N−1uαsαN2xkt2−xkt,k=0,1,2,.…
Then, all of the componentsxkt can be obtained as follows:(51)x1t=N−1uαsαN2x0t2−x0t=N−112α−1−1u2αs2α+1=12α−1−1t2αΓ2α+1,x2t=12α−1−1122α−1−1t3αΓ3α+1,x3t=12α−1−1122α−1−1123α−1−1t4αΓ4α+1.
Continuing the above recurrence relation yields(52)xt=tαΓα+1+∑i=1∞∏l=1i12lα−1−1ti+1αΓi+1α+1.
For the caseα=1, the second term in (52) is equal to zero; therefore, xt=t, which is the exact solution of problem (45). For M=5, the approximate solution is(53)xt≃tαΓα+1+∑i=15∏l=1i12lα−1−1ti+1αΓi+1α+1.
In Figure1, the convergence of approximate solutions to exact solution as α⟶1 is shown.Figure 1
Approximate solution for different values ofα.Example 2.
Consider the following fractional pantograph problem (see [29]):(54)D0αxt=t2−1−56xt+4xt2+9xt3,x0=1.
In the caseα=1, the exact solution is xt=1+67/7t+1675/72t2+12175/1296t3. According to the proposed method for α=1, we obtain(55)x^0t=1+13t3−t,x^1t=−112t25t−146,x^2t=−25216t27t−219,x^3t=127751296t3,x^4t=0,x^5t=0,x^6t=0.
Then, from (43), we obtain(56)Φ3t=∑k=03x^kt=1+677t+167572t2+121751296t3,which is the exact solution. The approximate solutions for different values of α and exact solution are shown in Figure 2 for M=10. Note that as α⟶1, the numerical solutions tends to the exact solution xt=1+67/7t+1675/72t2+12175/1296t3.Figure 2
Approximate solution for different values ofα.Example 3.
We consider the following nonpolynomial coefficients problem (see [29]):(57)D0αxt=12expt2xt2+12xt,x0=1.
The exact solution forα=1 is xt=expt. Let(58)a1t=12expt2≃a1,σ1t=∑l=0σ1al,1tlα,a2t=a2,σ2t=12.
Then, the perturbed problem is as follows:(59)D0αx^t=a1,σ1txt2+a2,σ2txt,x^0=1.
The approximate solutions forα=0.75,0.85,and0.95 and exact solution for σ1=5, σ2=0, and M=5 are shown in Figure 3. Table 2 denotes the behavior of approximate solution x^10t for different values of σ1=5,10,and15 with α=1. These results are consistent with Theorem 2 which shows convergence of the solution of perturbed problem (59) to (57).Figure 3
Approximate solution for different values ofα.Table 2
The absolute error for different values ofσ1.
t
Exact. Sol
App. Sol
Abs. Err
Forσ1=5
0.0
1.0000000000
1.0000000000
0
0.1
1.1051709181
1.1051873941
1.647604
×
10
−
5
0.2
1.2214027582
1.2214234795
2.072137
×
10
−
5
0.3
1.3498588076
1.3498811112
2.230360
×
10
−
5
0.4
1.4918246976
1.4918490746
2.437696
×
10
−
5
0.5
1.6487212707
1.6487484081
2.713742
×
10
−
5
0.6
1.8221188004
1.8221489350
3.013457
×
10
−
5
0.7
2.0137527075
2.0137858850
3.317755
×
10
−
5
0.8
2.2255409285
2.2255774493
3.652080
×
10
−
5
0.9
2.4596031112
2.4596436807
4.056953
×
10
−
5
1.0
2.7182818285
2.7183270070
4.517854
×
10
−
5
Forσ1=10
0.0
1.0000000000
1.0000000000
0
0.1
1.1051709181
1.1051709163
1.783752
×
10
−
9
0.2
1.2214027582
1.2214027562
1.959785
×
10
−
9
0.3
1.3498588076
1.3498588054
2.167082
×
10
−
9
0.4
1.4918246976
1.4918246952
2.394817
×
10
−
9
0.5
1.6487212707
1.6487212680
2.646726
×
10
−
9
0.6
1.8221188004
1.8221187975
2.925254
×
10
−
9
0.7
2.0137527075
2.0137527042
3.233969
×
10
−
9
0.8
2.2255409285
2.2255409249
3.578286
×
10
−
9
0.9
2.4596031112
2.4596031072
3.969453
×
10
−
9
1.0
2.7182818285
2.7182818240
4.431174
×
10
−
9
Forσ1=15
0.0
1.0000000000
1.0000000000
0
0.1
1.1051709181
2.7182818285
1.068110
×
10
−
13
0.2
1.2214027582
1.1051709181
1.180236
×
10
−
13
0.3
1.3498588076
1.3498588076
1.303292
×
10
−
13
0.4
1.4918246976
1.4918246976
1.415607
×
10
−
13
0.5
1.6487212707
1.6487212707
1.286469
×
10
−
13
0.6
1.8221188004
1.8221188004
5.534341
×
10
−
14
0.7
2.0137527075
2.0137527075
1.086449
×
10
−
12
0.8
2.2255409285
2.2255409285
5.438117
×
10
−
12
0.9
2.4596031112
2.4596031112
2.074457
×
10
−
11
1.0
2.7182818285
2.7182818285
6.767812
×
10
−
11


## 6. Future Work

In our future work, the PDNTM method will be studied for the general class of problems such as the system of fractional pantograph delay differential equations of the form(60)D∗0αixit=fit+∑ℓ=1m∑j=1naijℓtxjqℓt,i=1,…,n,subject to the initial conditions(61)xi0=βi,i=1,…,n,where 0<αi<1 and 0<qi<1. To this end, we can approximate the coefficients fit and aijℓt by polynomials of fractional order 0<α<1 as follows:(62)fit≃∑l=0ρflitlα,aijℓt≃∑l=0σi,jali,jtlα,and extend all processes of Section 4.1.


## 7. Conclusion

In this paper, a new method based on the combination of the natural transform method, Adomian decomposition method, and coefficient perturbation method is introduced to solve the fractional pantograph delay differential equations with nonconstant coefficients. From this method, we obtain an approximate or exact solution for the problem with some simple iterative calculation that is easy in use and computer programming. The numerical results confirm the accuracy of the method.



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*Source: 3913840-2019-11-03.xml*