Efficient method for solving variable-order pantograph models

Pramana – J. Phys. (2021) 95:183 © Indian Academy of Scienceshttps://doi.org/10.1007/s12043-021-02218-6

Efficient method for solving variable-order pantograph models

HODA F AHMED1 and MARINA B MELAD2 ,∗

1Department of Mathematics, Faculty of Science, Minia University, Minia 61519, Egypt2Department of Mathematics, Faculty of Science, New Valley University, Kharga 72511, Egypt∗Corresponding author. E-mail: [email protected], [email protected]

MS received 28 April 2021; revised 10 June 2021; accepted 23 June 2021

Abstract. This paper shows how to solve variable-order pantograph-delay differential equations (VO-PDDEs)and variable-order pantograph Volterra delay integro-differential equations (VO-VDIDEs) using shifted fractionalGegenbauer operational matrices (SFGOMs) of differentiation and integration in conjunction with the spectralcollocation method. In these equations, the fractional derivatives of variable order are represented in the Caputosense. As a result of the proposed method, the considered problems are translated into an easy-to-solve systemof algebraic equations. The proposed technique’s error bound is examined. To demonstrate the utility of theproposed method, numerical test problems are introduced and compared with other numerical methods in theexisting literature.

Keywords. Fractional pantograph delay differential equations with variable-order; fractional pantograph Volterradelay integro-differential equations with variable-order; shifted fractional Gegenbauer polynomials; operationalmatrices; collocation method.

PACS Nos 02.60.−x; 02.70.Jn; 02.90.+p

1. Introduction

Delay differential equations (DDEs) are a type of dif-ferential equation (DE) with a delayed argument for thestate variable. DDEs with fractional order are known asfractional delay differential equations (FDDEs). Theyare crucial in the fields of control theory, electrical net-works, genetics and life science research [1].

In the 1960s, the British Railways aimed to improvethe speed of the electric locomotive. The pantograph,which absorbs current from an overhead cable, was asignificant invention. The pantograph’s job is to drawcurrent from an overhead wire so that the locomotivecan pass. There must be no interruptions in the currentcollection mechanism in order for the electric locomo-tive to move smoothly and at a high speed. For that, thepantograph should remain in contact with the overheadwire throughout, particularly when passing through theoverhead wire’s supports, which is a crucial passage.Ockendon and Tayler [2] investigated how an electriclocomotive’s pantograph head moves. In the process,they encountered a special DDE of the form

y′(t) = ay(t) + by(qt), t > 0,

where a, b are real constants and 0 < q < 1 forq ∈ R. This type of DDE is called the pantograph equa-tion (PE). The PEs, as a type of DDE, have been usedto model a variety of real-world phenomena, includingeconomics, biology, astrophysics, control and electro-dynamics [3]. It is a type of functional DE with aproportional delay that is unique. In the past, Kato andMcLeod [4] discussed the analytical solution of the PEsas well as their asymptotic properties. Many compu-tational methods were developed in an attempt to findan approximate solution for these types of equations.For multipantograph delay boundary value problems,Yang and Tohidi [5] used the Legendre spectral collo-cation process. For example, to obtain the numericalsolution of a multipantograph with initial conditions,Yüzbasi and Sezer [6] proposed the Legendre colloca-tion procedure, which is based on the residual correctiontechnique. Bahsi and Çevik [7] used a compound tech-nique that combined the perturbation approach withan iteration algorithm to solve pantograph DDEs. Tonumerically solve PEs, PEs with the neutral term andmultiple-delay Volterra integral equations with a widedomain, Heydari et al [8] relied on operational matricesfor the integration, product, and delay of the Chebyshev

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183 Page 2 of 20 Pramana – J. Phys. (2021) 95:183

cardinal functions. For multiorder fractional differen-tial equations with proportional delay, Ezz-Eldien et al[9] used Chebyshev polynomials. To obtain the numer-ical solution of fractional-order pantograph differentialequations (FO-PDEs) in a wide interval, Rahimkhaniet al [10] suggested fractional-order Bernoulli waveletfunctions. Nemati et al [11] used hat functions opera-tional matrices to solve the FO-PDEs and Jafari et al[12] discussed the numerical solution of a general classof pantograph-type nonlinear FIDEs with non-singularand non-local kernel based on shifted Legendre polyno-mials spectral collocation method.

Variable-order differential equations (VO-DEs), inwhich the order is a function of space and/or time, areone of the new subjects. It is a fresh approach to sci-ence and engineering [13]. The VO models have beenimpressive in their ability to accurately describe real-world phenomena as a powerful mathematical method.As most of the VO-DEs do not have exact solutions,approximate and numerical techniques must be used. Inthis direction, Kadkhoda [14] solved linear VO-DEs influid mechanics using the operational matrices (OMs) ofBernstein polynomials (BPs), Ganji et al [15] used OMsbased on the shifted Legendre polynomials for solving aclass of nonlinear integro-differential equations of VOand Jafari et al [16] suggested a numerical techniquebased on OMs of BPs for solving a class of nonlinearVO differential equations. So, the major goal of thisarticle is to find numerical solutions of the followingVO-pantograph models:

• Variable-order pantograph-delay differential equa-tions (VO-PDDEs)We consider the following general form of VO-PDDEs:∗0D

ν(t)t y(t) = F(t, y(t), y(q1t), y(q2t), . . . , y(qr t),

∗0D

β1(t)t y(p1t),

∗0D

β2(t)t y(p2t), . . . ,

∗0D

βl (t)t y(pl t)), (1)

with the initial conditions (ICs)

y(i)(0) = εi , i = 0, 1, . . . ,m − 1, (2)

where t ∈ [0, L] in which L is a positive constant,0 < qi < 1 for i = 1, 2, . . . , r is a constant; 0 <

p j < 1 and 0 < β j (t) ≤ ν(t) for j = 1, 2, . . . , lare constant and continuous function, respectively,ν(t) ∈ (m − 1,m] is a continuous function and εifor i = 0, 1, . . . ,m−1 is a real constant. Moreover,F : [0, L] × Rr+l+1 → R is a smooth function and∗0D

ν(t)t is the Caputo fractional differential operator

of order ν(t). Furthermore, it is assumed that β j (t) ∈(δ j −1, δ j ] where δ j for j = 1, 2, . . . , l is a positiveinteger constant.

• Variable-order pantograph Volterra delay integrodifferential equations (VO-VDIDEs)We consider the following general form of VO-VDIDEs:

∗0D

ν(t)t y(t) = F

(t, y(t), y(b1t), . . . , y(br t),

∫ t

0K1(t, s)y(s)ds, . . . ,

∫ t

0Kl(t, s)y(s)ds,

∫ q1t

0K1(t, s)y(s)ds, . . . ,

∫ qht

0Kh(t, s)y(s)ds

),

(3)

with the ICs

y(i)(0) = γi , i = 0, 1, . . . ,m − 1, (4)

where t ∈ [0, L] in which L is a positive constant,0 < bi < 1 for i = 1, 2, . . . , r and 0 < q j < 1for j = 1, 2, . . . , h are constants and γi for i =0, 1, . . . ,m − 1 is a real constant. Moreover, F :[0, L] × Rr+l+h+1 → R is a smooth function and∗0D

ν(t)t is the Caputo fractional differential operator

of order ν(t).

The proposed technique depends on the combinationof spectral methods and operational matrices (OMs) ofthe shifted fractional Gegenbauer polynomials (SFGPs).The spectral methods are quickly introduced due to theirhigh accuracy and exponential convergence rates. TheSFGPs have many advantages that made them a perfectchoice of our proposed algorithm. In the following welist some of these advantages:

• The Gegenbauer polynomials (GPs) depend on aparameter α > −0.5. Every choice of this parame-ter represents a new polynomial. So they are madeup of an infinite number of orthogonal polynomi-als, including the Chebyshev polynomials (CPs) ofthe first kind with the parameter α = 0, the CPs ofthe second kind with the parameter α = 1 and theshifted Legendre polynomials (LPs) with the param-eter α = 1/2. The same relations are satisfied forshifted Gegenbauer polynomials (SGPs) and SFGPs.

• Many studies have shown that the Gegenbauer poly-nomials (GPs) are very efficient in solving differentkinds of problems. Elgindy and Smith-Miles [17]used GPs to solve integral equations and integro-differential equations and certain members of theGegenbauer family of polynomials converge to solu-tions of problems faster than Chebyshev and Legen-dre polynomials for small/medium numbers of spec-tral expansion terms. Izadkhah and Saberi-Nadjafi[18] used spectral method based on GPs to study thenumerical solution to time-fractional partial differ-ential equations with variable coefficients. Ahmed

Pramana – J. Phys. (2021) 95:183 Page 3 of 20 183

[19] solved two-dimensional time-space fractionaldiffusion equation with variable coefficients by SGspectral collocation method combined with the SGGauss–Lobatto and SG Gauss–Radau quadratureformulas. Ahmed et al [20] suggested a new spec-tral collocation methodology for providing reliableapproximation solutions to the fractional coupledBurgers’ equations in one- and two-dimensionalspaces.

• The SGPs’ OM approaches have been demonstratedto be successful in many issues. El-Gindy et al[21] derived a new SGPs’ OM of fractional inte-gration for solving multiorder fractional differentialequations (FDEs) and systems of FDEs, Ahmedand Melad [22] used fractional integral OM ofSGPs to find the numerical solution of fractionaldifferential-algebraic equations, Ahmed and Melad[23] proposed the SGPs’ OM with the Rayleigh–Ritz method to solve the one- and two-dimensionalfractional optimal control problems, Ahmed [24]suggested the SGPs’ OM with the Lagrange multi-plier method to provide efficient numerical solutionsto the multidimensional fractional optimal controlproblems, Ozdemir et al [25] solved the coupledsystem of Burgers’ equations with a time-fractionalderivative using the Gegenbauer wavelet polynomi-als’ (GWPs’) OM of fractional integration with theGalerkin method and the collocation method and Sri-vastava et al [26] used the GWPs’ OM of fractionalintegration and block-pulse functions to solve theBagley–Torvik equation.

• The direct application of spectral methods withstandard orthogonal polynomials such as Legendre,Chebyshev and Jacobi with low convergence ratesis known due to the singular behaviour of fractionaldifferential and/or integral equations. As a result,the numerical solutions’ rate of convergence wouldbe unacceptable and insufficient. To solve this prob-lem, we use SFGPs (see [27]), which replace thevector t in standard GPs with tμ, μ > 0, t ∈ [0, L].Abdelkawy et al [28] illustrated the advantages ofthe fractional-order polynomials to deal with prob-lems with smooth solutions, where a highly accurateapproximate solution is obtained by any choice ofthe fractional factor μ; they also proved that thefractional-order polynomials can mitigate the lossof the order of convergence of the non-smoothnessproblems.

So, in this paper, we investigate the OM of thevariable-order derivative of the SFGPs together withthe OM of the integration with time delay and usethem with the spectral collocation method to present

an effective numerical technique. The proposed tech-nique transforms the VO-PDDEs and VO-VDIDEs intoa system of algebraic equations which is very easy tosolve with any iterative method. In this article, we usedthe Gauss quadrature collocation set of points (the zeropoints of the SFGPs)

S1 = {ti : C (α,μ)S, j (ti ) = 0, i = 0, 1, . . . , N }.

The paper is organised as follows: Preliminaries ofVO-fractional calculus and SFGPs are presented in §2.Various types of shifted fractional Gegenbauer opera-tional matrices (SFGOMs) are deduced in §3. Also theprocedures of the proposed techniques for solving theVO-PDDEs and VO-VDIDEs are discussed. Some illus-trative issues are presented in §4. At last, a conclusionis given in §5.

2. Preliminaries and definitions

2.1 Definition

For a continuously differentiable function y : [0, ∞] →R, the Caputo fractional derivative of order ν(t) isknown as [29,30]

∗0D

ν(t)t y(t)= 1

�(m − ν(t))

∫ t

0(t−ξ)m−ν(t)−1y(m)(ξ)dξ,

(5)

where ν(t) is a positive continuous bounded andm − 1 < ν(t) ≤ m.

We can derive the following formula from the abovedefinition:

I ν(t)t t k = �(k + 1)

�(k + 1 + ν(t))tk+ν(t), k ≥ 0. (6)

∗0D

ν(t)t t k =

{ �(k + 1)

�(k + 1 − ν(t))tk−ν(t),

for k ∈ N0 and k ≥ m,0, for k ∈ N0 and k < m,

(7)

where N0 = N⋃{0}.

2.2 Shifted fractional Gegenbauer polynomials andsome properties

2.2.1 Gegenbauer polynomials. We shall review somebasic knowledge about orthogonal GPs C (α)

j (x), of

degree j ∈ Z+, and related to the parameter α > −12 in

the finite domain [−1, 1]. See [19–21] for more details.

• Doha [31] implemented a convenient standardisationof the GPs C (α)

j (x) of degree j , which was aligned


with the parameter α > −12 , which was expressed

by

C (α)j (x) = j !�(α + 1

2 )

�( j + α + 12 )

P(α− 1

2 ,α− 12 )

j (x),

j = 0, 1, 2, . . . ,

or equivalently

C (α)j (1) = 1, j = 0, 1, 2, . . . ,

where P(α− 1

2 ,α− 12 )

j (x) is the Jacobi polynomial (JP)of degree j.

• As a consequence of this standardisation, the GPsnow include other polynomials, such as the first-order CPs Tj (x) at α = 0; the LPs L j (x) at α = 1

2 ;and the CPs of the second type 1

( j+1)Uj (x) at α = 1.

• The analytical form of GP is one of the most impor-tant forms, which we get from

C (α)j (x) =

j∑k=0

(−1) j−k

× j !�(α + 12 )�( j + k + 2α)

�(k + α + 12 )�( j + 2α)( j − k)!k!

(x + 1

2

)k

.

(8)

• The three-term recurrence relation is used to produceGPs

( j + 2α)C (α)j+1(x) = 2( j + α)xC (α)

j (x) − jC (α)j−1(x),

j ≥ 1, (9)

beginning with C (α)0 (x) = 1 and C (α)

1 (x) = x .• GPs have an orthogonality relation [32] with respect

to the weight function ω(α)(x) = (1−x2)α− 12 , which

is an even function, and their orthogonality relation-ship is defined by⟨C (α)i (x),C (α)

j (x)⟩

=∫ 1

−1C (α)i (x)C (α)

j (x)ω(α)(x)dx = λ(α)j δi, j ,

where

λ(α)j = ‖C (α)

j (x)‖2 = 22α−1 j !�2(α + 12 )

( j + α)�( j + 2α)

is the normalisation factor and δi, j is the Kroneckerdelta function.

2.2.2 Shifted Gegenbauer polynomials.

• Shifted Gegenbauer polynomials (SGPs) are definedin the interval [0, L]. We can get them by replacing

the variable x with z = 2xL − 1, 0 ≤ z ≤ L .

C (α)S, j (z) = C (α)

j

(2x

L− 1

).

• The analytical form of SGPs is given from

C(α)S, j (z) =

j∑k=0

(−1) j−k

× j !�(α + 12 )�( j + k + 2α)

�(k + α + 12 )�( j + 2α)( j − k)!k!Lk z

k . (10)

• This polynomial recaptures the shifted CPs of thefirst kind TS, j (z) ≡ C (0)

S, j (z), the shifted LPs

LS, j (z) ≡ C( 1

2 )

S, j (z) and the shifted CPs of the second

kind C (1)S, j (z) ≡ 1

j+1US, j (z).• SGPs have an orthogonal relationship from [23]

⟨C (α)S,i (z),C

(α)S, j (z)

⟩

=∫ L

0C (α)S,i (z)C

(α)S, j (z)ω

(α)S (z)dz = λ

(α)S, jδi, j ,

(11)

where ω(α)S (z) denotes the weight function derived

from the relationship

ω(α)S (z) = (zL − z2)α− 1

2 (12)

and

λ(α)S, j =

(L

2

)2α

λ(α)j . (13)

2.2.3 Shifted fractional Gegenbauer polynomials.

• By replacing the variable z in SGPs with tμ, μ >

0, t ∈ [0, L], we can get new polynomials which arecalled the SFGPs [27]

C (α,μ)S,i (t) = C (α)

S,i (tμ).

• The explicit analytic form of SFGPs is given as

C(α,μ)S, j (t) =

j∑k=0

(−1) j−k

× j !�(α + 12 )�( j + k + 2α)

�(k + α + 12 )�( j + 2α)( j − k)!k!Lμk

tμk . (14)

• We also have the SFGPs’ orthogonal relationship.∫ L

0C (α,μ)S,i (t)C (α,μ)

S, j (t)ω(α,μ)S (t)dt = λ

(α)S, jδi, j , (15)

where ω(α,μ)S (t) = μtμ(α+ 1

2 )−1(1− tμ)α− 12 and λ

(α)S, j

are obtained from relation (13).


• The SFGPs are made up of an infinite number oforthogonal polynomials, including the first-ordershifted fractional-order CPs T (μ)

S, j (t) ≡ C (0,μ)S, j (t),

the shifted fractional-order CPs of the second kindU (μ)

S, j (t) ≡ ( j+1)C (1,μ)S, j (t) and the shifted fractional-

order LPs L(μ)S, j (t) ≡ C

( 12 ,μ)

S, j (t).

2.3 Function approximation

LetC(α,μ)S,N = Span{C (α,μ)

S,0 (t),C (α,μ)S,1 (t), . . . ,C (α,μ)

S,N (t)}and y(t) be an arbitrary element in L2

ω(α,μ)S

[0, L]. Since

C(α,μ)S,N is a finite-dimensional vector space, y(t) has the

unique best approximation out of C(α,μ)S,N like yN (t) ∈

C(α,μ)S,N , such that

∀g(t) ∈ C(α,μ)S,N , ‖y(t) − yN (t)‖2 ≤ ‖y(t) − g(t)‖2,

where ‖y(t)‖ = √〈y(t), y(t)〉.The square integrable function y(t) ∈ [0, L] can be

approximated by SGPs as

y(t) =∞∑j=0

yS, jC(α,μ)S, j (t),

where the coefficients yS, j are obtained from

yS, j = (λ(α,μ)S, j )−1

∫ L

0y(t)ω(α,μ)

S (t)C (α,μ)S, j (t)dt,

j = 0, 1, . . . . (16)

If we use the first (N + 1) terms to approximate y(t),we can write

y(t) =N∑j=0

yS, jC(α,μ)S, j (t). (17)

The vector form of the approximation of the functiony(t) is as follows:

y(t) = Y T�(t), (18)

where Y T = [Y0, Y1, . . . , YN ] is the shifted fractionalGegenbauer coefficient vector, and

�(t) =[C (α,μ)S,0 (t),C (α,μ)

S,1 (t), . . . ,C (α,μ)S,N (t)

]T = ATN (t)

(19)

is the SFG vector, A is a lower triangular matrix of order(N + 1) × (N + 1)

A =

⎛⎜⎜⎜⎜⎝

a(α)S,0,0 0 · · · 0

a(α)S,1,0 a(α)

S,1,1 · · · 0...

.... . .

...

a(α)S,N ,0 a(α)

S,N ,1 · · · a(α)S,N ,N

⎞⎟⎟⎟⎟⎠ , (20)

where

a(α)S, j,k = (−1) j−k j !�(α + 1

2 )�( j + k + 2α)

�(k + α + 12 )�( j + 2α)( j − k)!k!Lμk

,

and

TN (t) =

⎛⎜⎜⎜⎜⎝

1tμ

t2μ

...

t Nμ

⎞⎟⎟⎟⎟⎠ .

2.4 Convergence analysis

Theorem 2.1. Suppose that the function Dkμy(t) ∈C[0, 1], μ ∈ [0, 1] for k = 0, 1, . . . , N −1, (α +2N +52 )μ > 0. If yN (t) is the best approximation to y(t) from

C(α,μ)S,N , then the error bound is presented as follows:

‖y(t) − yN (t)‖2 ≤ Hμ

�((N + 1)μ + 1)

×√

�(α + 12 )�(α + 2N + 5

2 )

�(2α + 2N + 3), (21)

where ‖D(N+1)μy(t)‖ ≤ Hμ, t ∈ [0, 1].

Proof. We consider the generalised Taylor formula [33]

y(t) =N∑i=0

t iμ

�(iμ + 1)Diμy(0)

+ D(N+1)μy(ξ)

�((N + 1)μ + 1)t (N+1)μ,

with ξ ∈ [0, t], ∀t ∈ [0, 1]. Let

PN (t) =N∑i=0

t iμ

�(iμ + 1)Diμy(0).

Then,

|y(t) − PN (t)| =∣∣∣∣ D(N+1)μy(ξ)

�((N + 1)μ + 1)t (N+1)μ

∣∣∣∣.


We can benefit by using yN (t), which is the best squareapproximation function of y(t)

‖y(t) − yN (t)‖22 ≤ ‖y(t) − PN (t)‖2

2

=∫ 1

0ω

(α,μ)S (t)(y(t) − PN (t))2dt

=∫ 1

0ω

(α,μ)S (t)

(D(N+1)μy(ξ)

�((N + 1)μ + 1)t (N+1)μ

)2

dt

= (D(N+1)μy(ξ))2

�2((N + 1)μ + 1)

×∫ 1

0μtμ(α+ 1

2 )−1(1 − tμ)α− 12 t2(N+1)μdt

≤ H2μ

�2((N + 1)μ + 1)

�(α + 12 )�(α + 2N + 5

2 )

�(2α + 2N + 3).

The theorem can be proved by taking the square roots.

3. Methodology

3.1 Shifted fractional Gegenbauer operationalmatrices (SFGOM)

Theorem 3.1. The derivativematrix for theVO (ν(t) >

0) in the Caputo sense of the SFG vector�(t) is gettingfrom the relation

∗0D

ν(t)t �(t) � (ν(t))�(t), (22)

where P(ν(t)) is a square matrix of order (N + 1) ×(N + 1) of the following form:

(ν(t)) = ABA−1,

where A is defined in eq. (20) and B is an (N + 1) ×(N + 1) matrix and its components bi j , 0 ≤ i, j ≤ Nare given by

bi j=⎧⎨⎩

�(μi+1)

�(μi+1−ν(t))t−ν(t), for i= j, j=n, n+1, . . . , N ,

0, o.w.

Proof. From eq. (19), we can write

∗0D

ν(t)t �(t)

= ∗0D

ν(t)t

[C (α,μ)S,0 (t),C (α,μ)

S,1 (t), . . . ,

C (α,μ)S,N (t)

]T = ∗0D

ν(t)t (ATN (t)) = A∗

0Dν(t)t TN (t).

(23)

Using eq. (7) leads to

∗0D

ν(t)t TN (t) = ∗

0Dν(t)t

⎛⎜⎜⎜⎜⎝

1tμ

t2μ

...

t Nμ

⎞⎟⎟⎟⎟⎠

=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0...

�(nμ + 1)

�(nμ + 1 − ν(t))tnμ−ν(t)

�(nμ + 2)

�(nμ + 2 − ν(t))tnμ+1−ν(t)

...�(Nμ + 1)

�(Nμ + 1 − ν(t))t Nμ−ν(t)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

= BTN (t),

(24)

where B is an (N + 1) × (N + 1) matrix and it has thefollowing form:

B=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 . . . 0

0. . . 0 . . . 0

0 0�(nμ+1)t−ν(t)

�(nμ+1−ν(t)). . . 0

......

.... . .

...

0 0 0 . . .�(Nμ+1)t−ν(t)

�(Nμ+1−ν(t))

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(25)

Equations (24) and (25) are substituted into eq. (23)to produce

∗0D

ν(t)t �(t) = ABTN (t).

Since A is invertible, then

∗0D

ν(t)t �(t) = ABA−1�(t) � (ν(t))�(t), (26)

where (ν(t)) = ABA−1 is an upper triangular matrix,and this completes the proof.

Theorem 3.2. Let φ(t) be the SFG vector, then∫ qt

0�(ξ)�T (ξ)dξ = ϒ(qt), (27)

where ϒ(qt) is called OM of integration, and it is asquarematrix of order (N+1)×(N+1) of the followingform:

ϒ(qt) = A�(qt)AT ,


where A is defined in (20) and �(qt) is an (N + 1) ×(N + 1) matrix and its elements are given by

θi j = (qt)μi+μj+1

μi + μj + 1, 0 ≤ i, j ≤ N .

Proof. From eq. (19), we can write∫ qt

0�(ξ)�T (ξ)dξ =

(∫ qt

0ATN (ξ)(ATN (ξ))T dξ

)

= A

(∫ qt

0TN (ξ)T T

N (ξ)dξ

)AT

= A

⎛⎜⎜⎜⎝

∫ qt

0

⎛⎜⎜⎜⎝

1 ξμ . . . ξμN

ξμ ξ2μ . . . ξμN+μ

......

. . ....

ξμN ξμN+μ . . . ξ2μN

⎞⎟⎟⎟⎠ dξ

⎞⎟⎟⎟⎠ AT

= A

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

(qt)(qt)μ+1

μ + 1. . .

(qt)μN+1

μN + 1(qt)μ+1

μ + 1

(qt)2μ+1

2μ + 1. . .

(qt)μN+μ+1

μN + μ + 1...

.... . .

...

(qt)μN+1

μN + 1

(qt)μN+μ+1

μN + μ + 1. . .

(qt)2μN+1

2μN + 1

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

AT

= A�(qt)AT = ϒ(qt).

Here ϒ(qt) is called the operational matrix of∫ qt0 �(ξ)�T (ξ)dξ in terms of the SFGPs.

3.2 Description of the technique

• VO pantograph-delay differential equationsThe solution of eqs (1) and (2) can be approximatedas

yN (t) =N∑j=0

yS, jC(α,μ)S, j (t) = Y T�(t), (28)

and y(qi t), i = 0, 1, . . . , r is approximated by

yN (qi t) =N∑j=0

yS, jC(α,μ)S, j (qi t) = Y T�(qi t) (29)

and y(pi t), i = 0, 1, . . . , l is approximated by

yN (pi t) =N∑j=0

yS, jC(α,μ)S, j (pi t) = Y T�(pi t). (30)

Moreover, the Caputo fractional derivative of orderν(t) of yN (t) is then estimated as

∗0D

(ν(t))t yN (t)

= ∗0D

(ν(t))t

( N∑j=0

yS, jC(α,μ)S, j (t)

)

= Y T D(ν(t))t (�(t)).

By using Theorem 3.1

∗0D

(ν(t))t yN (t) = Y T (ν(t))�(t) (31)

and∗0D

(βi (t))t yN (pi t) = Y T (βi (t))�(pi t),

i = 0, 1, . . . , l. (32)

Substitute eqs (28)–(31) into eqs (1) and (2) we get

Y T (ν(t))�(t)

= F(t, Y T�(t), Y T�(q1t), YT�(q2t),

. . . , Y T�(qr t), YT (β1(t))�(p1t),

Y T (β2(t))�(p2t),

. . . , Y T (βl (t))�(pl t)), (33)

with the initial conditions

Y T ( j)L �(0) = ε j , j = 0, 1, . . . ,m − 1. (34)

Suppose the nodes tk ∈ S1. By substituting thesenodes in eqs (33) and (34), the collocation schemecan be written as

Y T (ν(tk))�(ti )

= F(tk, YT�(tk), Y

T�(q1tk), YT�(q2tk),

. . . , Y T�(qr tk), YT (β1(tk))�(p1tk),

Y T (β2(tk)�(p2tk),

. . . , Y T (βl (tk))�(pl tk)), (35)


Y T ( j)L �(0) = ε j , j = 0, 1, . . . ,m − 1. (36)

This results in a system of (N + 1) algebraic equa-tions with the necessary SFG coefficients yS, j , j =0, 1, . . . , N which can be solved using an appropri-ate iterative process. As a result, the approximatesolution yN (t) can be determined.

• VO pantograph delay integro-differential equationsThe solution of eqs (3) and (4) can be approximatedas

yN (t) =N∑j=0

yS, jC(α,μ)S, j (t) = Y T�(t) (37)

and y(bi t), i = 0, 1, . . . , r is approximated by

yN (bi t) =N∑j=0

yS, jC(α,μ)S, j (bi t) = Y T�(bi t). (38)


Moreover, the Caputo fractional derivative of orderν(t) of yN (t) is then estimated as

∗0D

(ν(t))t yN (t) =∗

0D(ν(t))t

( N∑j=0

yS, jC(α,μ)S, j (t)

)

= Y T D(ν(t))t (�(t)).

By using Theorem 3.1

∗0D

(ν(t))t yN (t) = Y T (ν(t))�(t). (39)

Now, we approximate K j (t, s), j = 1, . . . , l andKk(t, s), k = 1, . . . , h by the SFGPs as

K j (t, s) = �T (t)κ j�(s),

Kk(t, s) = �T (t)κk�(s), (40)

where

κ j = 1

λ(α)S,i λ

(α)S,q

∫ 1

0

∫ 1

0K j (t, s)C

(α,μ)S,i (t)C (α,μ)

S,q (s)

×ω(α,μ)S (t)ω(α,μ)

S (s)ds dt,

κk = 1

λ(α)S,i λ

(α)S,q

×∫ 1

0

∫ 1

0Kk(t, s)C

(α,μ)S,i (t)C (α,μ)

S,q (s)

×ω(α,μ)S (t)ω(α,μ)

S (s)ds dt.

By approximating the Volterra parts of eq. (3), using(37), (40), (27)∫ t

0K j (t, s)y(s)ds

�∫ t

0�T (t)κ j�(s)�T (s)Yds

= �T (t)κ j

( ∫ t

0�(s)�T (s)ds

)Y

= �T (t)κ jϒ(t)Y. (41)∫ qk t

0Kk(t, s)y(s)ds

�∫ qk t

0�T (t)κk�(s)�T (s)Y ds

= �T (t)κk

(∫ qk t

0�(s)�T (s)ds

)Y

= �T (t)κkϒ(qkt)Y. (42)

Substitute eqs (37)–(39) and eqs (41) and (42) intoeqs (3) and (4), we get

Y T (ν(t))�(t)

= F(t, Y T�(t), Y T�(b1t), . . . , YT�(br t),

�T (t)κ1ϒ(t)Y, . . . , �T (t)κlϒ(t)Y,

�T (t)κ1ϒ(q1t)Y, . . . , �T (t)κhϒ(qht)Y ),

(43)


Y T ( j)L �(0) = γ j , j = 0, 1, . . . ,m − 1. (44)

Suppose the nodes tk ∈ S1. By substituting thesenodes into eqs (43) and (44), the collocation schemecan be written as

Y T (ν(tk))�(t)

= F(tk, YT�(tk), Y

T�(b1tk), . . . , YT�(br tk),

�T (tk)κ1ϒ(tk)Y, . . . , �T (tk)κlϒ(tk)Y,

�T (t)κ1ϒ(q1tk)Y, . . . , �T (tk)κhϒ(qhtk)Y ),

(45)


Y T ( j)L �(0) = γ j , j = 0, 1, . . . ,m − 1. (46)

This results in a system of (N + 1) algebraicequations containing the requisite SFG coefficientsyS, j , j = 0, 1, . . . , N , which can be evaluated usingan appropriate iterative process. As a consequence,the approximate solution yN (t) can be given.

4. Results and discussion

Seven problems will be used to assess the applicabilityof our methodology in this section. The software waswritten in Mathematica version 12 on a Dell laptop withan Intel(R) Core(TM) i3 CPU M 370@ 2.40 GHz and3.00GB RAM configuration. The results obtained usingour methodology are compared with those obtainedusing other numerical techniques in the literature. Theresults obtained in this paper are measured by means ofthe absolute errors (AEs) given by

E(ti ) = |y(ti ) − yN (ti )|, 1 ≤ i ≤ N ,

where y(ti ) and yN (ti ) are the exact solution (ES) andthe approximate solution (AS), respectively, and themaximum absolute error (MAE) is

L∞ = max1≤i≤N

{E(ti ) : ∀ti ∈ [0, L]}.

4.1 VO pantograph-delay differential equations

Problem (1)Consider the following VO-PDDEs [34]:


Dν(t)t y(t) = ey(t) + y(qt) − tν1(t)−3

p31

×Dν1(t)t y(p1t) + tν2(t)−1

p2Dν2(t)t y(p2t) + f (t),

with the ICs,

y(0) = 0, y′(0) = 1, y′′(0) = 0

where t ∈ [0, 1], q = 13 , p1 = 1

2 , p2 = 13 , and ν(t) ∈

(2, 3], ν1(t) ∈ (1, 2] and ν2(t) ∈ (0, 1], and

f (t) = −t3−ν(t)E2,4−ν(t)(−t2)

−E2,4−ν1(t)(−p21t

2)

−E2,2−ν2(t)(−p22t

2) − esin (t) − sin (qt).

Ea,b(t) denotes the following Mittag–Leffler function:

Ea,b(t) =∞∑j=0

t j

�(aj + b), t ∈ R.

The corresponding ES is y(t) = sin (t).The numerical results of problem (1) are listed in

tables 1 and 2 and graphically illustrated in figures 1aand 1b. In table 1, we tabulate the comparison of the AEswhen α = 2, μ = 1 and distinct values of N using theSFGOMs method together with the results in [34]. Thesenumerical results demonstrate the effectiveness of theproposed technique. The AEs when μ = 1, N = 10 andfor different values of α are given in table 2. Figure 1ashows the AEs of y(t) when N = 10 and α = 1, μ = 1.

Figure 1b illustrates the harmony between the AS yN (t)and ES y(t).


Dν(t)t y(t) − 1

2Dν1(t)t y

(4

5t

)− 1

10y

(4

5t

)+ y(t)

=(

6

25t − 2

5

)e

−4t5 + e−t ,

ν(t), ν1(t) ∈ (0, 1], t ∈ [0, 10],with the IC

y(0) = 0,

where the ES at ν(t) = ν1(t) = 1 is y(t) = te−t .

The agreement between the ES and AS of problem(2) when N = 12, μ = 1, α = 1 and different choicesof ν(t), ν1(t) are shown in figures 2a and 2b. Figure 2cdescribes the AEs when ν(t) = ν1(t) = 1.


Dν(t)t y(t) = sin (y(t)) + (y(q1t))

2

+y(q2t) + 2Dν1(t)t y(p1t) + Dν2(t)

t y(p2t)

+ sin (t)Dν3(t)t y(p3t) + f (t),

with the ICs

y(0) = 1, y′(0) = −1, y′′(0) = 1

where t ∈ [0, 1], q1 = 12 , q2 = 1

3 , p1 = 12 , p2 =

13 , p3 = 3

4 and ν(t), ν1(t) ∈ (2, 3], ν2(t) ∈ (1, 2] andν3(t) ∈ (0, 1], and

f (t) = −t3−ν(t)E1,4−ν(t)(−t)

+2p31t

3−ν1(t)E1,4−ν1(t)(−p1t)

−p22t

2−ν2(t)E1,3−ν2(t)(−p2t)

+p3t1−ν3(t) sin (t)E1,2−ν3(t)(−p3t)

− sin (e−t ) − e−2q1t − e−q2t .

The corresponding ES is y(t) = e−t .

We have the choice of the following two cases of VO:

Case (A) Case (B)

ν(t) = 2.5 + 0.2 cos (π t), ν(t) = 2.5 + 0.2 cos (π t),ν1(t) = 2.5 − 0.3 cos (t), ν1(t) = 2.5 − 0.3 cos (t),ν2(t) = 1.7 + 0.2 cos (t), ν2(t) = 1.7 + 0.2 cos (t),ν3(t) = 0.7 − 0.2 cos (t), ν3(t) = 0.7 − 0.2 cos (t).

The numerical results of problem (3) are given intables 3 and 4 and in figures 3a and 3b. Table 3 displaysthe comparison of the AEs when α = 1, μ = 1 andfixed values of N using the SFGOMs method togetherwith the results in [34]. These numerical results demon-strate the effectiveness of the proposed technique. TheAEs of problem (3) when N = 8, μ = 1 and differentvalues of α are given in table 4. The AEs between theES and the AS are offered in figure 3a, while figure 3bdescribes the agreement between the ES and AS whenN = 8, μ = 1, α = 1.

Problem (4)Consider the following system of VO-PDDEs:

∗0D

ν(t)t y1(t) − y1(t) + y2(t) − y1(0.5t)

+e0.5tν(t) − e−tν(t) = f1(t),∗0D

ν(t)t y2(t) + y1(t) + y2(t) + y2(0.5t)

−e−0.5tν(t) − etν(t) = f2(t),

with the ICs,

y1(0) = 1, y2(0) = 1,

where


Table 1. Comparison of the AEs for α = 2 and different values of N of problem (1).

ν(t) = 2.5 + 0.2 sin(π t), ν1(t) = 1.5 + 0.2t2, ν2(t) = 0.6 + 0.2 sin(2π t)

t N = 4 N = 8

Our method Ref. [34] Our method Ref. [34]

0.2 2.47186 × 10−5 1.1324 × 10−4 1.59993 × 10−10 8.1494 × 10−9

0.4 7.35411 × 10−5 6.7530 × 10−4 3.53191 × 10−10 1.2619 × 10−8

0.6 3.04042 × 10−5 1.6412 × 10−3 1.08688 × 10−9 1.0898 × 10−8

0.8 4.08772 × 10−4 2.7019 × 10−3 3.55408 × 10−9 5.2044 × 10−8

1 8.49502 × 10−4 3.5625 × 10−3 6.18049 × 10−9 9.2193 × 10−8

Table 2. AEs of problem (1) for N = 10, μ = 1 and fixed values of α.

ν(t) = 2.5 + 0.2 sin(π t), ν1(t) = 1.5 + 0.2t2, ν2(t) = 0.6 + 0.2 sin(2π t)

t α = −0.3 α = 0 α = 0.5 α = 1

0.2 8.28559 × 10−13 1.94483 × 10−13 7.83817 × 10−14 9.27036 × 10−15

0.4 9.51794 × 10−13 9.38694 × 10−13 1.04189 × 10−12 1.08341 × 10−12

0.6 6.24212 × 10−12 9.32809 × 10−13 6.94333 × 10−13 1.60560 × 10−12

0.8 1.80141 × 10−11 4.50029 × 10−12 1.24023 × 10−12 6.85785 × 10−13

1 2.90752 × 10−11 8.73368 × 10−12 4.35774 × 10−12 1.62592 × 10−12

Table 3. Comparison of the AEs for α = 1 and different values of N of problem(3).

t Case A

N = 5 N = 10


0.2 2.93234 × 10−6 2.8147 × 10−5 2.54241 × 10−14 5.6835 × 10−11

0.4 1.08163 × 10−5 1.7480 × 10−4 1.37446 × 10−13 2.5971 × 10−10

0.6 1.78793 × 10−5 4.6819 × 10−4 2.49578 × 10−13 6.0271 × 10−10

0.8 2.94278 × 10−5 9.0964 × 10−3 4.26548 × 10−13 1.0918 × 10−9

1 4.80066 × 10−5 1.5135 × 10−3 6.17784 × 10−13 1.7494 × 10−9

t Case B

N = 5 N = 10


0.2 3.63733 × 10−6 5.5282 × 10−5 3.94873 × 10−12 2.3941 × 10−10

0.4 1.46462 × 10−5 6.4205 × 10−4 1.46003 × 10−11 1.1935 × 10−9

0.6 2.63412 × 10−5 1.8383 × 10−3 2.69352 × 10−11 2.9174 × 10−9

0.8 4.28024 × 10−5 3.7512 × 10−3 4.10571 × 10−11 5.4534 × 10−9

1 6.86898 × 10−5 6.4347 × 10−3 5.81747 × 10−11 5.4534 × 10−9

Table 4. AEs of problem (3) for N = 8, μ = 1 for Case A and different values of α.

t α = −0.3 α = 0 α = 0.5 α = 1

0.2 1.58812 × 10−10 1.35547 × 10−10 4.23642 × 10−11 8.98259 × 10−11

0.4 1.98087 × 10−10 3.34057 × 10−10 2.05370 × 10−10 2.02118 × 10−10

0.6 6.59668 × 10−11 4.81862 × 10−10 3.54766 × 10−10 4.46884 × 10−10

0.8 4.18909 × 10−10 1.03683 × 10−9 7.41562 × 10−10 6.95087 × 10−10

1 7.88802 × 10−11 1.08290 × 10−9 8.89792 × 10−10 1.20294 × 10−9


Figure 1. The numerical behaviour of the proposed method on problem (1). (a) AEs at N = 10 and α = 1 and (b) the graphof the ES and AS of y(t) on [0,1].

Figure 2. The numerical behaviour of the proposed method on problem (2).


Figure 3. The numerical behaviour of the proposed method on problem (3). (a) AEs at N = 8 and α = 1 and (b) the graphof the ES and AS of y(t) on [0,1].

Table 5. MAEs of problem (4) for α = 0.5 and different values of μ, N and CPU time (seconds).

t μ = 12 μ = 1

3

y1(t) y2(t) CPU time y1(t) y2(t) CPU time

3 4.20852 × 10−2 6.81743 × 10−3 0.577204 3.99902 × 10−2 4.76043 × 10−2 0.6240045 4.27964 × 10−4 1.29015 × 10−4 1.01401 1.92507 × 10−3 6.17226 × 10−4 0.9204067 2.41782 × 10−5 3.26674 × 10−6 3.51002 3.94557 × 10−4 7.14043 × 10−5 1.638019 2.62176 × 10−7 1.70863 × 10−7 4.18083 1.17947 × 10−5 3.37082 × 10−6 4.44603

11 2.14331 × 10−9 8.40649 × 10−10 10.2961 5.28158 × 10−7 2.98143 × 10−7 9.39126

t μ = 1 CPU time

y1(t) y2(t) μ = 1

3 1.81604 × 10−3 7.00919 × 10−4 0.6084045 5.64900 × 10−6 4.47277 × 10−7 0.7020057 2.10731 × 10−9 1.60172 × 10−9 1.279219 1.26787 × 10−12 1.25294 × 10−12 2.60522

11 2.48077 × 10−11 7.91095 × 10−12 7.45685

Table 6. AEs of problem (4) for N = 7, α = 0.5 and fixed values of ν(t).

t ν(t) = 0.3 ν(t) = 0.6

y1(t) y2(t) y1(t) y2(t)

0.2 8.48885 × 10−11 6.01114 × 10−10 2.23149 × 10−10 4.64484 × 10−10

0.4 2.84644 × 10−9 1.91724 × 10−10 1.5495 × 10−9 4.89566 × 10−10

0.6 1.68207 × 10−9 2.0468 × 10−9 5.36705 × 10−10 8.31278 × 10−10

0.8 1.14079 × 10−8 2.19161 × 10−9 2.12005 × 10−9 7.01108 × 10−10

1 2.68059 × 10−8 4.8493 × 10−9 4.0067 × 10−9 3.72171 × 10−10

t ν(t) = 1

Our method Ref. [9]

y1(t) y2(t) y1(t) y2(t)

0.2 1.19825 × 10−10 7.68405 × 10−11 5.383 × 10−11 4.634 × 10−10

0.4 2.14719 × 10−10 1.68594 × 10−10 1.637 × 10−9 2.434 × 10−10

0.6 1.74840 × 10−10 1.19850 × 10−10 1.963 × 10−9 3.913 × 10−10

0.8 1.90644 × 10−10 2.96776 × 10−10 8.020 × 10−10 5.076 × 10−10

1 5.78395 × 10−11 2.25153 × 10−11 2.379 × 10−9 1.056 × 10−9


Table 7. AEs of problem (4) for N = 9 and different values of α.

ν(t) = ν1(t) = 1 − sin (t)

t α = −0.3 α = 0

y1(t) y2(t) y1(t) y2(t)

0.1 4.01901 × 10−14 6.57252 × 10−14 3.34621 × 10−13 1.24345 × 10−14

0.3 6.53033 × 10−13 6.27054 × 10−13 1.17972 × 10−12 2.25042 × 10−13

0.5 1.17950 × 10−12 1.07336 × 10−12 8.37108 × 10−13 1.60649 × 10−13

0.7 9.19442 × 10−12 3.85736 × 10−12 2.46780 × 10−12 1.41342 × 10−12

0.9 1.03709 × 10−10 4.09222 × 10−11 1.35336 × 10−11 5.54029 × 10−12

t α = 0.5 α = 1

y1(t) y2(t) y1(t) y2(t)

0.1 3.86802 × 10−13 7.28306 × 10−14 1.99618 × 10−13 7.34968 × 10−14

0.3 7.03881 × 10−13 1.70308 × 10−13 4.84945 × 10−13 3.01870 × 10−13

0.5 9.17932 × 10−13 1.18794 × 10−14 2.36922 × 10−12 5.02820 × 10−13

0.7 5.01821 × 10−13 5.31020 × 10−13 3.37774 × 10−12 7.93088 × 10−13

0.9 1.26787 × 10−12 1.25294 × 10−12 1.14206 × 10−11 4.71390 × 10−12

Table 8. AEs at α = 0.5 and different values of N of problem (4).

ν(t) = ν1(t) = 1 − sin (t)

t N = 5 N = 7

y1(t) y2(t) y1(t) y2(t)

0.1 3.07839 × 10−7 4.31963 × 10−8 6.42867 × 10−10 1.93405 × 10−10

0.3 1.29929 × 10−7 1.03016 × 10−7 8.54314 × 10−10 1.41156 × 10−10

0.5 9.46331 × 10−7 2.72515 × 10−12 8.83383 × 10−10 1.08279 × 10−10

0.7 1.57286 × 10−6 4.42875 × 10−7 2.10731 × 10−9 3.27650 × 10−10

0.9 5.64900 × 10−6 4.27518 × 10−7 1.62643 × 10−9 1.60172 × 10−9

t N = 9 N = 11

y1(t) y2(t) y1(t) y2(t)

0.1 3.86802 × 10−13 7.28306 × 10−14 5.55112 × 10−15 1.4433 × 10−15

0.3 7.03881 × 10−13 1.70308 × 10−13 1.22125 × 10−14 1.22127 × 10−15

0.5 9.17932 × 10−13 1.18794 × 10−14 1.34781 × 10−13 3.33069 × 10−16

0.7 5.01821 × 10−13 5.31020 × 10−13 2.14539 × 10−12 3.96350 × 10−13

0.9 1.26787 × 10−12 1.25294 × 10−12 2.48077 × 10−11 7.91095 × 10−12

f1(t) =∞∑k=1

tk−ν(t)

�(k − ν(t) + 1)

−et + e−t − et2 + e

12 t

ν(t) − e−tν(t),

f2(t) =∞∑k=1

(−1)ktk−ν(t)

�(k − ν(t) + 1)+ et + e−t

+e−t2 − e

−12 tν(t) − et

ν(t)

and ESs are y1(t) = et , y2(t) = e−t .

The numerical results of problem (4) are listed intables 5–8 and graphically illustrated in figures 4a–4d.

In table 5, we tabulate the MAEs and the computa-tional time for different values of N using the SFGOMsmethod with various choices of FGP μ. Also, in table 6,we introduce the AEs of y(t) at different values ofν(t) = 0.3, 0.6, 0.9, 1 and compared our results atν(t) = 1 with those introduced in [9]. But, table 7 pre-sented the AEs of problem (4) when N = 9, ν(t) =1 − sin (t), and different values of α. Additionally table8 introduced the AEs at ν(t) = 1 − sin (t), and distinctvalues of N . Furthermore, in figure 4a the AEs betweenthe ES and AS at N = 9, μ = 1 and α = 0.5 are offered,while figure 4b describes the agreement between the ESand AS.


Figure 4. The numerical behaviour of the proposed method on problem (4). (a) AEs of y1(t) at N = 9 and α = 0.5, (b) thegraph of the ES and AS y1(t) on [0,1], (c) the AEs of y2(t) at N = 9 and α = 0.5 and (d) the graph of the ES and AS of y2(t)on [0,1].

4.2 VO pantograph delay integro-differentialequations

Problem (5)Consider the following VO-VDIDEs:

yν(t)(t) − 1

100(qt − t − 10)y(qt) − 1

100

×∫ t

0y(ξ)dξ − 1

1000

×∫ qt

0(t − ξ)y(ξ)dξ = f (t),

with the IC,

y(0) = e−1, 0 ≤ t ≤ 20,

where q = 0.05,

f (t) = e−1∞∑k=1

( 110 )k�(k + 1)tk−ν(t)

k!�(k + 1 − ν(t))

− 1

100(qt − t − 10)e

qt10 −1 −

(−1 + e

t10

10e

)

−(

−10 + t + eqt10 (−10 + (−1 + q)t)

100e

)

and the ES is y(t) = et

10 −1.

The numerical results of problem (5) are listed intables 9–11 and graphically illustrated in figures 5a and5b. In table 9, we tabulate the MAEs for different valuesof N using the SFGOMs method with various choicesof μ. Also, in table 10, a comparison of the proposedmethod for the AEs of problem (5) with distinct valuesof ν(t) = 0.75, 0.85, 0.95 and N = 2, 4, 6, 8 togetherwith the results in [35] at ν(t) = 1 are tabulated. Thesenumerical results show the effectiveness of the pro-posed technique. The AEs at different values of α aregiven in table 11. To demonstrate the convergence of the


Table 9. MAEs of problem (5) for α = 0.5, ν(t) = 1 − sin (t)2 and different values of N , and CPU

time (seconds) for μ = 1.

N μ = 12 CPU time μ = 1

3 CPU time μ = 1 CPU time

2 7.83407 × 10−5 0.889206 1.44119 × 10−3 0.748805 2.16590 × 10−6 0.8736064 5.72811 × 10−7 2.58962 1.54420 × 10−5 2.18401 9.48124 × 10−11 2.168416 1.14892 × 10−9 9.21966 1.05236 × 10−7 8.17445 1.16573 × 10−15 10.51458 1.56064 × 10−12 35.6462 1.16685 × 10−9 36.5647 7.77156 × 10−15 37.9706

Table 10. Comparison of MAEs for α = 0.5, μ = 1 and different values of N of problem (5).

N ν(t) = 1 ν(t) = 0.95 ν(t) = 0.85 ν(t) = 0.75

Our Method Ref. [35]

2 3.18392 × 10−6 1.9993 × 10−2 3.49003 × 10−6 3.76474 × 10−6 3.99711 × 10−6

4 9.10151 × 10−11 1.6187 × 10−4 9.50856 × 10−11 9.53060 × 10−11 1.06248 × 10−10

6 1.44329 × 10−15 8.2265 × 10−7 1.38778 × 10−15 1.55431 × 10−15 1.49880 × 10−15

8 1.88738 × 10−15 9.6985 × 10−9 2.33147 × 10−15 3.49720 × 10−15 4.88498 × 10−15

10 5.61218 × 10−14 2.8013 × 10−11 5.69544 × 10−14 6.92779 × 10−14 1.06859 × 10−13

Table 11. AEs of problem (5) for N = 6, ν(t) = 1 − sin (t)2 and different values of α.

t α = −0.3 α = 0 α = 0.5 α = 1

0.1 1.33227 × 10−15 1.22125 × 10−15 3.33067 × 10−16 8.32667 × 10−16

0.3 1.33227 × 10−15 1.16573 × 10−15 6.66134 × 10−16 9.43690 × 10−16

0.5 1.60982 × 10−15 3.33067 × 10−16 4.44089 × 10−16 2.22045 × 10−16

0.7 2.72005 × 10−15 1.77636 × 10−15 0 5.55112 × 10−16

0.9 4.77396 × 10−15 1.11022 × 10−16 6.66134 × 10−16 1.11022 × 10−15

proposed method, we plot the AEs between the ES andthe AS with N = 6, μ = 1 and α = 0.5 in figure 5a.Figure 5b describes the agreement between the ES andAS.

Problem (6)Consider the following VO-VDIDEs:

yν(t)(t) − y(t) + 1

2ln

(1 + t

2

)y

(t

2

)

−∫ t

0

t y(ξ)

1 + ξdξ −

∫ t2

0

y(ξ)

ξ + 1dξ = f (t),

Figure 5. The numerical behaviour of the proposed method on problem (5). (a) AEs at N = 6 and α = 0.5, ν(t) = 1 − sin (t)2

and (b) the graph of the ES and AS of y(t) on [0,1].


Table 12. MAEs of problem (6) for α = 0.5, ν(t) = 1 − t2 and fixed values of N ,

CPU time (seconds) at μ = 1.

N μ = 12 μ = 1

3 μ = 1 CPU time (μ = 1)

2 1.65347 × 10−2 2.89377 × 10−2 6.85341 × 10−3 7.831254 5.44975 × 10−4 4.90509 × 10−4 8.06323 × 10−5 12.77656 2.29411 × 10−6 2.50294 × 10−5 1.48387 × 10−6 25.41268 3.45093 × 10−7 1.27258 × 10−6 3.53855 × 10−8 58.0792

10 2.04601 × 10−8 6.95106 × 10−8 7.84667 × 10−10 159.417

Table 13. AEs of problem (6) for N = 10, μ = 1, α = 0.5 and different values ofν(t).

t ν(t) = 0.75 ν(t) = 0.85 ν(t) = 0.95 ν(t) = 1

0.1 1.88466 × 10−11 1.64069 × 10−10 2.95437 × 10−10 3.37872 × 10−10

0.3 9.16950 × 10−10 1.01739 × 10−9 1.05678 × 10−9 1.02795 × 10−9

0.5 3.95411 × 10−10 2.63408 × 10−10 7.17255 × 10−11 7.53949 × 10−11

0.7 7.95311 × 10−10 7.03963 × 10−10 6.69262 × 10−10 7.13275 × 10−10

0.9 2.12761 × 10−10 7.46889 × 10−11 4.42064 × 10−11 1.20476 × 10−10

Table 14. AEs of problem (6) for N = 10, ν(t) = 1 − t2 and different values of α.

t α = −0.3 α = 0 α = 0.5 α = 1

0.1 5.66009 × 10−10 6.46272 × 10−11 2.09570 × 10−10 4.29437 × 10−10

0.3 1.17900 × 10−9 1.21819 × 10−9 7.25830 × 10−10 5.27460 × 10−10

0.5 6.05815 × 10−10 6.55967 × 10−10 2.18317 × 10−10 1.20091 × 10−9

0.7 1.18510 × 10−9 7.44429 × 10−10 7.82129 × 10−10 2.33453 × 10−9

0.9 8.20205 × 10−10 8.42700 × 10−10 5.37686 × 10−10 3.12687 × 10−9

Figure 6. The numerical behaviour of the proposed method on problem (6). (a) AEs at N = 10 and α = 0.5 and (b) thegraph of the ES and AS of y(t) on [0,1].


Figure 7. The numerical behaviour of the proposed method on problem (7). (a) the AEs of y1(t) at N = 8 and α = 0.5, (b)the graph of the ES and AS y1(t) on [0,1], (c) the AEs of y2(t) at N = 8 and α = 0.5 and (d) the graph of the ES and AS ofy2(t) on [0,1].

Table 15. MAEs of problem (7) for α = 0.5 and fixed values of N , CPU time(seconds) at μ = 1.

N μ = 12 μ = 1

3

y1(t) y2(t) y1(t) y2(t)

2 6.32818 × 10−1 2.24492 × 10−1 4.66041 1.892174 2.44028 × 10−3 9.40087 × 10−3 2.06357 × 10−2 3.69893 × 10−2

6 5.69988 × 10−5 6.95229 × 10−6 6.15817 × 10−4 1.73933 × 10−3

8 8.81693 × 10−7 7.94743 × 10−8 3.35801 × 10−5 8.83710 × 10−6

10 1.30816 × 10−8 6.84460 × 10−10 1.21041 × 10−6 9.34182 × 10−8

N μ = 1

y1(t) y2(t) CPU time (μ = 1)

2 5.09939 × 10−2 3.45867 × 10−2 0.6240044 3.63959 × 10−5 1.38516 × 10−5 3.151226 5.24495 × 10−8 5.29370 × 10−9 19.79658 3.81748 × 10−11 3.68727 × 10−12 69.3268

10 1.13826 × 10−10 2.35015 × 10−10 212.598


Table 16. AEs of problem (7) at N = 8 and distinct values of α.

ν(t) = 1 − sin (t)2

t α = −0.3 α = 0

y1(t) y2(t) y1(t) y2(t)

0.1 1.49567 × 10−11 7.56529 × 10−12 5.54312 × 10−11 1.84761 × 10−11

0.3 1.11247 × 10−11 3.42505 × 10−11 1.16067 × 10−11 7.77530 × 10−12

0.5 9.57234 × 10−12 4.91364 × 10−11 3.94274 × 10−11 1.82288 × 10−11

0.7 1.16283 × 10−10 4.27178 × 10−11 1.56635 × 10−11 6.92415 × 10−11

0.9 2.90070 × 10−11 1.30422 × 10−10 1.73522 × 10−10 1.86682 × 10−10

t α = 0.5 ν(t) = 1

y1(t) y2(t) y1(t) y2(t)

0.1 2.93565 × 10−11 1.10428 × 10−12 1.15552 × 10−11 3.90152 × 10−12

0.3 7.87348 × 10−12 2.03009 × 10−12 6.29665 × 10−11 1.46558 × 10−11

0.5 1.18441 × 10−11 2.05080 × 10−12 9.90734 × 10−11 2.96503 × 10−11

0.7 2.91589 × 10−11 3.52268 × 10−12 1.38274 × 10−10 6.61675 × 10−11

0.9 3.81748 × 10−11 3.68727 × 10−12 3.29874 × 10−10 1.40899 × 10−10

Table 17. Comparison of MAEs for α = 0.5, ν(t) = 1, μ = 1 and fixed values ofN of problem (7).

N Our method Ref. [9]

y1(t) y2(t) y1(t) y2(t)

2 4.22399 × 10−2 4.52596 × 10−2 6.0777 × 10−2 1.1843 × 10−1

4 3.97436 × 10−5 6.60414 × 10−6 9.2227 × 10−4 8.2963 × 10−4

6 5.44873 × 10−8 2.82622 × 10−9 7.3973 × 10−7 1.8431 × 10−6

8 4.67899 × 10−11 1.43502 × 10−12 9.5045 × 10−10 6.6164 × 10−10

with the IC,

y(0) = 0, 0 ≤ t ≤ 1,

where

f (t) =∞∑k=1

(−1)k−1�(k + 1)

k�(k + 1 − ν(t))tk−ν(t) − ln (1 + t)

+1

2

(ln

(1 + t

2

))2

− 1

2t (ln (1 + t))2

−1

2

(ln

(2 + t

2

))2

and the ES [36] is y(x) = ln (1 + t).The numerical results of problem (6) are listed in

tables 12–14 and graphically illustrated in figures 5a and5b. In table 12, we tabulate the MAEs for different valuesof N using the SFGOMs method for different choicesof μ. Also, in table 13, the AEs of problem (6) withvarious values of ν(t) = 0.75, 0.85, 0.95, 1 at N = 10are tabulated. But, in table 14, we tabulate the AEs atdistinct values of α and N = 10. The AEs between theES and the AS for N = 10, α = 0.5, μ = 1 are offered

as in figure 6a while figure 6b describes the agreementbetween the ES and AS.

Problem (7)Consider the following system of VO-VDIDEs:

yν(t)1 (t) − sin (t)y1(t) + y2(t) − 2y1(qt)

− cos (t)y2(qt) −∫ t

0(y1(ξ) − y2(ξ))dξ

−∫ qt

0(t y1(ξ) + y2(ξ))dξ = f1(t),

yν(t)2 (t) − t y1(t) + 4y2(t) − y1(qt)

−et y2(qt) −∫ t

0(t2y1(ξ)

− sin (t)y2(ξ))dξ −∫ qt

0(3y1(ξ)

+y2(ξ))dξ = f2(t),

with the ICs,

y1(0) = 1, y2(0) = 0, 0 ≤ x ≤ 1,


where q = 0.8

f1(t) =∞∑k=1

�(k + 1)

k!�(k + 1 − ν(t))tk−ν(t)

− sin (t)et + t3 − 2eqt − cos (t)(qt)3 − (−1 + et )

+ t4

4− (−1 + eqt )t − q4t4

4,

f2(t) = �(4)

�(4 − ν(t))t3−ν(t)−tet+4t3−eqt − et (qt)3

−(−1 + et )t2 + t4

4sin (t) − 3(−1 + eqt ) − q4t4

4

and the ESs are y1(t) = et , y2(t) = t3.

The numerical results of problem (7) are given intables 15–17 and in figures 7a–7d. Table 15 displaysMAEs of y1(t), y2(t) for α = 0.5 and different val-ues of N , μ. Additionally, table 16 shows the AEs ofproblem (7) obtained with the proposed method fordifferent values of α. The AEs of problem (7) whenα = 0.5, μ = 1, ν(t) = 1 are given in table 17. TheAEs between the ES and the AS are given in figure 7a,while figure 7b describes the agreement between the ESand AS when N = 8, μ = 1, α = 0.5, ν(t) = 1− sin (t)

2 .

5. Conclusion

Our major goal was to develop a new computationaltechnique for solving VO-PDDEs and VO-VDIDEs.The proposed method used SFGOMs of differentiationand integration with the spectral collocation methodand to convert the problem into an algebraic structurethat can be solved using any iteration method. Variousnumerical problems are included to illustrate the valid-ity and practicability of the proposed technique. Fromthe problems discussed in this literature, we extractedthe following benefits:

• For any choice of shifted fractional Gegenbauerparameter μ, good numerical results are obtained.

• For different choices of shifted Gegenbauer param-eter α, good numerical results are gained and atα = 0, 0.5, 1 we obtained the results of CPs of thefirst kind, LPs and CPs of the second kind respec-tively.

• For a small value of the iteration N , a good approx-imation is gained, and by increasing N the AEsbecome better.

• The numerical results showed the efficiency andaccuracy of the new technique, compared to the AEsof other numerical methods in the fractional-ordercase.

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