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http://www.aimspress.com/journal/Math
AIMS Mathematics, 9(12): 34567–34587.
DOI: 10.3934/math.20241646
Received: 08 September 2024
Revised: 21 November 2024
Accepted: 03 December 2024
Published: 10 December 2024
Research article
Spectral tau technique via Lucas polynomials for the time-fractional
diffusion equation
Waleed Mohamed Abd-Elhameed1,∗, Abdullah F. Abu Sunayh2, Mohammed H. Alharbi2and
Ahmed Gamal Atta3
1Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt
2Department of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah, Saudi
Arabia
3Department of Mathematics, Faculty of Education, Ain Shams University, Roxy 11341, Cairo,
Egypt
*Correspondence: Email: waleed@cu.edu.eg.
Abstract: Here, we provide a new method to solve the time-fractional diffusion equation (TFDE)
following the spectral tau approach. Our proposed numerical solution is expressed in terms of a
double Lucas expansion. The discretization of the technique is based on several formulas about Lucas
polynomials, such as those for explicit integer and fractional derivatives, products, and certain definite
integrals of these polynomials. These formulas aid in transforming the TFDE and its conditions into a
matrix system that can be treated through a suitable numerical procedure. We conduct a study on the
convergence analysis of the double Lucas expansion. In addition, we provide a few examples to ensure
that the proposed numerical approach is applicable and efficient.
Keywords: time-fractional diffusion equation; Lucas polynomials; spectral methods; error bound
Mathematics Subject Classification: 35R11, 65N35, 40A05
1. Introduction
Both integrals and fractional derivatives are non-local operators because they include integration,
which is non-local. Because of this property, these operators help describe certain physical phenomena,
such as hereditary features, asymptotic scaling, and long-term memory effects. Fractional calculus
attracted many mathematicians, who worked hard to develop the subject further and devised distinctive
methods of defining fractional-order integrals and derivatives. Many researchers are starting to pay
more attention to the idea of fractional calculus because of all the places it might be useful, including
fluid dynamics, electrochemistry of corrosion, biology, optics, signal processing, fluid dynamics,
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regular variation in thermodynamics, aerodynamics, and many more [1–3]. Researchers conduct
extensive research using various numerical algorithms for fractional differential equations (FDEs).
For example, in [4], the Adomian decomposition method was used. In [5], a collocation algorithm was
used for a specific type of Korteweg–De Vries equation (KdV) equation. In [6], the authors applied
a physics-informed neural network-based scheme for treating some FDEs. The authors of [7] used a
collocation scheme and certain unified Chebyshev polynomials to solve the fractional heat equations.
In [8], a matrix algorithm was applied to solve certain FDEs. A Haar wavelets method was followed to
solve some pantograph FDEs in [9]. An approach based on an inverse Laplace transform was followed
in [10] to handle some FDEs. The authors of [11] used meshless analysis and the generalized finite
difference method to treat the fractional diffusion-wave equation. The same authors performed another
study in [12].
Lucas polynomials and their generalized sequences are important in many fields, such as number
theory, statistics, and computer science; see, for example, [13]. In numerical analysis, the roles of
Lucas polynomials and their generalized polynomials have increased. They are utilized in solving
differential equations (DEs) of all types. For example, the authors of [14] treated the time-fractional
Burgers equation using a finite difference approach. In [15], a numerical scheme was proposed to
treat the advection-diffusion equations using mixed Fibonacci Lucas polynomials. In [16], a spectral
method was utilized to solve an electro-hydrodynamics flow model. In [17], the authors employed
Lucas polynomials to solve certain multi-dimensional equations. The authors of [18] treated Cauchy
integral equations based on Lucas polynomials. In [19], the authors used modified Lucas polynomials
to treat some FDEs. Other multi-dimensional problems were treated via Lucas polynomials in [20].
High-order boundary value problems were handled using Lucas polynomials in [21].
To explain out-of-the-ordinary diffusion behaviors seen in a wide range of biological, financial, and
physical systems, the TFDE adds a fractional time derivative to the classical diffusion equation. The
traditional diffusion equation fails to effectively explain these behaviors because of nonlocal dynamics
and memory effects. The TFDE plays a crucial role in capturing these complicated processes by
providing a more precise and all-encompassing modeling framework. Types of TFDE were the focus
of numerous contributions. The authors of [22] proposed a numerical procedure using splines to solve
a class of TFDE. The authors of [23] used the Petrov-Galerkin method for treating the TFDE. In
[24], the authors applied a linearized numerical scheme for a class of nonlinear TFDE. In [25], the
authors utilized an implicit difference scheme for the TFDE. In [26], another difference scheme for the
generalized TFDE was followed. The authors of [27] used an exponential-sum algorithm for variable-
order TFDE.
For numerical analysis and scientific computing, spectral methods provide a collection of
approaches for solving DEs. Their distinguishing feature is the ability to describe the solution
throughout the entire domain of the problem using global basis functions, unlike approaches like
finite difference and finite element. These techniques can provide great precision with limited degrees
of freedom, particularly useful for problems with smooth solutions; see [28–30]. They exhibit high
convergence for the approximate solution. These advantages make their application a target for many
authors seeking solutions for various DEs. We assume that the solution of a given differential equation
is a combination of some basis functions, which could be orthogonal or non-orthogonal polynomials.
We extensively employ the various versions of spectral methods to solve different types of DEs.
Various papers have utilized the Galerkin method to solve some DEs, see [31–33]. The tau method
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is more popular than the Galerkin method. The main reason for this is that there are no restrictions in
choosing the trial and test functions if the tau method is utilized, unlike in the Galerkin method; see,
for example, [34, 35]. Various contributions extensively use the typical collocation method because
it can treat all types of DEs. For example, the authors of [36] followed a meshless superconvergent
stabilized collocation method to treat some linear and nonlinear elliptic problems. The collocation
method can also be applied to treat FDEs [37–39]. In addition, matrix methods together with the
collocation method were used in [40, 41]. For some other applications regarding the spectral methods
and their applications, one can refer to [42–44].
This article aims to analyze a new Lucas tau algorithm for handling the TFDE method. To tackle
this method, we need formulas concerning the Lucas polynomials, such as their product and specific,
definite integral formulas.
The structure of the rest of the paper is as follows: Section 2 presents some fundamentals and useful
preliminaries. A tau approach to solving the TFDE is presented in Section 3. The error bound is given
in Section 4. Four illustrative examples are provided in Section 5. Finally, Section 6 reports a few
conclusions.
2. Essentials and useful relations
This section introduces some basic definitions and fundamental formulas. The definition of
Caputo’s fractional derivative is given. An account of Lucas polynomials is given. Certain formulas of
Lucas polynomials that will be beneficial are also presented.
2.1. Caputo’s fractional derivative
Definition 2.1. Caputo’s fractional derivative is defined as [3]
Dζ
zY(z)=1
Γ(r−ζ)Zz
0
(z−t)r−ζ−1Y(r)(t)dt, ζ > 0,z>0,(2.1)
r−1< ζ ≤r,r∈Z+.
We also have
Dζ
zC=0,(Cis a constant),(2.2)
Dζ
zzk=
0,if k∈Z≥0and k <⌈ζ⌉,
k!
Γ(k+1−ζ)zk−ζ,if k∈Z≥0and k ≥ ⌈ζ⌉,(2.3)
where ⌈ζ⌉is the ceiling function.
2.2. An overview and some formulas of Lucas polynomials
This recursive formula is utilized to generate Lucas polynomials [23]:
Li(θ)=θLi−1(θ)+Li−2(θ),L0(θ)=2,L1(θ)=θ, i≥2,(2.4)
and they can be expressed as
Li(θ)=i
⌊i
2⌋
X
r=0
i−r
r
i−rθi−2r,i≥1,(2.5)
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that can be alternatively written as
Li(θ)=
i
X
k=0
Bk,iθk,i≥1,(2.6)
where
Bk,i=
2iδi+ki+k
2
i−k
2
i+k,(2.7)
and
δr=
1,if ris even,
0,if ris odd.(2.8)
In addition, Li(θ) can be expressed as
Li(θ)=θ+√θ2+4i+θ−√θ2+4i
2i,i≥0.(2.9)
In the following, we give some important formulas of Lucas polynomials that will be useful in deriving
our proposed algorithm.
Theorem 2.1. [45] Consider i,q∈Z+with i ≥q. In terms of Lucas polynomials, we can write
dqLi(θ)
dθ=iji−q
2k
X
m=0
ci−2m−q(−1)m m+q−1
m!(i−m−q+1)q−1Li−q−2m(θ),(2.10)
where
cr=
1
2,r=0,
1,r≥1.
Theorem 2.2. [46] Consider r and i to be two non-negative integers. The following linearization
formula holds:
Li(θ)Lr(θ)=(−1)iLr−i(θ)+Li+r(θ).(2.11)
Lemma 2.1. [47] For r ∈Z,Z1
0
Lr(θ)dθis given explicitly by the following formula:
Z1
0
Lr(θ)dθ=Mr,(2.12)
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where
Mr=
2,if r =0,
1
2,if r =1,
−1
2,if r =−1,
2F1
−1
2−r
2,−r
2
1−r
−4
1+r,if r is even, and r ≥0,
−4r+(−1+r)2F1
−1
2−r
2,−r
2
1−r
−4
−1+r2,if r is odd, and r >2,
(−1)r
2F1
−1
2+r
2,r
2
1+r
−4
1−r,if r is even, and r <0,
(−1)r
4r−(1+r)2F1
−1
2+r
2,r
2
1+r
−4
r2−1,if r is odd, and r <0.
(2.13)
Remark 2.1. It is worth mentioning here that many generalizations are established for the standard
Lucas polynomials. Among these generalizations, the generalized Lucas polynomials are generated by
the following recursive formula:
ψa,b
i(θ)=aθ ψa,b
i−1(θ)+bψa,b
i−2(θ), ψa,b
0(θ)=2, ψa,b
1(θ)=aθ, i≥2.(2.14)
It is clear that Li(θ)=ψ1,1
i(θ).
3. Tau approach for the TFDE
We confine this section to analyzing a numerical algorithm for solving the TFDE in one dimension.
We also account for another extended algorithm that treats a two-dimensional model effectively. We
apply the spectral tau approach to obtain the desired approximate solutions in one and two dimensions.
3.1. tau approximate solution for one-dimensional TFDE
Consider the following TFDE [22,23]:
Dζ
tξ(θ, t)−β ξθθ (θ, t)=f(θ, t),0< ζ ≤1,(3.1)
governed by the following conditions:
ξ(θ, 0) =g(θ),0< θ < 1,(3.2)
ξ(0,t)=h1(t), ξ(1,t)=h2(t),0<t< τ, (3.3)
where βis a positive constant, g(θ),h1(t),and h2(t) are given continuous functions, and f(θ, t) is the
source term.
Now, define
PN=span{Li(θ)Lj(t):1≤i,j≤ N +1},
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and thus, we can assume that any function ξN(θ, t)∈ PNmay be expressed as
ξN(θ, t)=N+1
X
i=1
N+1
X
j=1
ci j Li(θ)Lj(t)=L(θ)C L(t)T,(3.4)
where L(θ)=[L1(θ),L2(θ),...,LN+1(θ)],L(t)T=[L1(t),L2(t),...,LN+1(t)]T,and C=(ci j )1≤i,j≤N+1
is the matrix of unknowns of dimension (N+1)2.
The residual Res(θ, t) of Eq (3.1) has the following form:
Res(θ, t)=Dζ
tξN(θ, t)−β ξN
θθ (θ, t)−f(θ, t).(3.5)
As a result of applying tau method, we get
(Res(θ, t),Lr(θ)Ls(t)) =0,1≤r≤ N − 1,1≤s≤ N.(3.6)
Now, consider the following matrices:
F=(fr,s)(N−1)×N ,fr s =(f(θ, t),Lr(θ)Ls(t)),(3.7)
A=(ai,r)(N+1)×(N−1) ,ai,r=(Li(θ),Lr(θ)),(3.8)
H=(hir)(N+1)×(N−1),hir = d2Li(θ)
dθ2,Lr(θ)!,(3.9)
K=(kj,s)(N+1)×N,kj,s=Dζ
tLj(t),Ls(t),(3.10)
where
(g1(θ),g2(θ)) =Z1
0
g1(θ)g2(θ)dθ,
(g1(t),g2(t)) =Zτ
0
g1(t)g2(t)dt,
(g1(θ, t),g2(θ, t)) =Zτ
0Z1
0
g1(θ, t)g2(θ, t)dθdt.
(3.11)
Therefore, Eq. (3.6) can be rewritten as
N+1
X
i=1
N+1
X
j=1
ci j ai,rkj,s−βN+1
X
i=1
N+1
X
j=1
ci j hi,raj,s=fr,s,1≤r≤ N − 1,1≤s≤ N,(3.12)
or in the following matrix form:
ATCK−βHTCA=F.(3.13)
In addition, the conditions in (3.2) and (3.3) lead to the following equations:
N+1
X
i=1
N+1
X
j=1
ci j ai,rLj(0) =(g(θ),Lr(θ)),1≤r≤ N +1,
N+1
X
i=1
N+1
X
j=1
ci j aj,sLi(0) =(h1(t),Ls(t)),1≤s≤ N,
N+1
X
i=1
N+1
X
j=1
ci j aj,sLi(1) =(h2(t),Ls(t)),1≤s≤ N.
(3.14)
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Now, a suitable approach may be used to solve the resulting algebraic system of equations of order
(N+1)2, which includes Eqs (3.13) and (3.14).
Theorem 3.1. The elements ai,r,hi,r, and k j,sthat appear in the system (3.12) can be expressed in the
following closed forms:
ai,r=Z1
0
Li(θ)Lr(θ)dθ=Mr+i+(−1)iMr−i,(3.15)
hi,r=Z1
0
d2Li(θ)
dθ2Lr(θ)dθ=i⌊i−2
2⌋
X
n=0
ci−2n−2(−1)n(n+1)(i−n−1)
×(Mi−2−2n+r+(−1)rMi−2−2n−r),(3.16)
kj,s=Z1
0
Dζ
tLj(t)Ls(t)dt =
j
X
k=1
s
X
n=0
k!Bk,jBn,s
Γ(k−ζ+1)(−ζ+k+n+1),(3.17)
where Br,jand Mrare given by (2.7) and (2.12), respectively.
Proof. To find the elements ai,r, we make use of (2.11) to write
air =Z1
0(−1)iLr−i(θ)+Lr+i(θ)dθ. (3.18)
The last formula together with the integral in (2.12) leads to (3.15).
Now, to find the elements hi,r, we make use of formula (2.10), after putting q=2,to get
Z1
0
d2Li(θ)
dθLr(θ)dθ=Z1
0
i⌊i−2
2⌋
X
n=0
ci−2n−2(−1)n(n+1) (i−n−1) Li−2−2n(θ)
Lr(θ)dθ, (3.19)
which, after applying the product formula (2.11), may be transformed into
hi,r=i⌊i−2
2⌋
X
n=0
ci−2n−2(−1)n(n+1) (i−n−1) ×
(Mi−2−2n+r+(−1)rMi−2−2n−r).
This shows formula (3.16). To obtain kj,s, we use formula (2.1) to get
kj,s=Z1
0
Dζ
tLj(t)Ls(t)dt
=
j
X
k=1
s
X
n=0
Bk,jBn,sk!
Γ(k−ζ+1) Z1
0
tk−ζ+ndt,
(3.20)
which immediately gives the following result:
kj,s=
j
X
k=1
s
X
n=0
k!Bk,jBn,s
Γ(k−ζ+1) (−ζ+k+n+1).
This shows formula (3.17), and thus the proof of Theorem 3.1 is now complete. □
Remark 3.1. In Algorithm 1, we outline the methodology for addressing the TFDE with our spectral
tau approach.
AIMS Mathematics Volume 9, Issue 12, 34567–34587.
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Algorithm 1 Coding algorithm for the proposed technique
Input ζ, K,g(θ),h1(t),h2(t),and f(θ, t).
Step 1. Assume an approximate solution ξN(θ, t) as in (3.4).
Step 2. Apply the tau method to obtain the system in (3.13) and (3.14).
Step 3. Use Theorem 3.1 to get the elements of ai,r,hi,r,and kj,s.
Step 4. Use the NSolve command to solve the system in (3.13) and (3.14) to get ci j .
Output ξN(θ, t).
3.2. An extension of our algorithm
Our algorithm can be extended to solve 2D models. Similar steps can be followed to obtain
the proposed numerical solution. As an extension to our one-dimensional model, we can solve the
following two-dimensional TFDE:
Dζ
tξ(θ, y,t)−β1ξθθ (θ, y,t)−β2ξyy(θ, y,t)=G(θ, y,t),(3.21)
governed by the following conditions:
ξ(θ, y,0) =ξ1(θ, y),0< θ, y≤1,(3.22)
and
ξ(0,y,t)=ξ2(y,t), ξ(1,y,t)=ξ3(y,t),0<y≤1,0<t< τ,
ξ(θ, 0,t)=ξ4(θ, t), ξ(θ, 1,t)=ξ5(θ, t),0< θ ≤1,0<t< τ, (3.23)
where β1>0, β2>0, and G(θ, y,t) is the source term.
In this case, we may assume the approximate solution of the form
ξN(θ, y,t)=N+1
X
i=1
N+1
X
j=1
N+1
X
k=1
ci jk Li(θ)Lj(y)Lk(t).
We follow similar steps to those followed in Section 3.1 to get a linear system of algebraic equations of
dimension (N+1)3in the unknown expansion coefficients cijk , which can be solved using the Gaussian
elimination procedure.
4. Error bound
This section discusses the error analysis of the proposed double Lucas expansion.
Theorem 4.1. [48] Let u(θ)be a function defined on [0,1], such that |u(i)(0)| ≤ fifor i ≥0, where
f>0, and let it have the expansion u(θ)=∞
X
i=0
ˆuiψa,b
i(θ), where ψa,b
i(θ)denotes the generalized Lucas
polynomials generated by (2.14). Then, we have
•|ˆui| ≤ |a|−ificosh(2 |a|−1|b|1
2f)
i!.
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•The series converges absolutely.
Corollary 4.1. Let u(θ)be a function defined on [0,1], with |u(i)(0)| ≤ fi, i ≥0, and f >0, and let it
have the expansion u(θ)=∞
X
i=0
ˆuiLi(θ).Then, we have
•|ˆui| ≤ ficosh(2 f)
i!.
•The series converges absolutely.
Proof. It is a special case of Theorem 4.1 only by setting a=b=1. □
Theorem 4.2. Consider a function ξ(θ, t)=ξ1(θ)ξ2(t)defined on [0,1] ×[0,1], where |ξ(i)
k(0)| ≤ ϕi
kfor
k=1,2, i ≥0, and ϕk>0. Assume that ξ(θ, t)has the expansion ξ(θ, t)=∞
X
i=0
∞
X
j=0
ci j Li(θ)Lj(t). Then,
we have
•|ci j| ≤ ϕi
1ϕj
2cosh(2 ϕ1) cosh(2 ϕ2)
i!j!.
•∞
X
i=0
∞
X
j=0
ci j Li(θ)Lj(t)converges absolutely.
Proof. Based on Lemma 1 of [48], and setting a=b=1, the following relation can be obtained:
ξ1(θ)=∞
X
i=0
∞
X
k=0
(−1)kδiξ(i+2k)
1(0)
k! (i+k)! Li(θ).(4.1)
According to the assumption ξ(θ, t)=ξ1(θ)ξ2(t), we have
ξ(θ, t)=∞
X
i=0
∞
X
j=0
∞
X
k=0
∞
X
s=0
(−1)k+sδiδjξi+2k
1(0) ξj+2s
2(0)
k!s! (i+k)! ( j+s)! Li(θ)Lj(t).(4.2)
Now, the expansion coefficients ci j can be written as
ci j =∞
X
k=0
∞
X
s=0
(−1)k+sδiδjξi+2k
1(0) ξj+2s
2(0)
k!s! (i+k)! ( j+s)! .(4.3)
Taking into consideration that |ξ(i)
k(0)| ≤ ϕi
k, it is possible to rewrite the final equation in the following
form:
|ci j| ≤ ∞
X
k=0
ϕi+2k
1
k! (i+k)! ×∞
X
s=0
ϕj+2s
2
s! ( j+s)! .(4.4)
Now, based on Corollary 4.1 and after using similar steps as in [49], we get the following estimation:
|ci j| ≤ ϕi
1ϕj
2cosh(2 ϕ1) cosh(2 ϕ2)
i!j!.
This finalizes the proof. □
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Theorem 4.3. Assuming that ξ(θ, t)meets the conditions of Theorem 4.2, we have
|ξ(θ, t)−ξN(θ, t)| ≤
λh(√3ϕ1)N+1+(√3ϕ2)N+1i
(N+1)! ,(4.5)
where λ=cosh (2ϕ1)cosh (2ϕ2)e√3 (ϕ1+ϕ2).
Proof. From the definitions of ξ(θ, t) and ξN(θ, t),we can write
|ξ(θ, t)−ξN(θ, t)|=
∞
X
i=0
∞
X
j=0
ci j Li(θ)Lj(t)−N+1
X
i=1
N+1
X
j=1
ci j Li(θ)Lj(t)
≤
N+1
X
i=1
∞
X
j=N+2
ci j Li(θ)Lj(t)
+
∞
X
i=N+2
∞
X
j=1
ci j Li(θ)Lj(t)
+
∞
X
j=0
c0jL0(θ)Lj(t)
+
∞
X
i=1
ci0Li(θ)L0(t)
.
(4.6)
Theorem 4.2 along with the inequality |Li(θ)| ≤ 2 (3) i
2yields [49]
|ξ(θ, t)−ξN(θ, t)| ≤ 4 cosh (2ϕ1)cosh (2ϕ2)
×
N+1
X
i=1
(√3ϕ1)i
i!
∞
X
j=N+2
(√3ϕ2)j
j!+∞
X
i=N+2
(√3ϕ1)i
i!
∞
X
j=1
(√3ϕ2)j
j!+∞
X
j=0
(√3ϕ2)j
j!+∞
X
i=1
(√3ϕ1)i
i!
≤4 cosh (2ϕ1)cosh (2ϕ2)
×
N+1
X
i=1
(√3ϕ1)i
i!
∞
X
j=N+2
(√3ϕ2)j
j!+∞
X
i=N+2
(√3ϕ1)i
i!
∞
X
j=1
(√3ϕ2)j
j!
+∞
X
i=N+2
(√3ϕ2)i
i!
∞
X
j=0
(√3ϕ2)j
j!+∞
X
i=1
(√3ϕ1)i
i!
∞
X
j=N+2
(√3ϕ2)j
j!
.
(4.7)
Using the following inequalities:
N+1
X
i=1
(√3ϕ1)i
i!=−1+e√3ϕ1Γ(N+2, ϕ1)
(N+1)! <e√3ϕ1,
∞
X
j=N+2
(√3ϕ2)j
j!=e√3ϕ1
1−Γ(N+2,√3ϕ2)
(N+1)!
<e√3ϕ2(√3ϕ2)N+2
(N+1)! ,
∞
X
i=N+2
(√3ϕ1)i
i!=e√3ϕ1
1−Γ(N+2,√3ϕ1)
(N+1)!
<e√3ϕ1(√3ϕ1)N+2
(N+1)! ,
∞
X
j=1
(√3ϕ2)j
j!=−1+e√3ϕ1<e√3ϕ1,
∞
X
j=0
(√3ϕ2)j
j!=e√3ϕ1,
(4.8)
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we get the following estimation:
|ξ(θ, t)−ξN(θ, t)| ≤
λh(√3ϕ1)N+1+(√3ϕ2)N+1i
(N+1)! ,(4.9)
where λ=8 cosh (2ϕ1)cosh (2ϕ2)e√3 (ϕ1+ϕ2),Γ(.) and Γ(., .) denote, respectively, gamma and upper
incomplete gamma functions [50]. This completes the proof. □
5. Some numerical tests
To show the usefulness and effectiveness of our suggested numerical algorithms described in
Sections 3.1 and 3.2, we offer four test problems here.
Example 5.1. Consider the following equation:
Dζ
tξ(θ, t)−ξθθ (θ, t)=f(θ, t),0< ζ ≤1,(5.1)
controlled by
ξ(θ, 0) =0,0< θ < 1,
ξ(0,t)=ξ(1,t)=0,0<t<1,(5.2)
with the exact solution ξ(θ, t)=t2sin(π θ).
Table 1 displays the absolute errors (AEs) at ζ=0.5and N=7. Also, Table 2 presents the AE at
ζ=0.7and N=6. Figure 1 shows the AEs (left) and the numerical solution (right) at ζ=0.9and
N=6. Finally, Table 3 presents the maximum absolute error (MAE) at N=6and various values of ζ
when 0<t<1.
Table 1. The AEs of Example 5.1 (ζ=0.5,N=7).
θt=1
10 t=4
10 t=8
10
0.1 1.5646 ×10−61.2713 ×10−55.0240 ×10−5
0.2 3.0120 ×10−64.2200 ×10−61.7060 ×10−5
0.3 5.4282 ×10−62.0123 ×10−57.9654 ×10−5
0.4 6.6875 ×10−62.7045 ×10−51.0672 ×10−5
0.5 6.1593 ×10−67.0292 ×10−62.6161 ×10−5
0.6 5.1758 ×10−61.8614 ×10−67.6500 ×10−5
0.7 5.5230 ×10−62.1705 ×10−58.8191 ×10−5
0.8 7.3537 ×10−63.4306 ×10−81.5175 ×10−7
0.9 8.5392 ×10−71.2490 ×10−65.2775 ×10−5
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Table 2. The AEs of Example 5.1 (ζ=0.7,N=6).
θt=3
10 t=6
10 t=9
10
0.1 9.4838 ×10−63.3169 ×10−58.0517 ×10−5
0.2 5.1520 ×10−61.1391 ×10−63.7064 ×10−5
0.3 1.0661 ×10−55.5192 ×10−51.0579 ×10−5
0.4 1.5736 ×10−57.7288 ×10−51.4897 ×10−5
0.5 2.6243 ×10−62.5008 ×10−52.7776 ×10−5
0.6 1.4857 ×10−54.6215 ×10−51.3155 ×10−5
0.7 1.7015 ×10−55.7268 ×10−51.5129 ×10−5
0.8 1.4300 ×10−61.9167 ×10−61.0640 ×10−5
0.9 8.5156 ×10−63.8106 ×10−57.8419 ×10−5
0
0.00005
0.00010
0.00015
0.00020
0
0.2
0.4
0.6
0.8
1.0
Figure 1. The AEs (left) and the numerical solution (right) for Example 5.1 (ζ=0.9,N=6).
Table 3. The MAEs of Example 5.1 at N=6 and different values of ζwhen 0 <t<1.
θ ζ =0.1ζ=0.6ζ=0.8
0.1 3.6721 ×10−63.4345 ×10−53.0277 ×10−6
0.2 1.3840 ×10−51.6076 ×10−51.9753 ×10−5
0.3 1.8318 ×10−51.7565 ×10−51.6503 ×10−5
0.4 3.4566 ×10−53.6419 ×10−53.9254 ×10−5
0.5 4.8302 ×10−54.7205 ×10−55.0927 ×10−5
0.6 7.7501 ×10−57.7268 ×10−57.6865 ×10−5
0.7 1.0071 ×10−51.0355 ×10−51.0843 ×10−4
0.8 1.2146 ×10−51.2238 ×10−61.2584 ×10−4
0.9 1.5667 ×10−51.5172 ×10−51.5297 ×10−4
Remark 5.1. We can demonstrate that the theoretical results of the error bound given in Section 4
agree with the numerical results presented in Section 5. As an example, if we set ϕ1=ϕ2=0.5in Eq.
(4.5) of Theorem 4.3, then we can see from the results of Tables 1–3 that the values of the MAEs do not
exceed those from the theoretical bound given in Table 4.
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Table 4. Theoretical error of Example 5.1.
N N =6N=7
Error in 4.5 10−410−5
Example 5.2. [22] Consider the following equation:
Dζ
tξ(θ, t)−ξθθ (θ, t)=f(θ, t),0< ζ ≤1,(5.3)
controlled by
ξ(θ, 0) =0,0< θ < 1,
ξ(0,t)=ξ(1,t)=0,0<t<1,(5.4)
with the exact solution ξ(θ, t)=sin(πt) sin(π θ).
In Table 5, we compare the MAE of our method and the method in [22] at various values of ζ. Also,
Figure 2 shows the AEs (left) and the numerical solution (right) at ζ=0.2and N=6. Finally, Table
6 presents the AEs (left) and the numerical solution (right) at ζ=0.6and N=6.
Table 5. Comparison of the MAEs for Example 5.2.
ζMethod in [22] (M=32 and ∆x=0.001) Our method (N=6)
0.5 1.10 ×10−32.73506 ×10−4
0.7 3.21 ×10−33.04668 ×10−4
Table 6. The absolute error of Example 5.2 at ζ=0.6,N=6.
θt=2
10 t=5
10 t=9
10
0.1 8.2979 ×10−58.4645 ×10−51.0131 ×10−5
0.2 7.3977 ×10−51.6564 ×10−52.6708 ×10−5
0.3 1.1797 ×10−51.7440 ×10−51.0040 ×10−5
0.4 3.1963 ×10−52.3901 ×10−51.3058 ×10−5
0.5 5.9211 ×10−59.3774 ×10−58.8532 ×10−5
0.6 1.6995 ×10−51.0700 ×10−52.1170 ×10−5
0.7 1.7218 ×10−51.4253 ×10−51.4246 ×10−6
0.8 5.2457 ×10−51.6448 ×10−53.3389 ×10−5
0.9 3.3454 ×10−51.1176 ×10−44.8945 ×10−5
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0
0.00005
0.00010
0.00015
0.00020
0.00025
0
0.2
0.4
0.6
0.8
1.0
Figure 2. The AEs (left) and the numerical solution (right) for Example 5.2 at ζ=0.2 and
N=6.
Example 5.3. [51] Consider the following equation:
Dζ
tξ(θ, t)−ξθθ (θ, t)=f(θ, t),0< ζ ≤1,(5.5)
controlled by
ξ(θ, 0) =eθ,0< θ < 1,
ξ(0,t)=t2+t+1, ξ(1,t)=(t2+t+1) e,0<t<1,(5.6)
with the exact solution ξ(θ, t)=t2+t+1eθ.
In Table 7, we compare the MAE between our method and the method in [51] at different values of
ζ. Also, Figure 3 shows the AEs (left) and the numerical solution (right) at ζ=0.4and N=6. Table
8 presents the MAEs at N=6and different values of ζwhen 0<t<1. Finally, Figure 4 shows the
AEs when θ=t at ζ=0.25,and different values of N.
Table 7. Comparison of the MAEs for Example 5.3.
ζMethod in [51] (N=64) Our method (N=6)
0.25 1.2889 ×10−55.14071 ×10−6
0.5 1.1727 ×10−54.73248 ×10−6
0.75 1.1645 ×10−57.98892 ×10−6
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0
2.5 ×10-6
5.0 ×10-6
7.5 ×10-6
0.0000100
0.0000125
0.0000150
2
4
6
8
Figure 3. The AEs (left) and the numerical solution (right) for Example 5.3 at ζ=0.4 and
N=6.
Table 8. The MAEs of Example 5.3 at N=6 and different values of ζwhen 0 <t<1.
θ ζ =0.3ζ=0.6ζ=0.9
0.1 4.1589 ×10−63.9766 ×10−63.4231 ×10−6
0.2 9.2989 ×10−61.0198 ×10−61.8535 ×10−5
0.3 4.8188 ×10−64.2917 ×10−64.5357 ×10−6
0.4 1.0242 ×10−51.2212 ×10−51.9936 ×10−5
0.5 4.3276 ×10−66.7577 ×10−61.7851 ×10−5
0.6 1.0885 ×10−51.1041 ×10−51.0999 ×10−5
0.7 9.0879 ×10−61.1921 ×10−52.3006 ×10−5
0.8 8.5030 ×10−67.1524 ×10−61.2442 ×10−5
0.9 7.6069 ×10−61.1040 ×10−52.0807 ×10−5
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Absolute errors
●=6■=5◆=4
Figure 4. The AEs for Example 5.3 at ζ=0.25.
Example 5.4. Consider the following equation:
Dζ
tξ(θ, y,t)−ξθθ (θ, y,t)−ξyy(θ, y,t)=G(θ, y,t),(5.7)
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governed by
ξ(θ, y,0) =0,0< θ, y≤1,(5.8)
and
ξ(0,y,t)=t2ey, ξ(1,y,t)=t2ey+1,0<y≤1,0<t<1,
ξ(θ, 0,t)=t2eθ, ξ(θ, 1,t)=t2eθ+1,0< θ ≤1,0<t<1,(5.9)
where
G(θ, y,t)=2eθ+y t2−α
Γ(3 −α)−t2!,(5.10)
and the exact solution of this problem is ξ(θ, y,t)=t2eθ+y.
Figure 5 shows the AEs when ζ=0.3and N=4at different values of y. Table 9 presents the AEs at
N=4and different values of t when ζ=0.9.
Figure 5. The AEs for Example 5.4 at ζ=0.3.
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Table 9. The AEs of Example 5.4 at N=4 and different values of t.
θ=y t =0.1t=0.3t=0.5t=0.7
0.1 6.5976 ×10−52.4075 ×10−45.1857 ×10−44.3326 ×10−4
0.2 1.0708 ×10−41.9143 ×10−41.2010 ×10−32.0530 ×10−3
0.3 1.5103 ×10−41.4615 ×10−42.0659 ×10−34.2141 ×10−3
0.4 1.9788 ×10−46.6388 ×10−53.2067 ×10−37.0446 ×10−3
0.5 2.4968 ×10−49.1615 ×10−54.7098 ×10−31.0617 ×10−2
0.6 3.0354 ×10−43.0591 ×10−44.2893 ×10−31.4472 ×10−2
0.7 3.5543 ×10−45.0045 ×10−47.9993 ×10−31.7623 ×10−2
0.8 4.0366 ×10−45.6085 ×10−48.8725 ×10−31.8615 ×10−2
0.9 4.3481 ×10−42.3525 ×10−48.0376 ×10−31.4990 ×10−2
6. Conclusions
In this article, we offered two numerical algorithms for treating the TFDE in one and two
dimensions. The tau method was used to propose the desired numerical solutions. The suggested
numerical solution was an expansion of the double basis of the Lucas polynomials. Some specific
formulas of the Lucas polynomials were the backbone for transforming the TFDE with their conditions
into a solvable system of equations. We believe that the method may be useful in treating other DEs.
The accuracy of the double expansion of Lucas polynomials was tested from a theoretical point of
view. In addition, some examples were presented to test the algorithm from a numerical point of
view. As an expected future work, we aim to employ this paper’s developed theoretical results together
with suitable spectral methods to treat some other problems. All codes were written and debugged by
Mathematica 11 on HP Z420 Workstation, Processor: Intel(R) Xeon(R) CPU E5-1620 v2 - 3.70GHz,
16 GB Ram DDR3, and 512 GB storage.
Author contributions
WMA contributed to Conceptualization, Methodology, Validation, Formal analysis, Investigation,
Project administration, Supervision, Writing – Original draft, and Writing - review & editing. AFAS
contributed to Methodology and validation. MHA contributed to Validation and Supervision. AGA
contributed to Conceptualization, Methodology, Validation, Formal analysis, Investigation, Software,
Writing – Original draft, and Writing - review & editing. All authors have read and agreed to the
published version of the manuscript.
Acknowledgments
The authors would like to express their sincere gratitude to the reviewers for their valuable
comments and suggestions, which have improved the paper in its present form.
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Conflict of interest
The authors declare that they have no competing interests.
References
1. A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential
equations, Elsevier, 2006.
2. S. G. Samko, Fractional integrals and derivatives: theory and applications, Gordon and Breach
Science Publishers, 1993.
3. I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional
differential equations, to methods of their solution and some of their applications, Elsevier, 1998.
4. D. Albogami, D. Maturi, H. Alshehri, Adomian decomposition method for solving
fractional time-Klein-Gordon equations using Maple, Appl. Math.,14 (2023), 411–418.
https://doi.org/10.4236/am.2023.146024
5. K. Sadri, K. Hosseini, E. Hinc¸al, D. Baleanu, S. Salahshour, A pseudo-operational collocation
method for variable-order time-space fractional KdV–Burgers–Kuramoto equation, Math. Methods
Appl. Sci.,46 (2023), 8759–8778. https://doi.org/10.1002/mma.9015
6. S. M. Sivalingam, V. Govindaraj, A novel numerical approach for time-varying impulsive fractional
differential equations using theory of functional connections and neural network, Expert Syst.
Appl.,238 (2024), 121750. https://doi.org/10.1016/j.eswa.2023.121750
7. W. M. Abd-Elhameed, H. M. Ahmed, Spectral solutions for the time-fractional heat differential
equation through a novel unified sequence of Chebyshev polynomials, AIMS Math.,9(2024),
2137–2166. https://doi.org/10.3934/math.2024107
8. M. Izadi, S¸ . Y¨
uzbas¸ı, W. Adel, A new Chelyshkov matrix method to solve linear and nonlinear
fractional delay differential equations with error analysis, Math. Sci.,17 (2023), 267–284.
https://doi.org/10.1007/s40096-022-00468-y
9. H. Alrabaiah, I. Ahmad, R. Amin, K. Shah, A numerical method for fractional variable order
pantograph differential equations based on Haar wavelet, Eng. Comput.,38 (2022), 2655–2668.
https://doi.org/10.1007/s00366-020-01227-0
10. Kamran, S. Ahmad, K. Shah, T. Abdeljawad, B. Abdalla, On the approximation of fractal-fractional
differential equations using numerical inverse Laplace transform methods, Comput. Model. Eng.
Sci.,135 (2023), 3. https://doi.org/10.32604/cmes.2023.023705
11. L. Qing, X. Li, Meshless analysis of fractional diffusion-wave equations by
generalized finite difference method, Appl. Math. Lett.,157 (2024), 109204.
https://doi.org/10.1016/j.camwa.2024.08.008
12. L. Qing, X. Li, Analysis of a meshless generalized finite difference method for the
time-fractional diffusion-wave equation, Comput. Math. Appl.,172 (2024), 134–151.
https://doi.org/10.1016/j.camwa.2024.08.008
AIMS Mathematics Volume 9, Issue 12, 34567–34587.
34585
13. T. Koshy, Fibonacci and Lucas numbers with applications, John Wiley & Sons, 2011.
14. I. Ali, S. Haq, S. F. Aldosary, K. S. Nisar, F. Ahmad, Numerical solution of one-and two-
dimensional time-fractional Burgers equation via Lucas polynomials coupled with finite difference
method, Alex. Eng. J.,61 (2022), 6077–6087. https://doi.org/10.1016/j.aej.2021.11.032
15. I. Ali, S. Haq, K. S. Nisar, S. U. Arifeen, Numerical study of 1D and 2D advection-diffusion-
reaction equations using Lucas and Fibonacci polynomials, Arab. J. Math.,10 (2021), 513–526.
https://doi.org/10.1007/s40065-021-00330-4
16. M. N. Sahlan, H. Afshari, Lucas polynomials based spectral methods for solving the fractional
order electrohydrodynamics flow model, Commun. Nonlinear Sci. Numer. Simul.,107 (2022),
106108. https://doi.org/10.1016/j.cnsns.2021.106108
17. P. K. Singh, S. S. Ray, An efficient numerical method based on Lucas polynomials to solve multi-
dimensional stochastic Itˆ
o-Volterra integral equations, Math. Comput. Simul.,203 (2023), 826–
845. https://doi.org/10.1016/j.matcom.2022.06.029
18. A. M. S. Mahdy, D. Sh. Mohamed, Approximate solution of Cauchy integral equations by using
Lucas polynomials, Comput. Appl. Math.,41 (2022), 403. https://doi.org/10.1007/s40314-022-
02116-6
19. B. P. Moghaddam, A. Dabiri, A. M. Lopes, J. A. T. Machado, Numerical solution of mixed-
type fractional functional differential equations using modified Lucas polynomials, Comput. Appl.
Math.,38 (2019), 1–12. https://doi.org/10.1007/s40314-019-0813-9
20. O. Oruc¸, A new algorithm based on Lucas polynomials for approximate solution of 1D and
2D nonlinear generalized Benjamin–Bona–Mahony–Burgers equation, Comput. Math. Appl.,74
(2017), 3042–3057. https://doi.org/10.1016/j.camwa.2017.07.046
21. M. C¸ etin, M. Sezer, H. Kocayi˘
git, An efficient method based on Lucas polynomials for solving
high-order linear boundary value problems, Gazi Univ. J. Sci.,28 (2015), 483–496.
22. P. Roul, V. Goura, R. Cavoretto, A numerical technique based on B-spline for a class of
time-fractional diffusion equation, Numer. Methods Partial Differ. Equ.,39 (2023), 45–64.
https://doi.org/10.1002/num.22790
23. Y. H. Youssri, A. G. Atta, Petrov-Galerkin Lucas polynomials procedure for the time-fractional
diffusion equation, Contemp. Math.,4(2023), 230–248. https://doi.org/10.37256/cm.4220232420
24. P. Lyu, S. Vong, A fast linearized numerical method for nonlinear time-fractional diffusion
equations, Numer. Algorithms,87 (2021), 381–408. https://doi.org/10.1007/s11075-020-00971-0
25. X. M. Gu, H. W. Sun, Y. L. Zhao, X. Zheng, An implicit difference scheme for time-fractional
diffusion equations with a time-invariant type variable order, Appl. Math. Lett.,120 (2021), 107270.
https://doi.org/10.1016/j.aml.2021.107270
26. A. Khibiev, A. Alikhanov, C. Huang, A second-order difference scheme for generalized time-
fractional diffusion equation with smooth solutions, Comput. Methods Appl. Math.,24 (2024),
101–117. https://doi.org/10.1515/cmam-2022-0089
27. J. L. Zhang, Z. W. Fang, H. W. Sun, Exponential-sum-approximation technique for variable-
order time-fractional diffusion equations, J. Appl. Math. Comput.,68 (2022), 323–347.
https://doi.org/10.1007/s12190-021-01528-7
AIMS Mathematics Volume 9, Issue 12, 34567–34587.
34586
28. C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spectral methods in fluid dynamics,
Springer-Verlag, 1988.
29. J. Hesthaven, S. Gottlieb, D. Gottlieb, Spectral methods for time-dependent problems, Cambridge
University Press, 2007.
30. J. P. Boyd, Chebyshev and Fourier spectral methods, Courier Corporation, 2001.
31. W. M. Abd-Elhameed, M. M. Alsyuti, Numerical treatment of multi-term fractional differential
equations via new kind of generalized Chebyshev polynomials, Fractal Fract.,7(2023), 74.
https://doi.org/10.3390/fractalfract7010074
32. M. M. Alsuyuti, E. H. Doha, S. S. Ezz-Eldien, I. K. Youssef, Spectral Galerkin schemes for a
class of multi-order fractional pantograph equations, J. Comput. Appl. Math.,384 (2021), 113157.
https://doi.org/10.1016/j.cam.2020.113157
33. R. M. Hafez, M. A. Zaky, M. A. Abdelkawy, Jacobi spectral Galerkin method for distributed-order
fractional Rayleigh–Stokes problem for a generalized second grade fluid, Front. Phys.,7(2020),
240. https://doi.org/10.3389/fphy.2019.00240
34. W. M. Abd-Elhameed, A. M. Al-Sady, O. M. Alqubori, A. G. Atta, Numerical treatment
of the fractional Rayleigh-Stokes problem using some orthogonal combinations of Chebyshev
polynomials, AIMS Math.,9(2024), 25457–25481. https://doi.org/10.3934/math.20241243
35. A. A. El-Sayed, S. Boulaaras, N. H. Sweilam, Numerical solution of the fractional-order logistic
equation via the first-kind Dickson polynomials and spectral tau method, Math. Methods Appl. Sci.,
46 (2023), 8004–8017. https://doi.org/10.1002/mma.7345
36. H. Hou, X. Li, A meshless superconvergent stabilized collocation method for linear and
nonlinear elliptic problems with accuracy analysis, Appl. Math. Comput.,477 (2024), 128801.
https://doi.org/10.1016/j.amc.2024.128801
37. A. G. Atta, Two spectral Gegenbauer methods for solving linear and nonlinear time fractional cable
problems, Int. J. Mod. Phys. C,35 (2024), 2450070. https://doi.org/10.1142/S0129183124500700
38. M. M. Khader, M. Adel, Numerical and theoretical treatment based on the compact finite difference
and spectral collocation algorithms of the space fractional-order Fisher’s equation, Int. J. Mod.
Phys. C,31 (2020), 2050122. https://doi.org/10.1142/S0129183120501223
39. A. Z. Amin, M. A. Abdelkawy, I. Hashim, A space-time spectral approximation for solving
nonlinear variable-order fractional convection-diffusion equations with nonsmooth solutions, Int.
J. Mod. Phys. C,34 (2023), 2350041. https://doi.org/10.1142/S0129183123500419
40. H. M. Ahmed, New generalized Jacobi polynomial Galerkin operational matrices of
derivatives: An algorithm for solving boundary value problems, Fractal Fract.,8(2024), 199.
https://doi.org/10.3390/fractalfract8040199
41. ˙
I. Avcı, Spectral collocation with generalized Laguerre operational matrix for numerical solutions
of fractional electrical circuit models, Math. Model. Numer. Simul. Appl.,4(2024), 110–132.
https://doi.org/10.53391/mmnsa.1428035
42. H. Singh, Chebyshev spectral method for solving a class of local and nonlocal elliptic
boundary value problems, Int. J. Nonlinear Sci. Numer. Simul.,24 (2023), 899–915.
https://doi.org/10.1515/ijnsns-2020-0235
AIMS Mathematics Volume 9, Issue 12, 34567–34587.
34587
43. M. Abdelhakem, A. Ahmed, D. Baleanu, M. El-Kady, Monic Chebyshev pseudospectral
differentiation matrices for higher-order IVPs and BVP: Applications to certain types of real-life
problems, Comput. Appl. Math.,41 (2022), 253. https://doi.org/10.1007/s40314-022-01940-0
44. Priyanka, S. Sahani, S. Arora, An efficient fourth order Hermite spline collocation method for time
fractional diffusion equation describing anomalous diffusion in two space variables, Comput. Appl.
Math.,43 (2024), 193. https://doi.org/10.1007/s40314-024-02708-4
45. W. M. Abd-Elhameed, A. Napoli, Some novel formulas of Lucas polynomials via different
approaches, Symmetry,15 (2023), 185. https://doi.org/10.3390/sym15010185
46. W. M. Abd-Elhameed, N. A. Zeyada, New formulas including convolution, connection and radicals
formulas of k-Fibonacci and k-Lucas polynomials, Indian J. Pure Appl. Math.,53 (2022), 1006–
1016. https://doi.org/10.1007/s13226-021-00214-5
47. W. M. Abd-Elhameed, A. K. Amin, Novel identities of Bernoulli polynomials
involving closed forms for some definite integrals, Symmetry,14 (2022), 2284.
https://doi.org/10.3390/sym14112284
48. W. M. Abd-Elhameed, Y. H. Youssri, Generalized Lucas polynomial sequence approach
for fractional differential equations, Nonlinear Dynam.,89 (2017), 1341–1355.
https://doi.org/10.1007/s11071-017-3519-9
49. Y. H. Youssri, W. M. Abd-Elhameed, A. G. Atta, Spectral Galerkin treatment of linear one-
dimensional telegraph type problem via the generalized Lucas polynomials, Arab. J. Math.,11
(2022), 601–615. https://doi.org/10.1007/s40065-022-00374-0
50. G. J. O. Jameson, The incomplete gamma functions, Math. Gaz.,100 (2016), 298–306.
https://doi.org/10.1017/mag.2016.67
51. F. Zeng, C. Li, A new Crank–Nicolson finite element method for the time-fractional subdiffusion
equation, Appl. Numer. Math.,121 (2017), 82–95. https://doi.org/10.1016/j.apnum.2017.06.011
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AIMS Mathematics Volume 9, Issue 12, 34567–34587.
