EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 1, Article Number 5629
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Existence of Solutions to a System of Fractional
Differential Equations With Applications to Beam

Deflections

Hasanen A. Hammad1,2,∗, Manal Elzain Mohamed Abdalla3

1 Department of Mathematics, College of Science, Qassim University, Buraydah 51452,
Saudi Arabia
2 Department of Mathematics, Faculty of Science, Sohag University, Sohag 82524, Egypt
3 Department of Mathematics, College of Science and Arts, King Khalid University, Mahayil,
Saudi Arabia

Abstract. In this manuscript, we introduce a novel system of fractional differential equations in-
corporating both Caputo and conformable derivatives. We delve into the existence and uniqueness
of solutions for this system, employing fixed-point techniques under appropriate conditions. To
illustrate the practical applications of our theoretical findings, we investigate the traveling wave
solutions of a tripled system of conformable fractional differential equations. Our analysis pro-
vides valuable insights into the dynamics and behavior of complex systems modeled by fractional
differential equations.

2020 Mathematics Subject Classifications: 47H10, 30C45, 39B72, 39B82

Key Words and Phrases: Nonlinear system, fractional derivative, Tanh’s method, traveling
wave solution, evaluation metrics

1. Background materials

Fractional differential equations (DEs) have gained significant attention due to their
ability to model complex phenomena more accurately than classical integer-order DEs.
Several analytical techniques have been developed to solve fractional DEs, including the
Laplace transform, Fourier transform, Mellin transform, and operational calculus. These
methods often involve transforming the fractional DE into an algebraic equation, solving
the transformed equation, and then inverting the transform to obtain the solution. Ad-
ditionally, various iterative methods like the Adomian decomposition method, variational
iteration method, and homotopy perturbation method have been employed to find ap-
proximate solutions to nonlinear fractional DEs. These methods provide a powerful tool

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5629

Email addresses: h.abdelwareth@qu.edu.sa (H.A. Hammad), mabdalla@kku.edu.sa (M.E.M. Abdalla)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



H.A. Hammad, M.E.M. Abdalla / Eur. J. Pure Appl. Math, 18 (1) (2025), 5629 2 of 20

for investigating the behavior of systems described by fractional DEs, for more details;
see[1, 3, 26, 30, 31].

The classical DEs can be extended to fractional DEs, where the order of differentiation
might be non-integer or fractional. Since integer-order DEs and systems are unable to suf-
ficiently represent complex systems with memory and hereditary qualities, these equations
and systems have drawn a great deal of attention [2, 7, 10, 13, 14, 17, 28, 29, 37, 41, 46].
One of the formulations of fractional calculus (FC) that is frequently employed, the Ca-
puto fractional derivative (CFD) [12, 28], is very helpful for initial value problems, which
makes it appropriate for physical applications.

Fixed point (FP) theorems are a powerful tool in the study of fractional differential
equations. By transforming fractional DEs into equivalent integral equations, FP tech-
niques, such as the Banach contraction mapping principle or the Schauder FP theorem,
can be applied to establish the existence and uniqueness of solutions. These techniques are
particularly useful for nonlinear fractional DEs, where analytical solutions may be difficult
to obtain. By employing FP methods, researchers can gain insights into the qualitative
behavior of solutions, including their stability, boundedness, and asymptotic properties.
See [4, 8, 9, 18–25, 32, 34, 38] for more information.

Beam deflection equations and systems are a significant area of study for fractional
DEs. The ability of beams to sustain loads applied laterally to their axis makes them
structural components. The bending and deflection of beams under varied stresses are
described by the classical beam theory, which is frequently represented by fourth-order
ordinary DEs [5, 6, 12, 29, 36, 43, 45]. Fractional derivatives (FDs), on the other hand,
can give these models a more realistic depiction of materials with non-local behavior and
viscoelastic characteristics, which are typical of sophisticated engineering materials and
intricate structures.

Nevertheless, the inclusion of FDs in these models can yield a more precise portrayal of
materials possessing viscoelastic characteristics and non-local behavior, which are preva-
lent in sophisticated engineering materials and intricate structures. These results in FD
models that provide deeper insights and enhanced design skills in applied physics and
engineering by more accurately predicting the deflection and dynamic response of beams.

Wang and Yang investigated in [43] whether positive solutions exist for the nonlinear
fourth-order system that describes the deformation of an elastic beam as follows:{

φ(4)(s) + γ1φ
′′(s)− η1φ(ζ) = ℓ1 (s, φ(s), ϕ(s)) , s ∈ V = [0, 1],

ϕ(4)(s) + γ2ϕ
′′(s)− η2ϕ(ζ) = ℓ2 (s, φ(s), ϕ(s)) , s ∈ V,

via the stipulations {
φ(0) = φ(1) = φ′′(0) = φ′′(1) = 0,
ϕ(0) = ϕ(1) = ϕ′′(0) = ϕ′′(1) = 0,

where ℓi : V ×R+×R+ → R+ are continuous functions, and γi, ηi ∈ R such that γi ≤ 2π2,
−γ2

i
4 ≤ ηi,

ηi
π4 +

γi
π4 < 1, i = 1, 2. The authors demonstrated the existence of positive solution

results by providing a cone Q in C(V ) × C(V ). They built another successful solution
outcome by building over a product cone.



H.A. Hammad, M.E.M. Abdalla / Eur. J. Pure Appl. Math, 18 (1) (2025), 5629 3 of 20

The authors of [42] discussed the existence and uniqueness of solutions for the system
that has Caputo sequential derivatives (CSDs) as the following:{ (

cDη + ϑ cDη−1
)
ζ (s) = ℓ1

(
s, ζ(s), ω(s), Ir1

0+
ζ(s), Ir2

0+
ω(s)

)
, s ∈ V,(

cDγ + θ cDγ−1
)
ω (s) = ℓ2

(
s, ζ(s), ω(s), Iz1

0+
ζ(s), Iz2

0+
ω(s)

)
, s ∈ V,

under the constraints{
ζ (0) = ζ ′ (0) = 0, ζ ′ (1) = 1, ζ (1) =

∫ 1
0 ζ (t) dR1(t) +

∫ 1
0 ω (t) dR2(t),

ω (0) = ω′ (0) = 0, ω′ (1) = 1, ω (1) =
∫ 1
0 ζ (t) dM1(t) +

∫ 1
0 ω (t) dM2(t),

where η, γ ∈ (3, 4], ϑ, θ > 0, r1, r2, z1, z2 > 0, ℓ1, ℓ2 : V × R4 → R are continuous func-
tions, Ir1

0+
, Ir2

0+
, Iz1

0+
, Iz2

0+
are Riemann-Liouville (RL) integrals of orders r1, r2, z1, z2, and

R1, R2,M1,M2 are Riemann-Stieltjes integrals. These systems have applications in bio-
sciences (see to [7]). The authors discovered that the system’s solutions exist and are
unique.

The existence, uniqueness, and stability of the system in the Ulam-Hyers sense were
examined by Bensassa et al. in [11]:{

Dη1Dη2φ(ζ) = ℓ1 (ζ, φ(ζ), ω(ζ)) + b1ℏ1 (ζ, φ(ζ)) + c1ϖ1 (ζ,D
ρφ(ζ)) ,

Dγ1Dγ2ω(ζ) = ℓ2 (ζ, φ(ζ), ω(ζ)) + b2ℏ2 (ζ, ω(ζ)) + c2ϖ2 (ζ,D
ρω(ζ)) ,

under the stipulations {
φ(0) = φ(1) = b, φ′(0) = φ′(1) = 0,
ω(0) = ω(1) = c, ω′(0) = ω′(1) = 0,

where Dηi , Dγi , Dρ are CFDs, ρ ∈ (0, 1], ℓ1, ℓ2 ∈ C ([0, 1]× R× R;R) , b1, b2, c1, c2 ∈ R,
and ℏ1, ℏ2, ϖ1, ϖ2 ∈ C ([0, 1]× R;R) .

Drawing on the previously mentioned research and specifically from paper [43], the
current study will address the following CSDs problem:{

Dη1Dγ1φ1(ζ) = ⅁1Z1

(
ζ, φ1(ζ), φ2(ζ), D

ρ−1φ1(ζ), D
ρφ1(ζ)

)
Dη2Dγ2φ2(ζ) = ⅁2Z2

(
ζ, φ1(ζ), φ2(ζ), D

ρ−1φ2(ζ), D
ρφ2(ζ)

) (1)

with the conditions {
φj(0) = φj(1) = bj ∈ R,
Dγjφj(0) = Dγjφj(1) = 0, j = 1, 2,

(2)

where Dη1 , Dγ1 , and Dρ are CFDs.
In order to ensure that there are no semi-group and commutativity properties on the

CFD and to derive, as a specific case, the aforementioned Wang and Yang fourth-order
system under various circumstances, we also assume that ηj ∈ (2, 3], γj ∈ (0, 1], and ⅁j > 0
j = 1, 2. It is also assumed that ρ ∈ (1, 2]. As a limiting case of (1), we can achieve the
aforesaid Wang and Yang [43] problem, where Zi : [0, 1]× R4 → R and j = 1, 2 are some
continuous functions.



H.A. Hammad, M.E.M. Abdalla / Eur. J. Pure Appl. Math, 18 (1) (2025), 5629 4 of 20

It is noteworthy that our problem is more substantial than the studies cited above,
and that most of them are not the same because we include CFD in the beginning circum-
stances. This inclusion strengthens our problem’s relevance and applicability in physical
environments by adding a layer of complexity and realism. These constraints are com-
patible with observable initial circumstances and effective numerical techniques, enabling
improved capturing of memory effects and non-local behaviors of the system.

In addition, our system has three generic functions, Z1, Z2 and Z3, which broaden the
system’s applicability beyond issues that have already been researched and offer a more
thorough framework for comprehending the dynamics of beam deflection. In engineering
and applied physics, this more comprehensive approach enables the modeling of more
complicated situations, leading to deeper insights and enhanced forecasting skills.

Furthermore, the problem (1) fills the gap that exists between contemporary viscoelas-
tic models and traditional beam deflection theories. In particular, our problem (1) may
be reduced to the above fourth-order ODEs that describe the deformation of an elastic
beam [43] if we consider the specific situation:

Zj

(
ζ, φ1(ζ), φ2(ζ), D

ρ−1φ1(ζ), D
ρφ1(ζ)

)
= ℓj (ζ, φ1(ζ), φ2(ζ))− bjφ

′′
j + ajφ

′
j ,

where ρ = 2, ηj = 3, γj = 1, for j = 1, 2. Furthermore, (1) offers a powerful tool for more
in-depth and accurate research on a variety of physical phenomena. Because of this, our
work is conceptually stimulating as well as practically important.

The Tanh approach [15, 16] will be applied in the second section of our study to get
new traveling wave solutions for the tripled beam issue with generalized conformable FDs
[27] as follows: {

ℑ2η
s φ+ ℑ4γ

ζ φ+ ⅁1K (φ, ω, · · · ) = 0,

ℑ2η
s ω + ℑ4γ

ζ ω + ⅁2N (φ, ω, · · · ) = 0.

The relationship and utility of FDs-both Caputo and conformable-in tackling and resolving
complex mathematical issues are illustrated via the application of fractional calculus to
expand and improve classical models. A range of complex systems can be studied and
solved with the help of different FDs, as demonstrated by this cohesive method.

The Tanh method was chosen because it is generally less computationally demanding
than other approaches, and it is also very simple to apply. Because of this, it’s a useful
tool for problem solving. It also offers explicit analytical solutions, which are helpful
for verifying numerical simulations and comprehending the qualitative behavior of the
solutions. The Tanh technique aids in understanding the physical processes, such as
wave propagation, solitons, and other localized structures, represented by the equations
by providing correct traveling wave solutions. The Tanh technique is a useful tool for
studying traveling waves in nonlinear equations because of these benefits. To obtain
additional information about this approach and other significant related methods, see
articles [33, 35, 44].

The results of this study have significant implications for engineering and applied
physics, particularly in areas where beam deflection models with fractional derivatives



H.A. Hammad, M.E.M. Abdalla / Eur. J. Pure Appl. Math, 18 (1) (2025), 5629 5 of 20

could offer a more accurate and comprehensive representation of real-world phenomena.
By considering the memory and hereditary effects inherent in materials, these models can
provide valuable insights into the long-term behavior of structures and systems. This
could lead to improved design and analysis techniques, enabling engineers to develop
more reliable and efficient structures. Additionally, the application of fractional calculus
to beam deflection problems can contribute to advancements in fields such as vibration
analysis, control systems, and materials science, ultimately leading to innovative solutions
and technological breakthroughs.

2. Basic facts

This section is devoted to presenting some definitions and lemmas concerning the FC.
We can find these results in [12, 28].

Definition 1. For any function ϖ ∈ C(V ), V = [0, 1], the fractional integral of order η
is described as

Iηϖ (s) =
1

Γ (η)

∫ s

0
(s− r)η−1ϖ (r) dr,

where Γ (.) is the Euler gamma function.

Definition 2. For the function ϖ ∈ Cm(V ), the CFD of order η is given by

Dηϖ (s) =
1

Γ (m− η)

∫ s

0
(s− r)m−η−1ϖ(m) (r) dr, m = [η] + 1,

where [η] represents the integer part of η.

Lemma 1. [28] Assume that η > 0, the solution of the equation

Dηϖ (s) = 0, s ∈ V

is written as
ϖ (s) = k0 + k1s+ k2s

2 + · · ·+ km−1s
m−1,

where kv ∈ R, v = 0, 1, 2, · · · ,m− 1, m = [η] + 1.

Lemma 2. [28] For any η > 0, we have

Iη
0+
Dηϖ (s) = ϖ (s) + k0 + k1s+ k2s

2 + · · ·+ km−1s
m−1,

where kv ∈ R, v = 0, 1, 2, · · · ,m− 1, m = [η] + 1.

Remark 1. For the function ϖ ∈ C(V ), we have the following results:

(i) For each η, γ > 0 and for all s ∈ V, IηIγϖ (s) = Iη+γϖ (s) ;

(ii) According to Definition 2 and the property (i), if 0 < η ≤ γ, we get DηIγϖ (s) =
Iη−γϖ (s) ;



H.A. Hammad, M.E.M. Abdalla / Eur. J. Pure Appl. Math, 18 (1) (2025), 5629 6 of 20

(iii) If η = γ in (ii), we have DηIγϖ (s) = ϖ (s) .

It is very important to present the equivalent integral eqution to the problem (1).

Lemma 3. Assume that R1, R2 ∈ C(V,R), ηj ∈ (2, 3], γj ∈ (0, 1], and ⅁j > 0, j = 1.2.
The problem {

Dη1Dγ1φ1(ζ) = ⅁1R1 (ζ) ,
Dη2Dγ2φ2(ζ) = ⅁2R2 (ζ) ,

(3)

via the conditions {
φj(0) = φj(1) = bj ∈ R,
Dλjφj(0) = Dλjφj(1) = 0,

is equivalent to

φ1(ζ) = b1 + ⅁1I
η1+γ1R1 (ζ)

+
(

2⅁1
γ1Γ(η1)

∫ 1
0 (1− r)η1−1R1 (r) dr − ⅁1Γ(γ1+3)

γ1Γ(η1+γ1)

∫ 1
0 (1− r)η1+γ1−1R1 (r) dr

)
ζγ1+1

+
(

⅁1Γ(γ1+3)
γ1Γ(η1+γ1)

∫ 1
0 (1− r)η1+γ1−1R1 (r) dr − ⅁1(γ1+2)

γ1Γ(η1)

∫ 1
0 (1− r)η1−1R1 (r) dr

)
ζγ1+2,

φ2(ζ) = b2 + ⅁2I
η2+γ2R2 (ζ)

+
(

2⅁2
γ2Γ(η2)

∫ 1
0 (1− r)η2−1R2 (r) dr − ⅁2Γ(γ2+3)

γ2Γ(η2+γ2)

∫ 1
0 (1− r)η2+γ2−1R2 (r) dr

)
ζγ2+1

+
(

⅁2Γ(γ2+3)
γ2Γ(η2+γ2)

∫ 1
0 (1− r)η2+γ2−1R2 (r) dr − ⅁2(γ2+2)

γ2Γ(η2)

∫ 1
0 (1− r)η2−1R2 (r) dr

)
ζγ2+2.

(4)
Proof. Applying Iηj+γj , (j = 1, 2) on (3) and using Lemma 2, we have{

φ1(ζ) = ⅁1I
η1+γ1R1 (ζ) +

k0ζγ1
Γ(γ1+1) +

k1ζγ1+1

Γ(γ1+2) +
k2ζγ1+2

Γ(γ1+3) + k3,

φ2(ζ) = ⅁2I
η2+γ2R2 (ζ) +

c0ζγ2
Γ(γ2+1) +

c1ζγ2+1

Γ(γ2+2) +
c2ζγ2+2

Γ(γ2+3) + c3.
(5)

Because φj(0) = bj (j = 1, 2), we conclude that k3 = b1, and c3 = b2. Applying Dγj

(j = 1, 2) on (5), we get{
Dγ1φ1(ζ) =

⅁1
Γ(η1)

∫ ζ
0 (ζ − r)η1−1R1 (r) dr + k0 + k1ζ + k2ζ

2,

Dγ2φ2(ζ) =
⅁2

Γ(η2)

∫ ζ
0 (ζ − r)η2−1R2 (r) dr + c0 + c1ζ + c2ζ

2.
(6)

Letting ζ = 0 in (5), we have k0 = c0 = 0. Taking ζ = 1 in (5) and (6), we obtain that
⅁1

Γ(η1+γ1)

∫ 1
0 (1− r)η1+γ1−1R1 (r) dr +

k1
Γ(γ1+2) +

2k2
Γ(γ1+3) = 0,

⅁1
Γ(η1)

∫ 1
0 (1− r)η1−1R1 (r) dr + k1 + k2 = 0,

⅁2
Γ(η2+γ2)

∫ 1
0 (1− r)η2+γ2−1R2 (r) dr +

c1
Γ(γ2+2) +

2c2
Γ(γ2+3) = 0,

⅁2
Γ(η2)

∫ 1
0 (1− r)η2−1R2 (r) dr + c1 + c2 = 0.

After solving the above equations, we have
k1 = − ⅁1Γ(γ1+3)

γ1Γ(η1+γ1)

∫ 1
0 (1− r)η1+γ1−1R1 (r) dr +

2⅁1
γ1Γ(η1)

∫ 1
0 (1− r)η1−1R1 (r) dr,

k2 =
⅁1Γ(γ1+3)
γ1Γ(η1+γ1)

∫ 1
0 (1− r)η1+γ1−1R1 (r) dr − ⅁1(γ1+2)

γ1Γ(η1)

∫ 1
0 (1− r)η1−1R1 (r) dr,

c1 = − ⅁2Γ(γ2+3)
γ2Γ(η2+γ2)

∫ 1
0 (1− r)η2+γ2−1R2 (r) dr +

2⅁2
γ2Γ(η2)

∫ 1
0 (1− r)η2−1R2 (r) dr,

c2 =
⅁2Γ(γ2+3)
γ2Γ(η2+γ2)

∫ 1
0 (1− r)η2+γ2−1R2 (r) dr − ⅁2(γ2+2)

γ2Γ(η2)

∫ 1
0 (1− r)η2−1R2 (r) dr.



H.A. Hammad, M.E.M. Abdalla / Eur. J. Pure Appl. Math, 18 (1) (2025), 5629 7 of 20

Substituting from the values of kj and cj (j = 1, 2) in (5), we get (4).

3. Fixed point approaches

In this section, we illustrate that the mechanism of the FP for discussing the existence
of the solution of our problem.

We define the Banach space (BS)

Ξ =
{
φ ∈ C (V,R) , Dρ−1φ ∈ C (V,R) , and Dρφ ∈ C (V,R)

}
under the norm

∥φ∥Ξ = ∥φ∥∞ +
∥∥Dρ−1φ

∥∥
∞ + ∥Dρφ∥∞ ,

where

∥φ∥∞ = sup
ζ∈V

|φ(ζ)| ,
∥∥Dρ−1φ

∥∥
∞ = sup

ζ∈V

∣∣Dρ−1φ(ζ)
∣∣ , and ∥Dρφ∥∞ = sup

ζ∈V
|Dρφ(ζ)| , V = [0, 1].

Consider the product space Ξ× Ξ defined on the norm

∥(φ, ω)∥Ξ×Ξ = ∥φ∥Ξ + ∥ω∥Ξ .

Clearly, the pair (Ξ× Ξ, ∥.∥) is a BS.
Now, in order to apply the technique of the FP, we consider the operator ℧ = (℧1,℧2)

such that ℧ : Ξ× Ξ → Ξ× Ξ. This operator can be written as

℧ (φ(ζ), ω(ζ)) = ℧1 (φ(ζ), ω(ζ)) + ℧2 (φ(ζ), ω(ζ)) ,

where

℧j (φ(ζ), ω(ζ))

= bj + ⅁jI
ηj+γjZj

(
ζ, φ(ζ), ω(ζ), Dρ−1φ(ζ), Dρω(ζ)

)
+

(
2⅁j

γjΓ (ηj)

∫ 1

0
(1− r)ηj−1 Zj

(
r, φ(r), ω(r), Dρ−1φ(r), Dρω(r)

)
dr

− ⅁jΓ (γj + 3)

γjΓ (ηj + γj)

∫ 1

0
(1− r)ηj+γj−1 Zj

(
r, φ(r), ω(r), Dρ−1φ(r), Dρω(r)

)
dr

)
ζγj+1

+

(
⅁jΓ (γj + 3)

γjΓ (ηj + γj)

∫ 1

0
(1− r)ηj+γj−1 Zj

(
r, φ(r), ω(r), Dρ−1φ(r), Dρω(r)

)
dr

− ⅁j (γj + 2)

γjΓ (ηj)

∫ 1

0
(1− r)ηj−1 Zj

(
r, φ(r), ω(r), Dρ−1φ(r), Dρω(r)

)
dr

)
ζγj+2

Therefore, the FP of the operator ℧ is equivalent to the solution of the system (1).



H.A. Hammad, M.E.M. Abdalla / Eur. J. Pure Appl. Math, 18 (1) (2025), 5629 8 of 20

4. Existence and uniqueness results

We begin this part with the following assertion:

(A1) For all ζ ∈ V and (φ, ω, δ, ϱ) ,
(
φ̃, ω̃, δ̃, ϱ̃

)
∈ R4, there exists a matrix of positive

functions Mlj , l = 1, 2, 3, 4, j = 1, 2 such that∣∣∣Zj (ζ, φ, ω, δ, ϱ)− Zj

(
ζ, φ̃, ω̃, δ̃, ϱ̃

)∣∣∣ ≤ M1j(ζ) |φ− φ̃|+M2j(ζ) |ω − ω̃|

+M3j(ζ)
∣∣∣δ − δ̃

∣∣∣+M1j(ζ) |ϱ− ϱ̃| ,

with mlj = supζ∈V {Mlj} .

The first main result of this part is as follows:

Theorem 1. Assume that the assertion (A1) is true. The system (1) under conditions
(3) admits a unique solution, provided that Ω1 +Ω2 < 1, where

Ωj = Bj max {(m1j +m3j) , (m2j +m4j)} , j = 1, 2,

and

Bj = ⅁j

{(
γj + 2Γ (γj + 3)

γjΓ (ηj + γj)
+

γj + 4

γjΓ (ηj)

)
+

1

Γ (ηj + γj − ρ+ 2)

+
Γ (γj + 2)

γjΓ (ηj + 1)Γ (γj − ρ+ 4)

[
2 (γj − ρ+ 3) + (γj + 2)2

]
+

Γ (γj + 2)Γ (γj + 3)

γjΓ (γj + ηj + 1)Γ (γj − ρ+ 4)
[2γj − ρ+ 5] +

1

Γ (ηj + γj − ρ+ 1)

+
Γ (γj + 2)

γjΓ (ηj + 1)Γ (γj − ρ+ 3)

[
2 (γj − ρ+ 2) + (γj + 2)2

]
+

Γ (γj + 2)Γ (γj + 3)

γjΓ (γj + ηj + 1)Γ (γj − ρ+ 3)
[2γj − ρ+ 4]

}
for j = 1, 2.
Proof. It is enough to prove that the mapping ℧ is a Banach contraction principle to

finish the proof.
Assume that (φ1, ω1) and (φ2, ω2) be arbitrary elements in Ξ×Ξ. Then for each ζ ∈ V

and for j = 1, 2, we have

|℧j (φ1(ζ), ω1(ζ))− ℧j (φ2(ζ), ω2(ζ))|

≤ ⅁j

Γ (ηj + γj)

∣∣∣∣∫ ζ

0
(ζ − r)ηj+γj−1 [Zj

(
r, φ1(r), ω1(r), D

ρ−1φ1(r), D
ρω1(r)

)
− Zj

(
r, φ2(r), ω2(r), D

ρ−1φ2(r), D
ρω2(r)

)]
dr
∣∣



H.A. Hammad, M.E.M. Abdalla / Eur. J. Pure Appl. Math, 18 (1) (2025), 5629 9 of 20

+
2⅁j

γjΓ (ηj)

∣∣∣∣∫ 1

0
(1− r)ηj−1 [Zj

(
r, φ1(r), ω1(r), D

ρ−1φ1(r), D
ρω1(r)

)
− Zj

(
r, φ2(r), ω2(r), D

ρ−1φ2(r), D
ρω2(r)

)]
dr
∣∣

+
⅁jΓ (γj + 3)

γjΓ (ηj + γj)

∣∣∣∣∫ 1

0
(1− r)ηj+γj−1 [Zj

(
r, φ1(r), ω1(r), D

ρ−1φ1(r), D
ρω1(r)

)
− Zj

(
r, φ2(r), ω2(r), D

ρ−1φ2(r), D
ρω2(r)

)]
dr
∣∣

+
⅁jΓ (γj + 3)

γjΓ (ηj + γj)

∣∣∣∣∫ 1

0
(1− r)ηj+γj−1 [Zj

(
r, φ1(r), ω1(r), D

ρ−1φ1(r), D
ρω1(r)

)
− Zj

(
r, φ2(r), ω2(r), D

ρ−1φ2(r), D
ρω2(r)

)]
dr
∣∣

+
⅁j (γj + 2)

γjΓ (ηj)

∣∣∣∣∫ 1

0
(1− r)ηj−1 [Zj

(
r, φ1(r), ω1(r), D

ρ−1φ1(r), D
ρω1(r)

)
− Zj

(
r, φ2(r), ω2(r), D

ρ−1φ2(r), D
ρω2(r)

)]
dr
∣∣

=
⅁jγj + 2⅁jΓ (γj + 3)

γjΓ (ηj + γj)

∣∣∣∣∫ ζ

0
(ζ − r)ηj+γj−1 [Zj

(
r, φ1(r), ω1(r), D

ρ−1φ1(r), D
ρω1(r)

)
− Zj

(
r, φ2(r), ω2(r), D

ρ−1φ2(r), D
ρω2(r)

)]
dr
∣∣

+
⅁j (γj + 4)

γjΓ (ηj)

∣∣∣∣∫ 1

0
(1− r)ηj−1 [Zj

(
r, φ1(r), ω1(r), D

ρ−1φ1(r), D
ρω1(r)

)
− Zj

(
r, φ2(r), ω2(r), D

ρ−1φ2(r), D
ρω2(r)

)]
dr
∣∣ .

Applying the hypothesis (A1), we have

∥℧j (φ1, ω1)− ℧j (φ2, ω2)∥∞

≤
(
⅁jγj + 2⅁jΓ (γj + 3)

γjΓ (ηj + γj)
+

⅁j (γj + 4)

γjΓ (ηj)

)
(M1j ∥φ1 − φ2∥∞ +M2j ∥ω1 − ω2∥∞

+M3j

∥∥Dρ−1φ1 −Dρ−1φ2

∥∥
∞ +M4j ∥Dρω1 −Dρω2∥∞

)
. (7)

Using the CFD, we get

Dρ−1 (℧j (φ(ζ), ω(ζ)))

= ⅁jI
ηj+γj−ρ+1Zj

(
ζ, φ(ζ), ω(ζ), Dρ−1φ(ζ), Dρω(ζ)

)
+

⅁j (γj + 2)

Γ (γj − ρ+ 3)
ζγj−ρ+2

(
2

γjΓ (ηj)

∫ 1

0
(1− r)ηj−1 Zj

(
r, φ(r), ω(r), Dρ−1φ(r), Dρω(r)

)
dr

− Γ (γj + 3)

γjΓ (ηj + γj)

∫ 1

0
(1− r)ηj+γj−1 Zj

(
r, φ(r), ω(r), Dρ−1φ(r), Dρω(r)

)
dr

)
+

⅁j (γj + 3)

Γ (γj − ρ+ 4)
ζγj−ρ+3

(
Γ (γj + 3)

γjΓ (ηj + γj)

∫ 1

0
(1− r)ηj+γj−1 Zj

(
r, φ(r), ω(r), Dρ−1φ(r), Dρω(r)

)
dr

− γj + 2

γjΓ (ηj)

∫ 1

0
(1− r)ηj−1 Zj

(
r, φ(r), ω(r), Dρ−1φ(r), Dρω(r)

)
dr

)
,



H.A. Hammad, M.E.M. Abdalla / Eur. J. Pure Appl. Math, 18 (1) (2025), 5629 10 of 20

and

Dρ (℧j (φ(ζ), ω(ζ)))

= ⅁jI
ηj+γj−ρZj

(
ζ, φ(ζ), ω(ζ), Dρ−1φ(ζ), Dρω(ζ)

)
+

⅁j (γj + 2)

Γ (γj − ρ+ 2)
ζγj−ρ+1

(
2

γjΓ (ηj)

∫ 1

0
(1− r)ηj−1 Zj

(
r, φ(r), ω(r), Dρ−1φ(r), Dρω(r)

)
dr

− Γ (γj + 3)

γjΓ (ηj + γj)

∫ 1

0
(1− r)ηj+γj−1 Zj

(
r, φ(r), ω(r), Dρ−1φ(r), Dρω(r)

)
dr

)
+

⅁j (γj + 3)

Γ (γj − ρ+ 3)
ζγj−ρ+2

(
Γ (γj + 3)

γjΓ (ηj + γj)

∫ 1

0
(1− r)ηj+γj−1 Zj

(
r, φ(r), ω(r), Dρ−1φ(r), Dρω(r)

)
dr

− γj + 2

γjΓ (ηj)

∫ 1

0
(1− r)ηj−1 Zj

(
r, φ(r), ω(r), Dρ−1φ(r), Dρω(r)

)
dr

)
.

By using the hypothesis (A1), and similar to (7), we get∥∥Dρ−1℧j (φ1, ω1)−Dρ−1℧j (φ2, ω2)
∥∥
∞

≤
(

⅁j

Γ (ηj + γj − ρ+ 2)
+

⅁jΓ (γj + 2)

γjΓ (ηj + 1)Γ (γj − ρ+ 4)

[
2 (γj − ρ+ 3) + (γj + 2)2

]
+

⅁jΓ (γj + 2)Γ (γj + 3)

γjΓ (γj + ηj + 1)Γ (γj − ρ+ 4)
[2γj − ρ+ 5]

)
[M1j ∥φ1 − φ2∥∞ +M2j ∥ω1 − ω2∥∞

+M3j

∥∥Dρ−1φ1 −Dρ−1φ2

∥∥
∞ +M4j ∥Dρω1 −Dρω2∥∞

]
(8)

Replacing ρ− 1 with ρ in (8) except in the term within the large parentheses, one has

∥Dρ℧j (φ1, ω1)−Dρ℧j (φ2, ω2)∥∞

≤
(

⅁j

Γ (ηj + γj − ρ+ 1)
+

⅁jΓ (γj + 2)

γjΓ (ηj + 1)Γ (γj − ρ+ 3)

[
2 (γj − ρ+ 2) + (γj + 2)2

]
+

⅁jΓ (γj + 2)Γ (γj + 3)

γjΓ (γj + ηj + 1)Γ (γj − ρ+ 3)
[2γj − ρ+ 4]

)
[M1j ∥φ1 − φ2∥∞ +M2j ∥ω1 − ω2∥∞

+M1j

∥∥Dρ−1φ1 −Dρ−1φ2

∥∥
∞ +M2j ∥Dρω1 −Dρω2∥∞

]
(9)

It follows from (7)-(9) and the definition of the norm on Ξ that

∥℧j (φ1, ω1)− ℧j (φ2, ω2)∥Ξ

≤ ⅁j

{(
γj + 2Γ (γj + 3)

γjΓ (ηj + γj)
+

γj + 4

γjΓ (ηj)

)
+

1

Γ (ηj + γj − ρ+ 2)

+
Γ (γj + 2)

γjΓ (ηj + 1)Γ (γj − ρ+ 4)

[
2 (γj − ρ+ 3) + (γj + 2)2

]
+

Γ (γj + 2)Γ (γj + 3)

γjΓ (γj + ηj + 1)Γ (γj − ρ+ 4)
[2γj − ρ+ 5] +

1

Γ (ηj + γj − ρ+ 1)

+
Γ (γj + 2)

γjΓ (ηj + 1)Γ (γj − ρ+ 3)

[
2 (γj − ρ+ 2) + (γj + 2)2

]



H.A. Hammad, M.E.M. Abdalla / Eur. J. Pure Appl. Math, 18 (1) (2025), 5629 11 of 20

+
Γ (γj + 2)Γ (γj + 3)

γjΓ (γj + ηj + 1)Γ (γj − ρ+ 3)
[2γj − ρ+ 4]

}
× [(M1j +M3j) ∥φ1 − φ2∥∞ + (M2j +M4j) ∥ω1 − ω2∥∞]

≤ Bj [(M1j +M3j) ∥φ1 − φ2∥∞ + (M2j +M4j) ∥ω1 − ω2∥∞] .

By the definition of the norm on Ξ× Ξ and Ω1, Ω2, we have

∥℧ (φ1, ω1)− ℧ (φ2, ω2)∥Ξ×Ξ ≤ (Ω1 +Ω2) ∥(φ1, ω1)− (φ2, ω2)∥ .

Since Ω1 + Ω2 < 1, we conclude that ℧ is a contraction mapping. This completes the
proof.

The example below support the above theorem.

Example 1. Consider the following problem:
Dη1Dγ1φ1(ζ) = ⅁1Z1

(
ζ, φ1(ζ), φ2(ζ), D

ρ−1φ1(ζ), D
ρφ1(ζ)

)
Dη2Dγ2φ2(ζ) = ⅁2Z2

(
ζ, φ1(ζ), φ2(ζ), D

ρ−1φ2(ζ), D
ρφ2(ζ)

)
φ1(0) = φ1(1) = φ2(0) = φ2(1) = 0,
Dγ1φ1(0) = Dγ1φ1(1) = Dγ2φ1(0) = Dγ2φ1(1) = 0.

We consider that

η1 =
7

3
, γ1 =

2

3
, η2 =

5

2
, γ2 =

1

2
, ⅁1 = ⅁2 = 1, and ρ =

3

2
.

Moreover,

Z1

(
ζ, φ1(ζ), φ2(ζ), D

ρ−1φ1(ζ), D
ρφ1(ζ)

)
=

1

205eζ
φ1(ζ) +

(
cos
(
2 + ζ2

)
40 (5 + ζ2)

)
φ2(ζ)

+

(
sin (ζ)

25 (ζ2 + 10)

)
D

1
2φ1(ζ) +

1

450eζ
D

3
2φ1(ζ),

Z2

(
ζ, φ1(ζ), φ2(ζ), D

ρ−1φ1(ζ), D
ρφ1(ζ)

)
=

1

30

(
1

10π
ln (1 + ζ)

)
φ1(ζ) +

(
sin (1 + ζ)

250eζ

)
φ2(ζ)

+

(
cos (ζ)

16 (eζ + 10)

)
D

1
2φ1(ζ) +

1

180πe4ζ
D

3
2φ1(ζ),

Clearly, for all ζ ∈ [0, 1] and φ, ω, δ, ϱ, φ̃, ω̃, δ̃, ϱ̃ ∈ R, we get∣∣∣Z1 (ζ, φ, ω, δ, ϱ)− Z1

(
ζ, φ̃, ω̃, δ̃, ϱ̃

)∣∣∣ ≤ M11(ζ) |φ− φ̃|+M21(ζ) |ω − ω̃|

+M31(ζ)
∣∣∣δ − δ̃

∣∣∣+M41(ζ) |ϱ− ϱ̃| ,

and ∣∣∣Z2 (ζ, φ, ω, δ, ϱ)− Z2

(
ζ, φ̃, ω̃, δ̃, ϱ̃

)∣∣∣ ≤ M12(ζ) |φ− φ̃|+M22(ζ) |ω − ω̃|



H.A. Hammad, M.E.M. Abdalla / Eur. J. Pure Appl. Math, 18 (1) (2025), 5629 12 of 20

+M32(ζ)
∣∣∣δ − δ̃

∣∣∣+M42(ζ) |ϱ− ϱ̃| ,

with
M11(ζ) =

1
205eζ

, M21(ζ) =

(
cos(2+ζ2)
40(5+ζ2)

)
, M31(ζ) =

(
sin(ζ)

25(ζ2+10)

)
, M41(ζ) =

1
450eζ

,

M12(ζ) =
1
30

(
1

10π ln (1 + ζ)
)
, M22(ζ) =

(
sin(1+ζ)
250eζ

)
, M32(ζ) =

(
cos(ζ)

16(eζ+10)

)
, M42(ζ) =

1
180πe4ζ

.

Since mlj = supζ∈V {Mlj}, we have{
m11 =

1
205 , m21 =

1
240 , m31 =

1
250 , m41 =

1
450 ,

m12 =
1

300π , m22(ζ) =
1

250 , m32(ζ) =
1

160 , m42(ζ) =
1

180π .

By simple calculations, and the definition of Bj (j = 1, 2), we can write B1 = 25.0783 and
B2 = 29.5855. Further,

Ω1 = 25.0783max

{(
1

205
+

1

250

)
,

(
1

240
+

1

450

)}
≈ 0.22265,

Ω2 = 29.5855max

{(
1

300π
+

1

160

)
,

(
1

250
+

1

180π

)}
≈ 0.634706,

Hence, Ω1+Ω2 < 1. Therefore, all requirements of Theorem 1 are fulfilled. So, the problem
has at least one solution.

5. Applications to beam systems

The technique of Tanh [15, 33, 39] will be applied in this section to discuss the traveling
wave solutions for a tripled system of conformable FDs in the form of ℵ2η

s φ (s, ζ) + ℵ4γ
ζ φ (s, ζ) + ⅁1K

(
φ, ω,ℵ2γ

ζ (φ, ω)
)
(s, ζ) = 0,

ℵ2η
s ω (s, ζ) + ℵ4γ

ζ ω (s, ζ) + ⅁2N
(
φ, ω,ℵ2γ

ζ φ (φ, ω)
)
(s, ζ) = 0,

(10)

where η, γ ∈ (0, 1], K,N are given function, and ℵη
sφ (s, ζ) refers to the conformable FD.

ℵη
sφ (s, ζ) is defined by Khalil as follows:

ℑη
sφ (s, ζ) =

∂ηφ (s, ζ)

∂sη
= lim

ε→0

(
φ
(
s+ εs1−η, ζ

)
− φ (s, ζ)

ε

)
, η ∈ (0, 1], (11)

where φ (s, ζ) is unknown function. Also, by using (11), we can present ℵγ
ζ .

Clearly, the classical coupled beam equations [40, 47] can be obtained if we put η =
γ = 1 and ⅁1 = ⅁2 = 1 in (10). It takes the shape φss + φζζζζ + ⅁1K

(
φ, ω, (φ, ω)ζζ

)
= 0,

ωss + ωζζζζ + ⅁1K
(
φ, ω, (φ, ω)ζζ

)
= 0.



H.A. Hammad, M.E.M. Abdalla / Eur. J. Pure Appl. Math, 18 (1) (2025), 5629 13 of 20

5.1. Steps of the Tanh method

In this part, we summarize the important steps of the Tanh technique involving con-
formable FDs as follows:

(i) We begin with the following coupled system: Φ1

(
φ, ω,ℵη

sφ,ℵγ
ζφ,ℵ

η
sω,ℵγ

ζω,ℵ
2η
s φ,ℵη

s

(
ℵγ
ζφ
)
,ℵ2γ

ζ φ,ℵ
2η
s ω,ℵη

s

(
ℵγ
ζω
)
,ℵ2γ

ζ ω, · · ·
)
= 0,

Φ2

(
φ, ω,ℵη

sφ,ℵγ
ζφ,ℵ

η
sω,ℵγ

ζω,ℵ
2η
s φ,ℵη

s

(
ℵγ
ζφ
)
,ℵ2γ

ζ φ,ℵ
2η
s ω,ℵη

s

(
ℵγ
ζω
)
,ℵ2γ

ζ ω, · · ·
)
= 0.

(12)

(ii) We use the relation

ϕ =
k̃

η
sη +

ν

γ
sγ , (13)

for simplicity, the model (12) can be written as{
Ψ1 (W,E,W

′, E′,W ′′, E′′,W ′′′, E′′′, · · · ) = 0,
Ψ2 (W,E,W

′, E′,W ′′, E′′,W ′′′, E′′′, · · · ) = 0.

(iii) We apply the transformation
ℜ = tanh (ϕ) ,

d

dϕ
=

(
1−ℜ2

) d

dℜ
,

d2

dϕ2
= −2ℜ

(
1−ℜ2

) d

dℜ
+
(
1−ℜ2

)2 d2

dℜ2
,

d3

dϕ3
= −2

(
1−ℜ2

) (
3ℜ2 − 1

) d

dℜ
− 6ℜ

(
1−ℜ2

)2 d2

dℜ2
+
(
1−ℜ2

)3 d3

dℜ3
,

d4

dϕ4
= −8ℜ

(
1−ℜ2

) (
3ℜ2 − 2

) d

dℜ
+ 4

(
1−ℜ2

)2 (
9ℜ2 − 2

) d2
dϱ2

−12ℜ
(
1−ℜ2

)3 d3

dℜ3
+
(
1−ℜ2

)4 d4

dℜ4
. (14)

(iv) We consider that {
φ (ζ, s) =W (ϕ) = Q (ℜ) =

∑m
j=0 bjℜj ,

ω (ζ, s) = E (ϕ) = ℧ (ℜ) =
∑n

j=0 cjℜj .
(15)

(v) Lastly, we obtain the required solutions for the constants bj and cj by using Wazwaz
term-balancing [44].



H.A. Hammad, M.E.M. Abdalla / Eur. J. Pure Appl. Math, 18 (1) (2025), 5629 14 of 20

5.2. Traveling wave solutions

In this part, as a practical application, we suggest identifying traveling wave solutions
for the coupled problem{

ℵ4γ
ζ (φ) + ℵ2η

s (φ) + ℵ2γ
ζ (φ) + 2⅁1cℵγ

ζ

((
ℵγ
ζω
)
ω
)
= 0,

ℵ4γ
ζ (ω) + ℵ2η

s (ω) + ⅁2hℵ2γ
ζ (φ) (φω + eω) = 0,

(16)

where ⅁1,⅁2 are positive real constants and h, c ∈ R.
We apply the relation (13) to convert the problem (16) to the following integral equa-

tion: {
ν4Wϕϕϕϕ + k̃2Wϕϕ + ν2Wϕϕ + 2c⅁1ν

2 (EϕE)ϕ = 0,

ν4Eϕϕϕϕ + k̃2Vϕϕ + ⅁2hν
2 (WE)ϕϕ + eν2Vϕϕ = 0.

(17)

Integrating (17), one has{
ν4Wϕϕ + k̃2W + ν2W + c⅁1ν

2E2 = 0,

ν4Eϕϕ + k̃2V + ⅁2hν
2 (WE) + eν2V = 0.

(18)

The first equation of (18) becomes the following equation when (14) and (15) are substi-
tuted into (18):

ν4
[
−2ℜ

(
1−ℜ2

) dQ
dℜ

+
(
1−ℜ2

)2 d2Q
dℜ2

]
+ k̃2Q+ c⅁1ν

2℧2 + eν2Q = 0. (19)

It is possible to convert the second equation of (18) into

ν4
[
−2ℜ

(
1−ℜ2

) d℧
dℜ

+
(
1−ℜ2

)2 d2℧
dℜ2

]
+ k̃2℧+ h⅁2ν

2 (Q℧) + eν2℧ = 0. (20)

We balance ℜ4 d2Q
dℜ2 with ℧2 in (19) to 2 +m = 2n. Moreover, with (20), we can apply the

same method to get 2 + n = n+m. Consequently, we are able to write{
Q (ℜ) = b0 + b1ℜ+ b2ℜ2,
℧ (ℜ) = c0 + c1ℜ+ c2ℜ2.

(21)

We note that when we substitute (21) for (19),

−2ν4ℜ
(
1−ℜ2

)
(b1 + 2b2ℜ) + 2b2ν

4
(
1−ℜ2

)2
+ k̃2

(
b0 + b1ℜ+ b2ℜ2

)
+c⅁1ν

2
(
c0 + c1ℜ+ c2ℜ2

)2
+ eν2

(
b0 + b1ℜ+ b2ℜ2

)
= 0.

Moreover, when we change (21) into (20), we obtain

−2ν4ℜ
(
1−ℜ2

)
(c1 + 2c2ℜ) + 2c2ν

4
(
1−ℜ2

)2
+ k̃2

(
c0 + c1ℜ+ c2ℜ2

)
+h⅁2ν

2
(
c0 + c1ℜ+ c2ℜ2

) (
b0 + b1ℜ+ b2ℜ2

)
+ eν2

(
c0 + c1ℜ+ c2ℜ2

)
= 0.

As a result, we have the following two sets:



H.A. Hammad, M.E.M. Abdalla / Eur. J. Pure Appl. Math, 18 (1) (2025), 5629 15 of 20

Set 1: 

ℜ0 : c⅁1ν
2c20 − 2ν2b1 + 2ν4b2 + k̃2b0 + eν2b0 = 0,

ℜ1 : 2cν2c0c1 − 4ν2b2 + k̃2b1 + ν2b1 = 0,

ℜ2 : 2cν2c0c2 + c⅁1ν
2c21 + 2ν2b1 − 4ν2b2 + k̃2b2 + ν2b2 = 0,

ℜ3 : 2cν2c1c2 + 4ν2b2 = 0,
ℜ4 : c⅁1ν

2c22 + 2ν2b2 = 0,

(22)

Set 2: 

ℜ0 : hν2⅁2c0b0 − 2ν4c1 + 2ν4c2 + eν2c0 + k̃2c0 = 0,

ℜ1 : hν2⅁2c0b1 + hν2⅁2c1b0 − 4ν2c2 + eν2b1 + k̃2c1 = 0,

ℜ2 : hν2⅁2c0b2 + hν2⅁2c1b1 + hν2⅁2c2b0 + 2ν2c1 − 4ν4c2 + eν2c2 + k̃2c2 = 0,
ℜ3 : hν2⅁2c1b2 + hν2⅁2c2b1 + 4ν4c2 = 0,
ℜ4 : hν2⅁2c2b2 + 2ν4b2 = 0.

(23)
With the aid of Maple, for ⅁1 = ⅁1 = 1, the constants of (22) and (23) can be
obtained as follows:

Case 1: For the set 1, we have

ν = e, k̃ = ±e
√
4e2 − 1, b0 = c0 = 0, b1 = b2 = −2e2

h
, and c1 = c2 = ±

√
1

ch
.

(24)
From (24) in (21), the solution of (16) can be written as{

φ (ζ, s) = −2e2

h tanh (ϕ)− 2e2

h tanh2 (ϕ) ,

ω (ζ, s) = ±
√

1
ch tanh (ϕ)±

√
1
ch tanh2 (ϕ) .

(25)

Under particular conditions, we now trace the two parts of this traveling wave solu-
tion in Figure 1.

Figure 1: 3D shapes for the solutions of (25) with −10 ≤ ζ ≤ 10, 0 ≤ s ≤ 10, ν = e = 2, h = 1, η = 2
5
, and

γ = 1
5
.



H.A. Hammad, M.E.M. Abdalla / Eur. J. Pure Appl. Math, 18 (1) (2025), 5629 16 of 20

Case 2: For the set 2, we have

ν = e, k̃ = k̃, b0 =
4e2

h
, b1 = b2 = −2e2

h
, c0 = ±4e2

√
1

ch
and c1 = c2 = ±2e2

√
1

ch
.

(26)
From (26) in (21), the solution of (16) takes the form{

φ (ζ, s) = 4e2

h − 2e2

h tanh (ϕ)− 2e2

h tanh2 (ϕ) ,

ω (ζ, s) = ±4e2
√

1
ch ± 2e2

√
1
ch tanh (ϕ)± 2e2

√
1
ch tanh2 (ϕ) .

(27)

As previously mentioned, Figure 2 shows the two parts of this traveling wave solution
for a given set of parameters.

Figure 2: 3D shapes for the solutions of (27) with −10 ≤ ζ ≤ 10, 0 ≤ s ≤ 10, ν = e = 3
7
, h = 3, c = 3

2
,

η = 7
9
, and γ = 4

15
.

6. Abbreviations

DE→differential equation
FC→fractional calculus
CFD→Caputo fractional derivative
FD→Fractional derivative
CSD→Caputo sequential derivative
RL→Riemann-Liouville
FP→fixed point
BS→Banach space

7. Conclusion

In this work, we explored two interconnected mathematical problems. Firstly, we ana-
lyzed a fractional system involving Caputo derivatives, which generalizes a beam detection-
type system. Our primary objective was to establish the existence and uniqueness of solu-
tions for this system. Secondly, we investigated a system composed of two tripled evolution



H.A. Hammad, M.E.M. Abdalla / Eur. J. Pure Appl. Math, 18 (1) (2025), 5629 17 of 20

equations utilizing conformable FDs as defined by Khalil. We obtained an ordinary dif-
ferential system with moving waves by changing this conformable fractional system. The
application of FC to expand and improve classical models is what connects these two sec-
tions, showcasing the adaptability and strength of FDs both Caputo and conformable in
tackling and resolving complex mathematical problems. This integrated method enhances
our knowledge and proficiency in mathematical modeling and analysis by demonstrating
how various FDs can be used to investigate and resolve a range of complicated problems.

While this study offers a solid theoretical framework for understanding fractional dif-
ferential equations and their applications, it’s essential to acknowledge certain limitations
and assumptions. One significant limitation lies in the analytical nature of the solutions,
which might not always be practical for complex systems. Additionally, the choice of frac-
tional derivative, such as the Caputo or conformable derivative, can significantly influence
the system’s behavior. Future research could concentrate on numerical implementations
of the proposed fractional systems, enabling the exploration of a wider range of parameter
values and initial conditions. Furthermore, empirical validation of the traveling wave so-
lutions through experiments or simulations would allow for a direct comparison between
theoretical predictions and real-world observations. This could lead to refinements in the
model and its parameters. Another potential area for future research is the investigation
of fractional systems with more intricate boundary conditions and external forcing terms.
These extensions could have significant implications for diverse physical and engineering
applications. Additionally, exploring the stability analysis of fractional systems would
provide valuable insights into their long-term behavior.

Acknowledgements

The authors extend their appreciation to the Deanship of Research and Graduate
Studies at King Khalid University for funding this work through Large Research Project
under grant number R. G. P. 2/217/45.

Funding

This work was supported by the Deanship of Research and Graduate Studies at King
Khalid University under Grant R. G. P. 2/217/45.

Data Availability

No data is associated with this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.



H.A. Hammad, M.E.M. Abdalla / Eur. J. Pure Appl. Math, 18 (1) (2025), 5629 18 of 20

Authors Contributions

All authors contributed equally and significantly in writing this article.

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