EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 4, 2024, 3464-3491
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Conformable Sumudu Transform Based Adomian
Decomposition Method for Linear and Nonlinear

Fractional-Order Schrödinger Equations

Muhammad Imran Liaqat1,2,∗, Hussam Aljarrah3

1 Abdus Salam School of Mathematical Sciences, Government College University, Lahore
54600, Pakistan
2 National College of Business Administration & Economics, 54000 Lahore, Pakistan
3 Ahmed bin Mohammed Military College, P.O. Box 22988, Doha, Qatar

Abstract. Fractional-order Schrödinger differential equations extend the classical Schrödinger
equation by incorporating fractional calculus to describe more complex physical phenomena. In the
literature, the Schrödinger equation is mostly solved using fractional derivatives expressed through
the Caputo derivative. However, there is limited research on exact and approximate solutions
involving conformable fractional derivatives. This study aims to fill this gap by employing a
hybrid approach that combines the Sumudu transform with the decomposition technique to solve
the Schrödinger equation with conformable fractional derivatives, considering zero and nonzero
trapping potentials. The efficiency of this approach is evaluated through the analysis of relative and
absolute errors, confirming its accuracy. Moreover, the obtained results are compared with other
techniques, including the homotopy analysis method (HAM) and the residual power series method
(RPSM). The comparison demonstrates strong consistency with these methods, suggesting that
our approach is a viable alternative to Caputo derivative-based methods for solving time-fractional
Schrödinger equations. Furthermore, we can conclude that the conformable fractional derivative
serves as a suitable substitute for the Caputo derivative in modeling Schrödinger equations.

2020 Mathematics Subject Classifications: 74S40, 65D15

Key Words and Phrases: Fractional calculus, approximate solutions, exact solutions, con-
formable fractional derivatives

1. Introduction

Fractional derivatives form the foundation of fractional calculus (FC), providing an
effective means to describe and analyze systems with non-local or memory-driven behav-
iors. In contrast to integer-order derivatives, which focus solely on the instantaneous rate

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5456

Email addresses: imran liaqat 22@sms.edu.pk (M. I. Liaqat),
hussam aljarrah@abmmc.edu.qa (H.Aljarrah)

https://www.ejpam.com 3464 Copyright: © 2024 The Author(s). (CC BY-NC 4.0)



M.I. Liaqat, H.Aljarrah / Eur. J. Pure Appl. Math, 17 (4) (2024), 3464-3491 3465

of change at a single point, fractional derivatives account for the function’s behavior over
an interval, thereby incorporating memory into the system’s dynamics. There are several
forms of fractional-order derivatives, each defined and characterized in its own way. Some
of the commonly used types include the Riemann–Liouville, Caputo, Grünwald–Letnikov,
Caputo-Fabrizio, and conformable fractional operators [1, 4, 16, 19, 32, 34]. The choice of
which definition to use depends on the specific problem at hand and the desired mathe-
matical properties of the derivative.

Recently, a large number of scholars have been actively investigating different kinds of
fractional-order derivatives. In the context of Caputo-Fabrizio, for instance, the authors
[30, 31] established highly significant results. Conformable fractional operators are a novel
method for solving fractional-order differential equations (FODEs) that was presented by
the authors [17]. Zhang et al. [33] developed solutions to differential equations (DEs)
within the framework of Caputo fractional derivatives (Cap-FD). Using the framework of
the Caputo derivative, Zhang and Xiong [35] showed that the periodic solution of FODEs
with semilinear impulses has unique solutions and global exponential stability. Atangana-
Baleanu derivatives are used by Syam and Al-Refai [24] to obtain numerous valuable results
about solutions to FODEs. Using Grünwald-Letnikov fractional derivatives, Li and Wang
[27] discovered solutions to fractional Rössler chaotic systems. For more study, refer to
[6, 9, 20, 21].

Conformable fractional derivatives (Con-FrD) provide a useful and intuitive way to
incorporate memory effects into differential equations (DEs). Con-FrD retains many prop-
erties of classical derivatives, making it easier to work with while offering the flexibility
of FC. For this reason, Con-FrD is very useful for simulating real-world phenomena in
physics, engineering, control systems, biology, and other fields.

The Con-FrD of a function δ(φ) of order µ is defined as [12]:

Tµ
φδ(φ) = lim

ϵ→0

δ⌈µ⌉−1(φ+ ϵφ⌈µ⌉−µ)− δ⌈µ⌉−1(φ)

ϵ
, (1)

where the Con-FrD with regard to time is denoted by Tµ
φ, and the smallest integer that is

equal to or greater than µ is ⌈µ⌉, κ ∈ N , and κ − 1 < µ ≤ κ, φ > 0.
In the specific case where 0 < µ ≤ 1, we obtain the following [18]:

Tµ
φδ(φ) = lim

ϵ→0

δ(φ+ ϵφ1−µ)− δ(φ)

ϵ
, φ > 0. (2)

The conformable fractional integral is defined as follows:

Iα̃µ(δ)(φ) =

∫ φ

α̃

δ(℘)(
℘− α̃

)1−µd℘, µ ∈ (0, 1]. (3)

Con-FrD provides an alternative framework for fractional differentiation that simplifies
the computation of fractional derivatives and maintains some key properties of classical
derivatives. The Con-FrD offers several advantages compared to other types of fractional
derivatives. Here are some of the key benefits [5, 15, 26, 28]:



M.I. Liaqat, H.Aljarrah / Eur. J. Pure Appl. Math, 17 (4) (2024), 3464-3491 3466

i. Simplicity and Intuition: Con-FrD are designed to be a straightforward generalization
of the classical derivative, making them easier to understand and apply. They retain
many of the properties of integer-order derivatives, which can simplify the transition
from classical to FC.

ii. Consistency with Classical Calculus: Unlike some other fractional derivatives, the
conformable derivative maintains a closer relationship to classical calculus. It satisfies
properties like the product rule, chain rule, and power rule in a manner that is more
consistent with classical derivatives, making it more intuitive for those familiar with
traditional calculus.

iii. Clear Physical Interpretation: Conformable derivatives offer a clearer physical inter-
pretation compared to some other fractional derivatives, as they are often more closely
related to the original physical quantities and processes. This makes them useful in
modeling real-world phenomena where a straightforward interpretation is desired.

iv. Ease of Application: Due to their simpler form and consistency with classical calculus
rules, Con-FrD are often easier to apply in various mathematical and engineering
problems. They can be used in solving DEs, modeling systems, and analyzing dynamic
behaviors without requiring complex modifications to existing methods.

v. Flexibility in Modeling: Conformable derivatives provide a flexible framework for
modeling systems that exhibit non-integer order dynamics, such as memory effects or
anomalous diffusion, without the complexity that comes with some other fractional
derivatives. This makes them a practical choice for applications in physics, engineer-
ing, biology, and finance.

vi. Compatibility with Numerical Methods: The simpler and more intuitive nature of
conformable derivatives often leads to easier implementation in numerical methods.
This can be beneficial for computational modeling and simulation, where the goal is
to approximate solutions to FODEs.

The FODEs are an advanced mathematical tool that extends the concept of tradi-
tional DEs to non-integer orders of differentiation and integration. This generalization
provides a more flexible and accurate framework for modeling complex systems that ex-
hibit memory, hereditary properties, or anomalous diffusion, which are not adequately
captured by traditional integer-order models. The power of FODEs lies in their ability to
model processes where the rate of change is not merely dependent on the current state
but also influenced by the history of the system. This is particularly relevant in fields like
physics, engineering, biology, and finance, where systems often have memory effects, such
as viscoelastic materials, anomalous transport phenomena, and financial markets with
long-term dependencies.

In the field of FODEs, several well-known models have been developed to describe
complex systems with memory and hereditary properties. Among these models, the frac-
tional Schrödinger differential equation (FSDEs) stands out as particularly important.



M.I. Liaqat, H.Aljarrah / Eur. J. Pure Appl. Math, 17 (4) (2024), 3464-3491 3467

This equation extends the classical Schrödinger differential equation (SDE) by incorpo-
rating FC, allowing for a more accurate description of quantum systems with anomalous
diffusion and other non-standard behaviors. Typically, the fractional derivative is intro-
duced into either the time or spatial components of the SDE, leading to modifications in
the wave function’s behavior, with potential applications in quantum mechanics and other
fields such as statistical mechanics and condensed matter physics.

FSDEs have diverse applications in various fields of science and engineering, primarily
due to their ability to model systems with memory and non-local interactions. Some of
the key applications include [8, 23]:

i. Quantum Mechanics: FSDEs are used to model quantum systems where the standard
SDEs may not be sufficient, such as systems with fractal geometries, complex poten-
tials, or in scenarios involving anomalous diffusion. They help in understanding the
quantum behavior in disordered or complex materials.

ii. Condensed Matter Physics: In systems with fractal or irregular structures, like cer-
tain types of semiconductors, polymers, or amorphous materials, FSDEs can provide
more accurate descriptions of the electronic properties, enabling better predictions of
material behavior.

iii. Quantum Optics: FSDEs are applied in the study of light-matter interactions, particu-
larly in media with non-standard refractive indices or in systems exhibiting anomalous
dispersion. This can lead to new insights into photon transport and light localization.

iv. Statistical Mechanics: In statistical mechanics, FSDEs describe systems with long-
range interactions or memory effects. This is particularly useful in modeling complex
systems like glasses, spin systems, or in understanding anomalous transport phenom-
ena.

v. Signal Processing and Image Analysis: FSDEs have been employed in signal process-
ing, particularly in the analysis of signals with fractal characteristics or in the context
of time-series analysis. Similarly, they are used in image processing, especially for
edge detection or texture analysis in images with irregular patterns.

vi. Biophysics: In biological systems where anomalous diffusion or memory effects are
present, such as in the transport of molecules within cellular environments or in mod-
eling neuron dynamics, FSDEs can provide more accurate models.

vii. Financial Mathematics: Fractional order models, including those derived from the
SDEs, are used in financial mathematics to describe markets with memory or to
model the dynamics of prices and options in more complex financial environments.

The solutions to FSDEs are crucial for understanding, modeling, and predicting the
behavior of complex systems that exhibit memory effects, non-local interactions, anoma-
lous diffusion, and other phenomena. In the literature, the SDE has been solved using
fractional derivatives represented in the form of the Cap-FD. For example, FSDEs have



M.I. Liaqat, H.Aljarrah / Eur. J. Pure Appl. Math, 17 (4) (2024), 3464-3491 3468

been solved using various methods [7, 11, 13, 22, 25, 29, 36]. However, the algorithms of
each of these methods have been applied in the sense of Caputo derivatives. No research
work has yet solved FSDEs using the Adomian decomposition method (ADM) with the
Sumudu transform (ST) in the sense of Con-FrD. In this research, we addressed this gap
by solving linear and nonlinear FSDEs using a hybrid approach that combines ADM and
ST, applied in the sense of Con-FrD. We refer to it as the Sumudu Adomian decomposition
method (SADM). Moreover, the correctness of SADM is assessed through an examination
of absolute errors (Abs-E) and relative errors (Rel-E) presented in both numerical and
graphical representations. It is observed that the approximate solution (App-S) rapidly
approaches the exact solution (Ex-S), as evidenced by the evaluation of 2D and 3D graphs
across various fractional-order values. The numerical and graphical findings confirm the
notable precision and effectiveness of SADM. In addition, the results obtained using SADM
are compared with other methods, such as the RPSM [37] and HAM [14]. The comparison
shows excellent agreement with these methods, indicating that SADM is a suitable alter-
native to Cap-FD-based approaches for solving FSDEs. Furthermore, we can conclude
that Con-FD is a suitable alternative to Cap-FD in modeling FSDEs.

Our method also has an assumption. To obtain the solution in the original space,
SADM first requires finding the ST of the target equations and then performing the inverse
ST. Consequently, for nonhomogeneous equations, the source functions must be piecewise
continuous and of exponential order, and the inverse ST must exist after computations.

The ST is defined for exponentially ordered functions. We examine functions that are
defined within the set Q.

Q = {δ(φ)|∃P,H1,H2 > 0, |δ(φ) < P exp
|φ|
Hℓ if δ ∈ (−1)ℓ × [0,∞)}.

The constant P for a given function in the set Q must be a finite value.
The ST is defined as follows [10]:

Sµ[δ(φ)] = δ∗(Ω) =
1

Ω

∫ ∞

0
δ(φ) exp

−φµ

µΩ dµφ, Ω ∈ (−H1,H2).

The ADM is a powerful technique for solving DEs, including FODEs. It is particularly
useful for handling non-linear DEs and can be adapted to the fractional-order case. ADM
involves decomposing the solution of a DE into a series of simpler functions, making it
easier to solve the equation iteratively. The main idea is to break down the problem
into manageable components. The ADM holds significant importance in solving FODEs
due to several key advantages and features. FODEs often include non-linear terms that
can be challenging to solve using traditional methods. ADM effectively decomposes these
non-linear terms into simpler components using Adomian polynomials, making it easier to
find solutions even when the equations are highly non-linear. ADM provides a systematic
framework for solving FODEs by breaking down the problem into a series of simpler
problems. This approach allows for an iterative solution where each term in the series is
computed sequentially, making it easier to handle complex DEs. The method is flexible
and can be applied to a wide range of FODEs, including those with complex boundary



M.I. Liaqat, H.Aljarrah / Eur. J. Pure Appl. Math, 17 (4) (2024), 3464-3491 3469

conditions and variable coefficients. This adaptability makes ADM suitable for a diverse
set of problems across different scientific and engineering disciplines.

We take into account the following: nonlinear SDE with respect to a conformable
operator and a non-zero trapping potential that is time-fractional.

ιTµ
φδ(ϱ, φ) + ζδϱϱ(ϱ, φ) + ℶ(ϱ)δ(ϱ, φ) + ℵ|δ(ϱ, φ)|2δ(ϱ, φ) = 0, (4)

with the initial condition (I-C):
δ(ϱ, 0) = δ0(ϱ), (5)

here ζ,ℵ ∈ R, 0 < µ ≤ 1, ι2 = −1; Tµ
φ is Con-FrD; δ(ϱ, φ) complex-valued function, ϱ ∈ R;

φ ≥ 0; ℶ(ϱ) represents trapping potential; |δ(ϱ, φ)| is the modulus of δ(ϱ, φ).
The subsequent sections are structured as follows: Section 2 presents the main char-

acteristics and results that form the foundation of our study. Section 3 provides the
algorithm of the SADM. In Section 4, we solve three types of FSDEs using the SADM. In
Section 5, we discuss the results obtained in Section 4 through graphical and numerical
outcomes and analyze the correctness of our approach. Finally, Section 6 presents the
conclusion.

2. Preliminaries

In this section, we go over some useful characteristics of Con-FrD and ST that will be
useful in this paper.

Theorem 1. [2] Let 0 < µ ≤ 1, A1(ϱ, φ) and A2(ϱ, φ) be µ-differentiable at a point φ > 0.
Then,

i. Tµ
φ(ν1A1(ϱ, φ) + ν2A2(ϱ, φ)) = ν1T

µ
φA1(ϱ, φ) + ν2T

µ
φA2(ϱ, φ), ∀ ν1, ν2 ∈ R.

ii. Tµ
φ(φν) = νφν−µ, ∀ ν ∈ R.

iii. Tµ
φ(ν) = 0, ν ∈ R.

iv. Tµ
φA1(ϱ, φ)A2(ϱ, φ)) = A1(ϱ, φ)T

µ
φA2(ϱ, φ) + A2(ϱ, φ)T

µ
φA1(ϱ, φ).

v. Tµ
φ
A1(ϱ, φ)

A2(ϱ, φ)
=

A2(ϱ, φ)T
µ
φA1(ϱ, φ)− A1(ϱ, φ)T

µ
φA2(φ)

A2
2(φ)

.

Lemma 1. [3] Assume that A1(ϱ, φ) and A2(ϱ, φ) satisfies the axioms of existence of
ST, Sµ[A1(ϱ, φ)] = A∗

1(ϱ,Ω),Sµ[A2(ϱ, φ)] = A∗
2(ϱ,Ω) and B1,B2 are constants. Then, the

following axioms hold:

i. Sµ[B1A1(ϱ, φ) +B2A2(ϱ, φ)] = B1A∗
1(ϱ,Ω) +B2A∗

2(ϱ,Ω).

ii. S−1
µ [B1A∗

1(ϱ,Ω) +B2A∗
2(ϱ,Ω)] = B1A1(ϱ, φ) +B2A2(ϱ, φ).

iii. Sµ[Tµ
φA(ϱ, φ)] = S[A(ϱ,φ)]

Ω − A(ϱ,0)
Ω .

iv. Sµ[Tnµ
φ A(ϱ, φ)] = S[A(ϱ,φ)]

Ωn − A(ϱ,0)
Ωn .



M.I. Liaqat, H.Aljarrah / Eur. J. Pure Appl. Math, 17 (4) (2024), 3464-3491 3470

3. Analysis of the SADM

This section presents the main steps for solving FSDEs using ST and ADM. We apply
these steps to solve FSDEs in the standard structure. For this, take δ(ϱ, φ) = δ1(ϱ, φ) +
ιδ2(ϱ, φ) in Eq. (4) and we get:

Tµ
φδ1(ϱ, φ) =− ζDϱϱδ2(ϱ, φ)− ℶ(ϱ)δ2(ϱ, φ)− ℵ

(
δ21(ϱ, φ)δ2(ϱ, φ) + δ32(ϱ, φ)

)
,

Tµ
φδ2(ϱ, φ) =ζDϱϱδ1(ϱ, φ) + ℶ(ϱ)δ1(ϱ, φ) + ℵ

(
δ22(ϱ, φ)δ1(ϱ, φ) + δ31(ϱ, φ)

)
, (6)

with I-Cs:
δ1(ϱ, 0) = δ1,0(ϱ), δ2(ϱ, 0) = δ2,0(ϱ), (7)

here, δ(ϱ, 0) = δ1,0(ϱ) + ιδ2,0(ϱ).
By applying Sµ on the above system.

Sµ[Tµ
φδ1(ϱ, φ)] =− Sµ

[
ζDϱϱδ2(ϱ, φ) + ℶ(ϱ)δ2(ϱ, φ)+

ℵ
(
δ21(ϱ, φ)δ2(ϱ, φ) + δ32(ϱ, φ)

)]
,

Sµ[Tµ
φδ2(ϱ, φ)] =Sµ

[
ζDϱϱδ1(ϱ, φ) + ℶ(ϱ)δ1(ϱ, φ)+

ℵ
(
δ22(ϱ, φ)δ1(ϱ, φ) + δ31(ϱ, φ)

)]
. (8)

Through Lemma 1(iii) and performing various computations, we acquire the following:

Sµ[δ1(ϱ, φ)] =δ1(0)− ΩSµ
[
ζDϱϱδ2(ϱ, φ)

]
− ΩSµ

[
ℶ(ϱ)δ2(ϱ, φ)

]
− ΩSµ

[
ℵ
(
δ21(ϱ, φ)δ2(ϱ, φ) + δ32(ϱ, φ)

)]
,

Sµ[δ2(ϱ, φ)] =δ2(0) + ΩSµ
[
ζDϱϱδ1(ϱ, φ)

]
+ΩSµ

[
ℶ(ϱ)δ1(ϱ, φ)

]
+ΩSµ

[
ℵ
(
δ22(ϱ, φ)δ1(ϱ, φ) + δ31(ϱ, φ)

)]
. (9)

Implementing the S−1
µ on the system mentioned above:

δ1(ϱ, φ) =S−1
µ

[
δ1(0)

]
− S−1

µ

[
ΩSµ

[
ζDϱϱδ2(ϱ, φ)

]]
− S−1

µ

[
ΩSµ

[
ℶ(ϱ)δ2(ϱ, φ)

]]
− S−1

µ

[
ΩSµ

[
ℵ
(
δ21(ϱ, φ)δ2(ϱ, φ) + δ32(ϱ, φ)

)]]
,

δ2(ϱ, φ) =S−1
µ

[
δ2(0)

]
+ S−1

µ

[
ΩSµ

[
ζDϱϱδ1(ϱ, φ)

]]
+ S−1

µ

[
ΩSµ

[
ℶ(ϱ)δ1(ϱ, φ)

]]
+ S−1

µ

[
ΩSµ

[
ℵ
(
δ22(ϱ, φ)δ1(ϱ, φ) + δ31(ϱ, φ)

))]
. (10)

Consider the solution of Eq. (6) in the following fractional power series (FPS) using the
idea of ADM.

δ1(ϱ, φ) =

∞∑
ρ=0

δ1,ρ(ϱ, φ),

δ2(ϱ, φ) =

∞∑
ρ=0

δ2,ρ(ϱ, φ). (11)



M.I. Liaqat, H.Aljarrah / Eur. J. Pure Appl. Math, 17 (4) (2024), 3464-3491 3471

The nonlinear terms Z1(δ1, δ2) and Z2(δ1, δ2) represented as

Z1(δ1, δ2) =

∞∑
ρ=0

A1,ρ(δ1, δ2),

Z2(δ1, δ2) =
∞∑
ρ=0

A2,ρ(δ1, δ2), (12)

here, A1,ρ(δ1, δ2) and A2,ρ(δ1, δ2) are the Adomian polynomials (A-Ps) for the nonlinear
term.

A1,ρ(δ1, δ2) =
1

Γ(ρ+ 1)

∂ρ

∂χρ

[
Z1

( ρ∑
σ=0

χσδ1,σ(ϱ, φ)
)]

χ=0
, ρ = 0, 1, 2, . . . ,

A2,ρ(δ1, δ2) =
1

Γ(ρ+ 1)

∂ρ

∂χρ

[
Z2

( ρ∑
σ=0

χσδ2,σ(ϱ, φ)
)]

χ=0
, ρ = 0, 1, 2, . . . . (13)

From Eqs. (11), (12), and (10).

∞∑
ρ=0

δ1,ρ(ϱ, φ) =S−1
µ

[
δ1(0)

]
− S−1

µ

[
ΩSµ

[
ζDϱϱ

∞∑
ρ=0

δ2,ρ(ϱ, φ)
]]
−

ℵ−1
ϖ

[
Ωℵϖ

[
ℶ(ϱ)

∞∑
ρ=0

δ2,ρ(ϱ, φ)
]]

− S−1
µ

[
ΩSµ

[
ℵ

∞∑
ρ=0

A1,ρ(δ1, δ2)
]
,

∞∑
ρ=0

δ2,ρ(ϱ, φ) =S−1
µ

[
δ2(0)

]
+ S−1

µ

[
ΩSµ

[
ζDϱϱ

∞∑
ρ=0

δ1,ρ(ϱ, φ)
]]
+

S−1
µ

[
ΩSµ

[
ℶ(ϱ)

∞∑
ρ=0

δ1,ρ(ϱ, φ)
]]

+ S−1
µ

[
ΩSµ

[
ℵ

∞∑
ρ=0

A2,ρ(δ1, δ2)
]
. (14)

From Eq. (14), we have as

δ1,0(ϱ, φ) = S−1
µ

[
δ1(0)

]
,

δ2,0(ϱ, φ) = S−1
µ

[
δ2(0)

]
. (15)

The 2nd term of the FPS is as

δ1,1(ϱ, φ) =− S−1
µ

[
ΩSµ

[
ζDϱϱδ2,0(ϱ, φ)

]]
− S−1

µ

[
ΩSµ

[
ℶ(ϱ)δ2,0(ϱ, φ)

]]
−

S−1
µ

[
ΩSµ

[
ℵA1,0(δ1, δ2)

]
,

δ2,1(ϱ, φ) =S−1
µ

[
ΩSµ

[
ζDϱϱδ1,0(ϱ, φ)

]]
+ S−1

µ

[
ΩSµ

[
ℶ(ϱ)δ1,0(ϱ, φ)

]]
+

S−1
µ

[
ΩSµ

[
ℵA2,0(δ1, δ2)

]
. (16)

The following are the 3rd and 4th terms of the FPS:

δ1,2(ϱ, φ) =− S−1
µ

[
ΩSϱ

[
ζDϱϱδ2,1(ϱ, φ)

]]
− S−1

µ

[
ΩSµ

[
ℶ(ϱ)δ2,1(ϱ, φ)

]]
−



M.I. Liaqat, H.Aljarrah / Eur. J. Pure Appl. Math, 17 (4) (2024), 3464-3491 3472

S−1
µ

[
ΩSµ

[
ℵA1,1(δ1, δ2)

]
,

δ2,2(ϱ, φ) =S−1
µ

[
ΩSµ

[
ζDϱϱδ1,1(ϱ, φ)

]]
+ S−1

µ

[
ΩSµ

[
ℶ(ϱ)δ1,1(ϱ, φ)

]]
+

S−1
µ

[
ΩSµ

[
ℵA2,1(δ1, δ2)

]
. (17)

δ1,3(ϱ, φ) =− S−1
µ

[
ΩSµ

[
ζDϱϱδ2,2(ϱ, φ)

]]
− S−1

µ

[
ΩSµ

[
ℶ(ϱ)δ2,2(ϱ, φ)

]]
−

S−1
µ

[
ΩSµ

[
ℵA1,2(δ1, δ2)

]
,

δ2,3(ϱ, φ) =S−1
µ

[
ΩSµ

[
ζDϱϱδ1,2(ϱ, φ)

]]
+ S−1

µ

[
ΩSµ

[
ℶ(ϱ)δ1,2(ϱ, φ)

]]
+

S−1
µ

[
ΩSµ

[
ℵA2,2(δ1, δ2)

]
. (18)

By making generalizing we get the following:

δ1,ρ+1(ϱ, φ) =− S−1
µ

[
ΩSµ

[
ζDϱϱδ2,ρ(ϱ, φ)

]]
− S−1

µ

[
ΩSµ

[
ℶ(ϱ)δ2,ρ(ϱ, φ)

]]
−

S−1
µ

[
ΩSµ

[
ℵA1,ρ(δ1, δ2)

]
,

δ2,ρ+1(ϱ, φ) =S−1
µ

[
ΩSµ

[
ζDϱϱδ1,ρ(ϱ, φ)

]]
+ S−1

µ

[
ΩSµ

[
ℶ(ϱ)δ1,ρ(ϱ, φ)

]]
+

S−1
µ

[
ΩSµ

[
ℵA2,ρ(δ1, δ2)

]
. (19)

Lastly, the FPS is obtained in the following ways:

δ1(ϱ, φ) = lim
σ→∞

σ∑
ρ=0

δ1,ρ(ϱ, φ),

δ2(ϱ, φ) = lim
σ→∞

σ∑
ρ=0

δ2,ρ(ϱ, φ). (20)

4. Numerical Problems

Using the method outlined in the preceding section, we find the App-Ss and Ex-Ss of
linear and nonlinear FSDEs in this section. The solutions to the Schrödinger equations
encompass a wealth of information regarding the probabilistic nature of quantum sys-
tems, their energy states, and their dynamic behavior. This information is foundational
for understanding the principles of quantum mechanics and their applications in various
fields, including quantum chemistry, condensed matter physics, and quantum information
science.
Problem 4.1. Examine the subsequent linear FSDEs:

ιTµ
φδ(ϱ, φ) + Dϱϱδ(ϱ, φ) = 0, φ ≥ 0, 0 < µ ≤ 1, (21)

with the I-C:
δ(ϱ, 0) = 1 + cosh(2ϱ), ϱ ∈ R.

By utilizing δ(ϱ, φ) = δ1(ϱ, φ) + ιδ2(ϱ, φ) and δ(ϱ, 0) = δ1,0(ϱ) + ιδ2,0(ϱ), we have the
following:

Tµ
φδ1(ϱ, φ) =Dϱϱδ2(ϱ, φ),



M.I. Liaqat, H.Aljarrah / Eur. J. Pure Appl. Math, 17 (4) (2024), 3464-3491 3473

Tµ
φδ2(ϱ, φ) =− Dϱϱδ1(ϱ, φ), (22)

with I-Cs:

δ1,0(ϱ, 0) =1 + cosh(2ϱ),

δ2,0(ϱ, 0) =0. (23)

By applying Sµ to the above system and implementing some calculations, we get the
subsequent:

Sµ[δ1(ϱ, φ)] = δ1(0) + ΩSµ
[
Dϱϱδ2(ϱ, φ)

]
,

Sµ[δ2(ϱ, φ)] = δ2(0)− ΩSµ
[
Dϱϱδ1(ϱ, φ)

]
. (24)

Taking S−1
µ .

δ1(ϱ, φ) =S−1
µ

[
δ1(0)

]
+ S−1

µ

[
ΩSµ

[
Dϱϱδ2(ϱ, φ)

]]
,

δ2(ϱ, φ) =S−1
µ

[
δ2(0)

]
− S−1

µ

[
ΩSµ

[
Dϱϱδ1(ϱ, φ)

]]
. (25)

We derive the following result:

∞∑
ρ=0

δ1,ρ(ϱ, φ) =S−1
µ

[
δ1(0)

]
+ S−1

µ

[
ΩSµ

[
Dϱϱ

∞∑
ρ=0

δ2,ρ(ϱ, φ)
]]
,

∞∑
ρ=0

δ2,ρ(ϱ, φ) =S−1
µ

[
δ2(0)

]
− S−1

µ

[
ΩSµ

[
Dϱϱ

∞∑
ρ=0

δ1,ρ(ϱ, φ)
]]
. (26)

We extract the following from the above:

δ1,0(ϱ, φ) =S−1
µ

[
δ1,0(ϱ, 0)

]
,

δ2,0(ϱ, φ) =S−1
µ

[
δ2,0(ϱ, 0)

]
. (27)

Below is the first term:

δ1,0(ϱ, φ) =1 + cosh(2ϱ),

δ2,0(ϱ, φ) =0. (28)

The next term expressions are as follows:

δ1,1(ϱ, φ) =S−1
µ

[
ΩSµ

[
Dϱϱδ2,0(ϱ, φ)

]]
,

δ2,1(ϱ, φ) =S−1
µ

[
ΩSµ

[
Dϱϱδ1,0(ϱ, φ)

]]
. (29)

From the above system, we have

δ1,1(ϱ, φ) =S−1
µ

[
ΩSµ

[
Dϱϱ(0)

]]
,

δ2,1(ϱ, φ) =S−1
µ

[
ΩSµ

[
Dϱϱ(1 + cosh(2ϱ))

]]
. (30)



M.I. Liaqat, H.Aljarrah / Eur. J. Pure Appl. Math, 17 (4) (2024), 3464-3491 3474

As a result,

δ1,1(ϱ, φ) =0,

δ2,1(ϱ, φ) =− 4

Γ(2)
cosh(2ϱ)

φµ

µ
. (31)

The third-term expressions are as follows:

δ1,2(ϱ, φ) =S−1
µ

[
ΩSµ

[
Dϱϱδ2,1(ϱ, φ)

]]
,

δ2,2(ϱ, φ) =S−1
µ

[
ΩSµ

[
Dϱϱδ1,1(ϱ, φ)

]]
. (32)

We have the following from above:

δ1,2(ϱ, φ) =S−1
µ

[
ΩSµ

[
Dϱϱ

( 4

Γ(2)
cosh(2ϱ

)φµ

µ
)
]]
,

δ2,2(ϱ, φ) =S−1
µ

[
ΩSµ

[
Dϱϱ(0)

]]
. (33)

The third term is as follows:

δ1,2(ϱ, φ) =− 8

Γ(3)
cosh(2ϱ)

φ2µ

µ2
,

δ2,2(ϱ, φ) =0. (34)

The next terms are as follows:

δ1,3(ϱ, φ) =0,

δ2,3(ϱ, φ) =
64

Γ(4)
cosh(2ϱ)

φ3µ

µ3
. (35)

δ1,4(ϱ, φ) =
256

Γ(5)
cosh(2ϱ)

φ4µ

µ4
,

δ2,4(ϱ, φ) =0. (36)

The procedure is repeated to achieve the results for the sixth term.

δ1,5(ϱ, φ) =0,

δ2,5(ϱ, φ) =− 1024

Γ(6)
cosh(2ϱ)

φ5µ

µ5
. (37)

As a result,

δ(ϱ, φ) = 1 + cosh(2ϱ)

( ∞∑
ρ=0

(−1)ρ

Γ(2ρ+ 1)

(4φµ

µ

)2ρ − ι
∞∑
ρ=0

(−1)ρ

Γ(2ρ+ 2)

(4φµ

µ

)2ρ+1
)
. (38)



M.I. Liaqat, H.Aljarrah / Eur. J. Pure Appl. Math, 17 (4) (2024), 3464-3491 3475

The Ex-S when µ = 1.0 is 1 + cosh(2ϱ) exp−4ιφ. A similar result was obtained by [10].
Problem 4.2. Take into consideration the subsequent nonlinear FSDE:

ιTµ
φδ(ϱ, φ) + δϱϱ(ϱ, φ) + 2|δ(ϱ, φ)|2δ(ϱ, φ) = 0, (39)

with the I-C:
δ(ϱ, 0) = expiϱ, ϱ ∈ R. (40)

Suppose δ(ϱ, φ) = δ1(ϱ, φ) + ιδ2(ϱ, φ) so that δ(ϱ, 0) = δ1,0(ϱ) + ιδ2,0(ϱ). As a result,

Tµ
φδ1(ϱ, φ) = −Dϱϱδ2(ϱ, φ)− 2

(
δ21(ϱ, φ)δ2(ϱ, φ) + δ32(ϱ, φ)

)
,

Tµ
τ δ2(ϱ, φ) = Dϱϱδ1(ϱ, φ) + 2

(
δ22(ϱ, φ)δ1(ϱ, φ) + δ31(ϱ, φ)

)
. (41)

with the I.Cs:

δ1(ϱ, 0) = cos(ϱ),

δ2(ϱ, 0) = sin(ϱ), (42)

Applying ST to Eq. (41), using the linear property of ST and Lemma 1(iii) and making
some calculations, we get

Sµ[δ1(ϱ, φ)] =δ1(0)− ΩSµ
[
Dϱϱδ2(ϱ, φ)

]
− ΩSµ

[
2
(
δ21(ϱ, φ)δ2(ϱ, φ) + δ32(ϱ, φ)

)]
,

Sµ[δ2(ϱ, φ)] =δ2(0) + ΩSµ
[
Dϱϱδ1(ϱ, φ)

]
+ΩSµ

[
2
(
δ22(ϱ, φ)δ1(ϱ, φ) + δ31(ϱ, φ)

)]
. (43)

Through S−1
µ on the system mentioned before.

δ1(υ, τ) =S−1
µ

[
δ1(0)

]
− S−1

µ

[
ΩSµ

[
Dυυδ2(ϱ, φ)

]]
− S−1

µ

[
ΩSµ

[
2
(
δ21(ϱ, φ)δ2(ϱ, φ) + δ32(ϱ, φ)

)]]
,

δ2(ϱ, φ) =S−1
µ

[
δ2(0)

]
+ S−1

µ

[
ΩSµ

[
Dϱϱδ1(ϱ, φ)

]]
+ S−1

µ

[
ΩSµ

[
2
(
δ22(ϱ, φ)δ1(ϱ, φ) + δ31(ϱ, φ)

)]
. (44)

We get the following:

∞∑
ρ=0

δ1,ρ(ϱ, φ) =S−1
µ

[
δ1(0)

]
− S−1

µ

[
ΩSµ

[
Dϱϱ

∞∑
ρ=0

δ2,ρ(ϱ, φ)
]]
−

S−1
µ

[
ΩSµ

[
2

∞∑
ρ=0

A1,ρ(δ1, δ2)
]
,

∞∑
ρ=0

δ2,ρ(ϱ, φ) =S−1
µ

[
δ2(0)

]
+ S−1

µ

[
ΩSµ

[
Dϱϱ

∞∑
ρ=0

δ1,ρ(ϱ, φ)
]]
+



M.I. Liaqat, H.Aljarrah / Eur. J. Pure Appl. Math, 17 (4) (2024), 3464-3491 3476

S−1
µ

[
ΩSµ

[
2

∞∑
ρ=0

A2,ρ(δ1, δ2)
]
. (45)

From the above system, we obtain the first terms of the FPS solutions.

δ1,0(ϱ, φ) = cos(ϱ),

δ2,0(ϱ, φ) = sin(ϱ). (46)

We extract the following terms of the FPS as follows:

δ1,1(ϱ, φ) = − sin(ϱ)
φµ

µΓ(2)
,

δ2,1(ϱ, φ) = cos(ϱ)
φµ

µΓ(2)
. (47)

δ1,2(ϱ, φ) = − φ2µ

µ22!
cos(ϱ),

δ2,2(ϱ, φ) = − sin(ϱ)
φ2µ

µ2Γ(3)
. (48)

δ1,3(ϱ, φ) = sin(ϱ)
φ3µ

µ3Γ(4)
,

δ2,3(ϱ, φ) = − cos(ϱ)
φ3µ

µ3Γ(4)
. (49)

δ1,4(ϱ, φ) = sin(ϱ)
φ3µ

µ3Γ(5)
,

δ2,4(ϱ, φ) = sin(ϱ)
φ3µ

µ3Γ(5)
. (50)

The procedure is repeated to achieve the results for the sixth terms.

δ1,5(ϱ, φ) = − sin(ϱ)
φ5µ

µ5Γ(6)
,

δ2,5(ϱ, φ) = cos(ϱ)
φ5µ

µ5Γ(6)
. (51)

The FPS solution is as follows:

δ(ϱ, φ) = (cos(ϱ) + ι sin(ϱ))

( ∞∑
ρ=0

1

Γ(ρ+ 1)

( ιφµ

µ

)ρ)
. (52)



M.I. Liaqat, H.Aljarrah / Eur. J. Pure Appl. Math, 17 (4) (2024), 3464-3491 3477

The Ex-S when µ = 1 is δ(ϱ, φ) = expι
(
ϱ+φ

)
. The same solution was achieved by [2].

Problem 4.3. Consider the following nonlinear FSDE that follows:

ιTµ
φδ(ϱ, φ) +

1

2
δϱϱ(ϱ, φ)− cos2(ϱ)δ(ϱ, φ)− |δ(ϱ, φ)|2δ(ϱ, φ) = 0, (53)

with the I-C:
δ(ϱ, 0) = sin(ϱ), ϱ ∈ R. (54)

Suppose δ(ϱ, φ) = δ1(ϱ, φ) + ιδ2(ϱ, φ) then δ(ϱ, 0) = δ1,0(ϱ) + ιδ2,0(ϱ). As a result,

Tµ
φδ1(ϱ, φ) =− 1

2
Dϱϱδ2(ϱ, φ) + cos2(ϱ)δ2(ϱ, φ) +

(
δ21(ϱ, φ)δ2(ϱ, φ) + δ32(ϱ, φ)

)
,

Tµ
φδ2(ϱ, φ) =

1

2
Dϱϱδ1(ϱ, φ)− cos2(ϱ)δ1(ϱ, φ)−

(
δ22(ϱ, φ)δ1(ϱ, φ) + δ31(ϱ, φ)

)
, (55)

with the I-Cs:

δ1(ϱ, 0) = sin(ϱ),

δ2(ϱ, 0) =0. (56)

After applying ST to Eq. (55), utilizing Lemma 1(iii) and the linear property of ST,
and performing certain computations, we obtain

S[δ1(ϱ, φ)] =δ1(0)− ΩSµ
[1
2
Dϱϱδ2(ϱ, φ)

]
+ΩSµ

[
cos2(ϱ)δ2(ϱ, φ)

]
+ΩSµ

[(
δ21(ϱ, φ)δ2(ϱ, φ) + δ32(ϱ, φ)

)]
,

Sµ[δ2(ϱ, φ)] =δ2(0) + ΩSµ
[1
2
Dϱϱδ1(ϱ, φ)

]
− ΩSµ

[
cos2(ϱ)δ1(ϱ, φ)

]
− ΩSµ

[(
δ22(ϱ, φ)δ1(ϱ, φ) + δ31(ϱ, φ)

)]
. (57)

Assess S−1
µ in the system mentioned earlier.

δ1(ϱ, φ) =S−1
µ

[
δ1(0)

]
− S−1

µ

[
ΩS

[1
2
Dϱϱδ2(ϱ, φ)

]]
+ S−1

[
ΩS

[
cos2(ϱ)δ2(ϱ, φ)

]]
+ S−1

µ

[
ΩSµ

[(
δ21(ϱ, φ)δ2(ϱ, φ) + δ32(ϱ, φ)

)]]
,

δ2(ϱ, φ) =S−1
µ

[
δ2(0)

]
+ S−1

µ

[
ΩSµ

[1
2
Dϱϱδ1(ϱ, φ)

]]
− S−1

µ

[
ΩSµ

[
cos2(ϱ)δ1(ϱ, φ)

]]
− S−1

µ

[
ΩS

[(
δ22(ϱ, φ)δ1(ϱ, φ) + δ31(ϱ, φ)

)]]
. (58)

Using the method outlined in Section 3, we derive the following result from Eq. (58):

∞∑
ρ=0

δ1,ρ(ϱ, φ) =S−1
µ

[
δ1(0)

]
− S−1

µ

[
ΩSµ

[1
2
Dϱϱ

∞∑
ρ=0

δ2,ρ(ϱ, φ)
]]
+

S−1
µ

[
ΩSµ

[
cos2(ϱ)

∞∑
ρ=0

δ2,ρ(ϱ, φ)
]]

+ S−1
µ

[
ΩSµ

[ ∞∑
ρ=0

A1,ρ(δ1, δ2)
]
,



M.I. Liaqat, H.Aljarrah / Eur. J. Pure Appl. Math, 17 (4) (2024), 3464-3491 3478

∞∑
ρ=0

δ2,ρ(ϱ, φ) =S−1
µ

[
δ2(0)

]
+ S−1

µ

[
ΩS

[1
2
Dϱϱ

∞∑
ρ=0

δ1,ρ(ϱ, φ)
]]
−

S−1
µ

[
ΩSµ

[
cos2(ϱ)

∞∑
ρ=0

δ1,ρ(ϱ, φ)
]]

− S−1
µ

[
ΩSµ

[ ∞∑
ρ=0

A2,ρ(δ1, δ2)
]
. (59)

We extracted the first term of the FPS solution for the Eq. (55) via the equivalent on the
two ends of Eq. (59).

δ1,0(ϱ, φ) = sin(ϱ),

δ2,0(ϱ, φ) = 0. (60)

We extract the following second term of the FPS.

δ1,1(ϱ, φ) = 0,

δ2,1(ϱ, φ) = −3

2
sin(ϱ)

φµ

µΓ(2)
. (61)

Similarly, we found the third, fourth, and fifth terms.

δ1,2(ϱ, φ) = −9

4
sin(ϱ)

φ2µ

µ2Γ(3)
,

δ2,2(ϱ, φ) = 0. (62)

δ1,3(ϱ, φ) = 0,

δ2,3(ϱ, φ) =
27

8
sin(ϱ)

φ3µ

µ3Γ(4)
. (63)

δ1,4(ϱ, φ) =
81

16
sin(ϱ)

φ4µ

µ4Γ(5)
,

δ2,4(ϱ, φ) = 0. (64)

The procedure is repeated to achieve the results for the sixth term.

δ1,5(ϱ, φ) = 0,

δ2,5(ϱ, φ) = −243

32
sin(ϱ)

φ5µ

µ5Γ(6)
. (65)

Hence, the system described in Eq. (55) has an FPS solution, which is outlined below.

δ(ϱ, φ) = sin(ϱ)

( ∞∑
ρ=0

(−1)ρ

Γ(2ρ+ 1)
(
3φµ

2µ
)2ρ − ι

∞∑
ρ=0

(−1)ρ

Γ(2ρ+ 2)
(
3φµ

2µ
)2ρ+1

)
. (66)

The Ex-S when µ = 1.0 is δ(ϱ, φ) = sin(ϱ) exp−
3ιφ
2 . An identical result was obtained by

[10].



M.I. Liaqat, H.Aljarrah / Eur. J. Pure Appl. Math, 17 (4) (2024), 3464-3491 3479

5. Graphical and Numerical Outcome

In this section, we analyze the numerical and graphical results of the Ex-Ss and App-
Ss for the linear and nonlinear problems presented in the fourth section of this research
study. To assess the SADM’s accuracy, we use two error functions, the Abs-E and Rel-
E functions. Delineating the errors in the App-Ss is essential since SADM provides an
approximation expressed in terms of an infinite FPS.

First, we present some useful notation for Ex-S and App-S, along with formulas for
error functions, which we utilize in this section to analyze the reliability and correctness
of our approach.

δ(ϱ, φ) ≈δκ(ϱ, φ), κ = 1, 2, 3, · · · ,

where δ(ϱ, φ) and δκ(ϱ, φ) denote the Ex-S and App-S of the nonlinear problems 4.2 and
4.3 obtained by SADM.

Abs-E plays a crucial role in quantifying the accuracy of these App-Ss and is in-
dispensable for validating and refining numerical techniques. The Abs-E represents the
magnitude of the absolute difference between the App-S (δκ(ϱ, φ)) obtained through ap-
proximate methods and the Ex-S (δ(ϱ, φ)), if available. Smaller Abs-E values indicate
higher accuracy in the approximation. The Abs-E is defined as follows:

Abs.Eκ(ϱ, φ) =|δ(ϱ, φ)− δκ(ϱ, φ)|, κ = 1, 2, 3, · · · ,

when κ increases to infinity, it frequently happens that Abs.Eκ(ϱ, φ) gets decreasing,
eventually decreasing almost to zero.

The Rel-E serves as a powerful tool for evaluating the effectiveness of the approach
that generates App-Ss. It is calculated as the ratio of the Abs-E to the magnitude of the
Ex-S at each point within the solution domain. This provides valuable insights into the
degree of alignment between the App-S and the behavior of the Ex-S. A smaller Rel-E
indicates higher accuracy in the approximation. Mathematically, it is defined as follows:

Rel.Eκ(ϱ, φ) =
|δ(ϱ, φ)− δκ(ϱ, φ)|

|δ(ϱ, φ)|
, κ = 1, 2, 3, · · · ,

where the Rel-E for the κth-step App-S is represented by Rel.Eκ(ϱ, φ) for the Ex-S
(δ(ϱ, φ)). In fact, it often happens that as κ goes to infinity, Rel.Eκ(ϱ, φ) gets ever smaller
until it almost reaches zero.

In Figures 1-4 for Problems 4.2 and 4.3, 2D curves are employed to compare the App-Ss
and Ex-Ss in terms of Rel-E and Abs-E. The comparative analysis reveals a high degree
of similarity between the fifth-step App-Ss and the Ex-Ss. The Abs-E is presented on the
graphs to demonstrate the excellent precision of SADM.

The 2D graphs of the App-Ss obtained from five iterations and the Ex-Ss derived by
SADM for µ = 0.6, 0.7, 0.8, 0.9, and 1.0 are depicted in Figures 5 and 6 for Problems 4.2
and 4.3. These graphs illustrate how, as µ → 1.0, the App-Ss converge to the Ex-Ss. The
interaction between the App-Ss and Ex-Ss when µ = 1.0 demonstrates the accuracy of the
proposed approach.



M.I. Liaqat, H.Aljarrah / Eur. J. Pure Appl. Math, 17 (4) (2024), 3464-3491 3480

Figures 7-14 for Problems 4.2 and 4.3 display the 3D graphs of the App-Ss obtained
from five iterations and the Ex-Ss determined by SADM for µ = 0.7, 0.8, 0.9, 1.0 and Ex-
Ss. These graphs demonstrate how the App-Ss converge to the Ex-Ss as µ approaches 1.0.
The interaction between the App-Ss and Ex-Ss when µ = 1.0 illustrates the accuracy of
the proposed method.

The Abs-E and Rel-E for specified locations between the Ex-Ss and fifth-order App-Ss
derived by SADM in Problems 4.2 and 4.3 at µ = 1.0 are presented in Tables 1 and 2.
These tables demonstrate that the App-Ss and Ex-Ss are nearly in agreement, confirming
the accuracy of SADM. From these tables, it is observed that the Abs-E and Rel-E for
all problems in the fifth-step App-Ss is very small. The findings presented in this section,
depicted in both graphs and tables, demonstrate that SADM is a useful and effective
technique for solving FSDEs, requiring fewer calculations and iterations.

0.0 0.1 0.2 0.3 0.4 0.5

0.000000

5.×10-6

0.000010

0.000015

φ

R
el
.E
r

(a)

0.0 0.1 0.2 0.3 0.4 0.5

0

1.×10-6

2.×10-6

3.×10-6

4.×10-6

φ

R
el
.E
r

(b)

Figure 1: 2D graphs of Rel-E in the range φ ∈ [0, 0.5] comparing the fifth-step App-S and Ex-S for ϱ = 0.1
with µ = 1.0 in problem 4.2: (a) δ1(ϱ, φ); (b) δ2(ϱ, φ).

0.0 0.1 0.2 0.3 0.4 0.5

0

5.×10-12

1.×10-11

1.5 ×10-11

2.×10-11

φ

R
el
.E
r

(a)

0.0 0.1 0.2 0.3 0.4 0.5

0

2.×10-13

4.×10-13

6.×10-13

8.×10-13

1.×10-12

1.2 ×10-12

φ

R
el
.E
r

(b)

Figure 2: 2D graphs of Rel-E in the range φ ∈ [0, 0.5] comparing the fifth-step App-S and Ex-S for ϱ = 0.1
with µ = 1.0 in problem 4.3: (a) δ1(ϱ, φ); (b) δ2(ϱ, φ).



M.I. Liaqat, H.Aljarrah / Eur. J. Pure Appl. Math, 17 (4) (2024), 3464-3491 3481

0.0 0.1 0.2 0.3 0.4 0.5

0.000000

2.×10-6

4.×10-6

6.×10-6

8.×10-6

0.000010

0.000012

0.000014

φ

A
bs
.E
r

(a)

0.0 0.1 0.2 0.3 0.4 0.5

0

5.×10-7

1.×10-6

1.5 ×10-6

2.×10-6

φ

A
bs
.E
r

(b)

Figure 3: 2D graphs of Abs-E in the range φ ∈ [0, 0.5] comparing the fifth-step App-S and Ex-S for ϱ = 0.1
with µ = 1.0 in problem 4.2: (a) δ1(ϱ, φ); (b) δ2(ϱ, φ).

0.0 0.1 0.2 0.3 0.4 0.5

0

2.×10-13

4.×10-13

6.×10-13

8.×10-13

φ

A
bs
.E
r

(a)

0.0 0.1 0.2 0.3 0.4 0.5

0

1.×10-14

2.×10-14

3.×10-14

φ

A
bs
.E
r

(b)

Figure 4: 2D graphs of Abs-E in the range φ ∈ [0, 0.5] comparing the fifth-step App-S and Ex-S for ϱ = 0.1
with µ = 1.0 in problem 4.3: (a) δ1(ϱ, φ); (b) δ2(ϱ, φ).

0.0 0.2 0.4 0.6 0.8 1.0
-0.2

0.0

0.2

0.4

0.6

0.8

1.0

φ

δ
1

μ=0.6 μ=0.7 μ=0.8

μ=0.9 μ=1.0 Exact

(a)

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

φ

δ
2

μ=0.6 μ=0.7 μ=0.8

μ=0.9 μ=1.0 Exact

(b)

Figure 5: For problem 4.2, the 2D diagrams of App-S for various levels of µ and Ex-S in the range φ ∈ [0, 1.0]
at ϱ = 0.1 are shown for: (a) δ1(ϱ, φ); (b) δ2(ϱ, φ).



M.I. Liaqat, H.Aljarrah / Eur. J. Pure Appl. Math, 17 (4) (2024), 3464-3491 3482

0.0 0.2 0.4 0.6 0.8 1.0

-0.05

0.00

0.05

0.10

φ

δ
1

μ=0.6 μ=0.7 μ=0.8

μ=0.9 μ=1.0 Exact

(a)

0.0 0.2 0.4 0.6 0.8 1.0

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

φ

δ
2

μ=0.6 μ=0.7 μ=0.8

μ=0.9 μ=1.0 Exact

(b)

Figure 6: For problem 4.3, the 2D diagrams of App-Ss for various levels of µ and Ex-Ss in the range φ ∈ [0, 1.0]
at ϱ = 0.1 are shown for: (a) δ1(ϱ, φ); (b) δ2(ϱ, φ).

(a) (b)



M.I. Liaqat, H.Aljarrah / Eur. J. Pure Appl. Math, 17 (4) (2024), 3464-3491 3483

(c) (d)

Figure 7: The 3D diagrams of App-S for various levels of µ in the range φ ∈ [0, 2.0] and −3π ≤ ϱ ≤ 3π for
Problem 4.2 of δ1(ϱ, φ) are as follows: (a) µ = 0.7; (b) µ = 0.8; (c) µ = 0.9; and (d) µ = 1.0.

Figure 8: For problem 4.2, the 3D diagrams of Ex-S in the range φ ∈ [0, 2.0] and −3π ≤ ϱ ≤ 3π for δ1(ϱ, φ)
are shown.

(a) (b)



M.I. Liaqat, H.Aljarrah / Eur. J. Pure Appl. Math, 17 (4) (2024), 3464-3491 3484

(c) (d)

Figure 9: The 3D diagrams of App-S for various levels of µ in the range φ ∈ [0, 2.0] and −3π ≤ ϱ ≤ 3π for
Problem 4.2 of δ2(ϱ, φ) are as follows: (a) µ = 0.7; (b) µ = 0.8; (c) µ = 0.9; and (d) µ = 1.0.

Figure 10: For problem 4.2, the 3D diagrams of Ex-S in the range φ ∈ [0, 2.0] and −3π ≤ ϱ ≤ 3π for δ2(ϱ, φ)
are shown.

(a) (b)



M.I. Liaqat, H.Aljarrah / Eur. J. Pure Appl. Math, 17 (4) (2024), 3464-3491 3485

(c) (d)

Figure 11: The 3D diagrams of App-S for various levels of µ in the range φ ∈ [0, 2.0] and −3π ≤ ϱ ≤ 3π for
Problem 4.3 of δ1(ϱ, φ) are as follows: (a) µ = 0.7; (b) µ = 0.8; (c) µ = 0.9; and (d) µ = 1.0.

Figure 12: For problem 4.3, the 3D diagrams of Ex-S in the range φ ∈ [0, 2.0] and −3π ≤ ϱ ≤ 3π for δ1(ϱ, φ)
are shown.

(a) (b)



M.I. Liaqat, H.Aljarrah / Eur. J. Pure Appl. Math, 17 (4) (2024), 3464-3491 3486

(c) (d)

Figure 13: The 3D diagrams of App-S for various levels of µ in the range φ ∈ [0, 2.0] and −3π ≤ ϱ ≤ 3π for
Problem 4.3 of δ2(ϱ, φ) are as follows: (a) µ = 0.7; (b) µ = 0.8; (c) µ = 0.9; and (d) µ = 1.0.

Figure 14: For problem 4.3, the 3D diagrams of Ex-S in the range φ ∈ [0, 2.0] and −3π ≤ ϱ ≤ 3π for δ2(ϱ, φ)
are shown.

The numerical values of Abs-E and Rel-E are shown in Tables 1 and 2 for the 5th-
step App-S and Ex-S of the real (δ1(ϱ, φ)) and imaginary (δ2(ϱ, φ)) parts of δ(ϱ, φ) for
problems 4.2 and 4.3, respectively, with µ = 1 and ϱ = 0.2. These tables demonstrate
that the App-Ss and Ex-Ss are nearly in agreement, confirming the accuracy of SADM.



M.I. Liaqat, H.Aljarrah / Eur. J. Pure Appl. Math, 17 (4) (2024), 3464-3491 3487

Table 1: The Abs-E and Rel-E for δ1(ϱ, φ) and δ2(ϱ, φ) at µ = 1 for problem 4.2.

φ |δ1 − δ51 |
|δ1−δ51 |

|δ1|
|δ2 − δ52 |

|δ2−δ52 |
|δ2|

0.1 1.35702× 10−9 1.42046× 10−9 2.95323× 10−10 9.99334× 10−10

0.2 8.65506× 10−8 9.39683× 10−8 2.01346× 10−8 5.17042× 10−8

0.3 9.82114× 10−7 1.11911× 10−6 2.43305× 10−7 5.07492× 10−7

0.4 5.49515× 10−6 6.65808× 10−6 1.44488× 10−6 2.55892× 10−6

0.5 2.08672× 10−5 2.72831× 10−5 5.80614× 10−6 9.01270× 10−6

0.6 6.20037× 10−5 8.89954× 10−5 1.82078× 10−5 2.53818× 10−5

0.7 1.55526× 10−4 2.50199× 10−4 4.80863× 10−5 6.13872× 10−6

0.8 3.44589× 10−4 6.37771× 10−4 1.11933× 10−4 1.33020× 10−4

0.9 6.94386× 10−4 1.53085× 10−3 2.36508× 10−4 2.65379× 10−4

1.0 1.29829× 10−3 3.58289× 10−3 4.62838× 10−6 4.96586× 10−4

Table 2: The Abs-E and Rel-E for δ1(ϱ, φ) and δ2(ϱ, φ) at µ = 1 for problem 4.3.

φ |δ1 − δ51 |
|δ1−δ51 |

|δ1|
|δ2 − δ52 |

|δ2−δ52 |
|δ2|

0.1 2.77556× 10−17 1.41294× 10−16 3.46945× 10−18 1.16861× 10−16

0.2 2.22045× 10−16 1.16991× 10−15 0.00000 0.00000
0.3 2.85605× 10−14 1.59653× 10−13 9.99201× 10−16 1.15629× 10−14

0.4 9.01057× 10−13 5.49529× 10−12 4.16195× 10−14 3.71016× 10−13

0.5 1.30975× 10−11 7.98778× 10−11 7.55923× 10−13 5.58203× 10−12

0.6 1.16620× 10−10 9.44332× 10−10 8.07848× 10−12 5.19106× 10−11

0.7 7.40353× 10−10 7.48950× 10−9 5.98459× 10−11 3.47274× 10−10

0.8 3.66893× 10−9 5.09649× 10−8 3.39027× 10−10 1.83092× 10−9

0.9 1.50474× 10−8 3.45838× 10−7 1.56469× 10−9 8.07183× 10−9

1.0 5.31541× 10−8 3.78232× 10−6 6.14324× 10−9 3.09996× 10−8

6. Conclusions

To the best of the authors’ knowledge, no research has yet solved FSDEs using the
ADM with the ST in the sense of Con-FrD. In this research, we addressed this gap by
solving linear and nonlinear FSDEs using a hybrid approach that combines ADM and ST,
applied in the sense of Con-FrD. The effectiveness of the SADM is demonstrated through
graphical and numerical results, showing that the App-Ss obtained via the SADM are
in complete agreement with the Ex-Ss. Tables 1 and 2 provide numerical evidence of
the correctness of our approach by comparing the App-Ss and Ex-Ss through Abs-E and
Rel-E. Furthermore, a comparison is made between the results obtained from SADM
and those from alternative techniques, including the HAM and RPSM. The comparison
demonstrates a high degree of agreement with these techniques, suggesting that SADM
is a good substitute for Cap-FD-based methods in solving FSDEs. Additionally, we can
conclude that Con-FD serves as a suitable alternative to Cap-FD for modeling FSDEs.

The key features of the SADM distinguish it from other methods that provide ap-
proximate solutions. This method eliminates the need for assumptions about physical
parameters, making it applicable to both weakly and strongly nonlinear problems and ad-
dressing some limitations of perturbation techniques. The SADM allows for the derivation



REFERENCES 3488

of FPS solutions for FODEs without requiring perturbation, linearization, or discretiza-
tion, unlike other approximate solution methods. The efficiency of the SADM underscores
its computational strength, making it a valuable alternative to Cap-FD-based methods for
solving FSDEs. We also found that the Con-FD effectively replaces the Cap-FD for mod-
eling time-FSDEs. Overall, the SADM proves to be user-friendly, accurate, and efficient.
Moreover, this method is versatile and can be applied to a range of ordinary and partial
FODEs.

In the future, we intend to employ SADM to solve various nonlinear fractional models
arising in biological systems.

Acknowledgements

. The authors express their gratitude to the editor and the reviewers for their valuable
comments.

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