EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 4, Article Number 6378
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Analytical Investigation for Some Problems Under
Different Fractional Differential and Integral Operators

Muhammad Sohail1,2,∗, Eiman1, Hassan Khan2,3, Muhammad Sarwar1,4,
Kamal Shah1,4, Thabet Abdeljawad4

1 Department of Mathematics, University of Malakand, Chakdara Dir (L) 18000, KPK,
Pakistan
2 Department of Mathematics, Abdul Wali Khan University Mardan 23200, Pakistan
3 Department of Mathematics, Near East University TRNC, 10, Mersin, Turkey
4 Department of Mathematics and Sciences, Prince Sultan University, 66833, 11586 Riyadh,
Saudi Arabia

Abstract. This article presents the three well-known fractional order differential operators in-
cluding the Caputo, Caputo-Fabrizio, and Atangana-Baleanu to describe the solutions of a class of
non-linear fractional partial differential equation (FPDE). A method called the Adomian decom-
position method (ADM) is used to find series solutions in a semi-analytical way, using different
transforms like Laplace, Elzaki, Sumudu, Aboodh, Mohand, Yang, Natural, and Shehu. The so-
lutions obtained by the proposed method have precision and a high rate of convergence. We then
verify the derived solutions numerically and graphically for both fractional and integer orders. Fur-
thermore, the solutions under these transformations are the same. The proposed simulations show
that as the number of iterations increases, the corresponding absolute error reduces. Moreover,
fractional order solutions are converging to integer order solutions.

2020 Mathematics Subject Classifications: 26A33, 34A08

Key Words and Phrases: Adomian decomposition method, integral transforms, Caputo deriva-
tive operator, Caputo-Fabrizio derivative operator, Atangana-Baleanu derivative operator

1. Introduction

In the past few decades, mathematicians have established many mathematical mod-
els utilizing the idea of fractional order derivatives. These fractional-order mathematical
models provided an extremely accurate match to the data from experiments of the associ-
ated problems as opposed to traditional models. The scientists and scholars have initiated
unique mathematical models employing fractional differential/integral equations. Due to

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6378

Email addresses: muhammad.sohail@awkum.edu.pk (M. Sohail, Eiman),
hassanmath@awkum.edu.pk (H. Khan), ehuzaifa@gmail.com (Eiman), sarwarswati@gmail.com (M.
Sarwar), kshah@psu.edu.sa (K. Shah), tabdeljawad@psu.edu.sa (T. Abdeljawad)

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



M. Sohail et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6378 2 of 20

the numerous uses of fractional differential equations (FDEs) in the fields of science and
engineering, mathematicians are increasingly focusing on this area of study. Fractional
calculus is widely used in mathematical biology [1, 2], quantum field theory [3], fluid me-
chanics [4], and plasma physics [5]. In control engineering, it is essential for fractional order
systems [6] and PID controllers [7]. Applications also encompass neural networks [8], op-
tical communications [9], and electrochemical processes [10]. Moreover, fractional calculus
is employed in the modeling of time-space fractional diffusion equations [11], groundwa-
ter contamination [12], heat conduction in materials [13], and acoustic wave propagation
[14]. Emerging applications encompass fractional dynamical systems in economics [15],
fractional modeling in image processing [16], and machine learning algorithms [17].

Nonlinear partial differential equations (PDEs) can be tricky to solve precisely. Conse-
quently, in recent decades, mathematicians and scholars have investigated and introduced
several numerical methods and methodologies for addressing nonlinear PDEs. Given the
absence of universal methods applicable to all equation types and the necessity to treat
each equation as an individual problem, solving nonlinear PDEs presents significant chal-
lenges. We can assess and understand the numerical methods and the physical phe-
nomena clarified through analytical solutions to FPDEs. Numerical and novel analytical
methodologies that exist for solving FPDEs are Numerical methods include the Grünwald-
Letnikov method [18], finite difference methods [19], fractional Adams-Bashforth and
Adams-Moulton methods [20], spectral methods [21], fractional order Runge-Kutta meth-
ods [22], wavelet methods [23], and fractional discrete Fourier transform [24]. Analytical
methods include the Laplace transform [25], series solutions [26], Fourier series technique
[27], the homotopy perturbation method [28], variational methods [29], and the method
of successive approximations [30]. Besides these, other methods are [31–35].

The primary objective is to illustrate the solutions of non-linear FPDEs with Ini-
tial Conditions (ICs) by employing the Caputo fractional derivative (CFD) operator,
the Caputo-Fabrizio fractional (CFFD) operator, and the Atangana-Baleanu fractional
(ABFD) operator in a mathematical model. Fractional derivatives of the proposed opera-
tors will be utilized to illustrate the positive effects of employing these operators to solve a
mathematical problem. Outcomes from these operators are analyzed to demonstrate their
efficacy and precision. In 1966, Caputo [36] proposed the Caputo Derivative, that includes
a singular kernel (power law kernel). In 2015, Caputo and Fabrizio [37] introduced the
Caputo-Fabrizio Fractional Derivative, characterized by a non-singular kernel featuring an
exponential function. In 2016, Atangana and Baleanu [38] presented the derivative with
a non-singular kernel utilizing the Mittag-Leffler function.

The ADM, developed by George Adomian between the 1970s and 1990s, is a semi-
analytical technique that decomposes the unknown term of equations both linear and
nonlinear differential and integral equations into components of varying degrees. These
components are subsequently summed, and the method seeks to determine the solutions for
each order, which constitutes the fundamental principle of ADM. The method’s essential
element is the utilization of ”Adomian Polynomials” which facilitate the convergence of
solutions for the nonlinear portion of the equation by simple linearization of the problem.
This study integrates ADM with various transforms and subsequently analyzes the results



M. Sohail et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6378 3 of 20

using tables and graphs.
Analytical methods have a significant role in fractional calculus because solutions ob-

tained by analytical methods are exact and closed-form. This ensures the accuracy of
the resulting quantitative predictions. On the other side, the numerical methods generate
errors that may lead to results significantly diverging from the exact solutions. ADM is a
semi-analytical method that has precision and a high rate of convergence to closed-form
solutions.

The Laplace [39], Elzaki [40], Aboodh [41], Sumudu[42],Yang[43], Mohand [44], Nat-
ural [45], and Shehu [46] transforms are integral transforms employed to solve FPDEs
across different fields, including engineering and mathematics. These transforms, essential
instruments, convert time-domain functions into the complex frequency domain.

There are some basic definitions along with some mathematical calculations for dif-
ferent transformations and derivative operators in section (2). A simple, straightforward,
and general methodology in section (3). In section (4), we solve a non-linear FPDE under
different derivative operators and transformations.

2. Preliminaries

Definition 1. The Laplace transform [39] of g(τ) is a function G(s) of another variable
s defined by

G(s) = L[g(τ)] =
∫ ∞

0
e−s τg(τ)dτ, where τ ≥ 0. (1)

Where g(τ) be a continuous or sectionally continuous function of τ .

Definition 2. The Elzaki transform [40] of g(τ) is defined by

G(s) = E[g(τ)] = s

∫ ∞

0
e−

τ
s g(τ)dτ, τ ≥ 0. (2)

Definition 3. The Aboodh transform [41] of g(τ) is defined by

G(s) = A[g(τ)] =
1

s

∫ ∞

0
e−s τg(τ)dτ, τ ≥ 0. (3)

Definition 4. The Sumudu transform [42] of g(τ) is defined by

G(s) = S[g(τ)] = 1

s

∫ ∞

0
e−

τ
s g(τ)dτ, τ ≥ 0. (4)

Definition 5. The Yang transform [43] of g(τ) is defined by

G(s) = Y[g(τ)] =

∫ ∞

0
e−

τ
s g(τ)dτ, τ ≥ 0. (5)

Definition 6. The Mohand transform [44] of g(τ) is defined by

G(s) = M[g(τ)] = s2
∫ ∞

0
e−sτg(τ)dτ, τ ≥ 0. (6)



M. Sohail et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6378 4 of 20

Definition 7. The Natural transform [45] of g(τ) is defined by

F (s, θ) = N [g(τ)] =
1

θ

∫ ∞

0
e−

sτ
θ g(τ)dτ, τ ≥ 0. (7)

Definition 8. The Shehu transform [46] of g(τ) is defined by

F (s, θ) = S[g(τ)] =

∫ ∞

0
e−

sτ
θ g(τ)dτ, τ ≥ 0. (8)

Definition 9. The kernel of the integral equation

w(θ) =

∫ q(θ)

p(θ)
κ(χ, θ)w(θ)dθ. (9)

is κ(χ, θ). If the kernel is singular, then equation ( 9 ) becomes a singular integral equation.
The Kernel is singular if one or both limits of integration become infinite, or if it becomes
infinite at one point or more points in the interval of integration, mathematically

κ(χ, θ) → ∞ with θ → χ.

Definition 10. Consider, g1 and g2 are two piecewise continuous functions of exponential
order on the interval [0,∞), then the convolution is defined on g1 and g2 as

g1 ∗ g2 =
∫ τ

0
g1(τ − τ)g2(τ)dτ =

∫ τ

0
g2(τ)g1(τ − τ)dτ = g2 ∗ g1, (10)

Showing, g1 ∗ g2 = g2 ∗ g1.

Further, let take LT

L(g1 ∗ g2) = L
[∫ τ

0
g1(τ − τ)g2(τ)dτ

]
= L[g1(τ)]L[g2(τ)] = G1(s)G2(s). (11)

Similarly, Convolution under different transformations; given in the Equation (12).

L[g1 ∗ g2] = L[g1(τ)]L[g2(τ)] = G1(s)G2(s).

E[g1 ∗ g2] = E[g1(τ)]E[g2(τ)]
s = G1(s)G2(s)

s .

A[g1 ∗ g2] = sA[g1(τ)]A[g2(τ)] = sG1(s)G2(s).

S[g1 ∗ g2] = sS[g1(τ)]S[g2(τ)] = sG1(s)G2(s).

Y[g1 ∗ g2] = Y[g1(τ)]Y[g2(τ)] = G1(s)G2(s).

M[g1 ∗ g2] = M[g1(τ)]M[g2(τ)]
s2

= G1(s)G2(s)
s2

.

N [g1 ∗ g2] = θN [g1(τ)]N [g2(τ)] = θ G1(s)G2(s).

S[g1 ∗ g2] = S[g1(τ)]S[g2(τ)] = G1(s)G2(s).

(12)



M. Sohail et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6378 5 of 20

Now,first order derivative under different transform are

L[g′(τ)] = sG(s)− g(0).

E[g′(τ)] = G(s)
s − s g(0).

A[g′(τ)] = sG(s)− g(0)
s .

S[g′(τ)] = G(s)
s − g(0)

s .

Y[g′(τ)] = G(s)
s − g(0).

M[g′(τ)] = sG(s)− s2 g(0).

N [g′(τ)] = sG(s,θ)
θ − g(0)

θ .

S[g′(τ)] = sG(s,θ)
θ − g(0).

(13)

Definition 11. The CFD [36] is defined as

CDα
θ χ(θ) =

1

Γ(1− α)

∫ θ

0
(θ − τ)−α ∂

∂τ
χ(τ) dτ. (14)

Taking LT, implies

L[CDα
θ χ(θ)] =L

[
1

Γ (1− α)

(
θ−α ∗ ∂

∂θ
χ (θ)

)]
=

1

Γ (1− α)

(
L[θ−α]L

[
∂

∂θ
χ (θ)

])
=

1

Γ (1− α)

(
Γ (1− α)

s1−α
(sX(s)− χ(0))

)
= sα−1 (sX(s)− χ(0))

= sαX(s)− sα−1χ(0).

(15)

Implies
L[ CDα

θ χ(θ) ] = sαX(s)− sα−1χ(0), where 0 < α ≤ 1. (16)

Similarly, under different transformations are given in the equation (17).

L[ CDα
θ χ(θ) ] = sαX(s)− sα−1χ(0).

E[ CDα
θ χ(θ) ] = s−αX(s)− s2−αχ(0).

A[ CDα
θ χ(θ) ] = sαX(s)− sα−2χ(0).

S[ CDα
θ χ(θ) ] = s−αX(s)− s−αχ(0).

X [ CDα
θ χ(θ) ] = s−αX(s)− s1−αχ(0).

M[ CDα
θ χ(θ) ] = sαX(s)− sα+1χ(0).

N [ CDα
θ χ(θ) ] =

(
s
ς

)α
X(s, ς)−

(
s
ς

)α 1
s χ(0).

S[ CDα
θ χ(θ) ] =

(
s
ς

)α
X(s, ς)−

(
s
ς

)α−1
χ(0).

(17)



M. Sohail et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6378 6 of 20

Definition 12. The CFFD [37] is defined as:

CFDα
θ χ(θ) =

N(α)

1− α

∫ θ

0
e−

α
1−α

(θ−τ) ∂

∂τ
χ(τ) dτ, (18)

where N(0) = N(1) = 1.

Taking LT

L[CFDα
θ χ(θ)] =L

[
N(α)

1− α

(
e−

α
1−α

θ ∗ ∂

∂θ
χ (θ)

)]
=

N(α)

1− α

(
L
[
e−

α
1−α

θ
]
L
[
∂

∂θ
χ (θ)

])
=

N(α)

1− α

1

s−
(
− α

1−α

)(sX(s)− χ(0))

=
N(α)

(1− α)(s+ α
1−α)

(sX(s)− χ(0)),

(19)

implies

L[ CFDα
θ χ(θ) ] =

N(α)

(1− α)s+ α
(sX(s)− χ(0)). (20)

Similarly, under different transformations in equation (21)

L[ CFDα
θ χ(θ) ] =

N(α)
s(1−α)+α(sX (s)− χ (0)).

E[ CFDα
θ χ(θ) ] =

N(α)
1+α(s−1)

(
X (s)− s2 χ (0)

)
.

A[ CFDα
θ χ(θ) ] =

N(α)
s2(1−α)+αs

(
s2X (s)− χ (0)

)
.

S[ CFDα
θ χ(θ) ] =

N(α)
1+α(s−1) (X (s)− χ (0)) .

X [ CFDα
θ χ(θ) ] =

N(α)
1+α(s−1)(X (s)− s χ (0)).

M[ CFDα
θ χ(θ) ] =

N(α)
s(1−α)+α(sX (s)− s2 χ (0)).

N [ CFDα
θ χ(θ) ] =

N(α)
s
ς
(1−α)+α

(
s
ςX (s, ς)− χ(0)

ς

)
.

S[ CFDα
θ χ(θ) ] =

N(α)
s
ς
(1−α)+α

(
s
ςX (s, ς)− χ (0)

)
.

(21)

Definition 13. The ABFD [38] defined as:

ABDα
θ χ(θ) =

AB(α)

1− α

∫ θ

0
Eα

(
− α

1− α
(θ − τ)α

)
∂

∂τ
χ(τ) dτ, (22)

where AB(0) = AB(1) = 1 and Eα(·) denoted the Mittag-Leffler function [47] i.e.

Eα(θ) = Eα,1(θ) =
∞∑
i=0

θi

Γ(iα+ β)

∣∣∣∣∣
β=1

α > 0. (23)



M. Sohail et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6378 7 of 20

Taking LT

L[ABDα
θ χ(θ)] =L

[
AB(α)

1− α

(
Eα

(
− α

1− α
θα
)
∗ ∂

∂θ
χ (θ)

)]
=

AB(α)

1− α

(
L
[
Eα

(
− α

1− α
θα
)]

L
[
∂

∂θ
χ (θ)

])

=
AB(α)

1− α

 sα−1

sα −
(
− α

1−α

)(sX(s)− χ(0))


=

sα−1AB(α)

sα(1− α) + α
(sX(s)− χ(0)),

(24)

implies

L[ABDα
θ χ(θ) ] =

sα−1AB(α)

sα(1− α) + α
(sX(s)− χ(0)). (25)

Similarly, the equation (26) for different transformations.

L[ABDα
θ χ(θ) ] =

sα−1AB(α)

(1−α)( α
1−α

+sα)
(sX (s)− χ (0)) .

E[ABDα
θ χ(θ) ] =

s AB(α)

(1−α)( α
1−α

sα+1)

(
X(s)
s − s χ (0)

)
.

A[ABDα
θ χ(θ) ] =

sα−1AB(α)

(1−α)( α
1−α

+sα)

(
sX (s)− χ(0)

s

)
.

S[ABDα
θ χ(θ) ] =

s AB(α)

(1−α)( α
1−α

sα+1)

(
X(s)
s − χ(0)

s

)
.

X [ABDα
θ χ(θ) ] =

s AB(α)

(1−α)( α
1−α

sα+1)

(
X(s)
s − χ (0)

)
.

M[ABDα
θ χ(θ) ] =

sα−1AB(α)

(1−α)( α
1−α

+sα)

(
sX (s)− s2χ (0)

)
.

N [ABDα
θ χ(θ) ] =

ςsα−1AB(α)

(1−α)( α
1−α

ςα+sα)

(
s
ςX (s)− χ(0)

ς

)
.

S[ABDα
θ χ(θ) ] =

ςsα−1AB(α)

(1−α)( α
1−α

ςα+sα)

(
s
ςX (s)− χ (0)

)
.

(26)

At last, we have calculated the transformations of some basic functions, which are given
in the table (1).



M. Sohail et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6378 8 of 20

Table 1: Basic functions under different transformations.

Functions Laplace Elzaki Aboodh Sumudu

1 1
s s2 1

s2
1

µ µ
s s2µ µ

s2
µ

θ 1
s2

s3 1
s3

s
θ2 2

s3
2 s4 2

s4
2 s2

θn Γ(n+1)
sn+1 s2+nΓ (n+ 1) Γ(n+1)

sn+2 Γ (n+ 1) sn

sin (µ θ) µ
µ2+s2

s3µ
µ2s2+1

µ
(µ2+s2)s

µ s
µ2s2+1

cos (µ θ) s
µ2+s2

s2

µ2s2+1
1

µ2+s2
1

µ2s2+1

sinh (µ θ) µ
−µ2+s2

− s3µ
µ2s2−1

µ
−µ2s+s3

− µ s
µ2s2−1

cosh (µ θ) s
−µ2+s2

− s2

µ2s2−1
1

s2−µ2
1

1−µ2s2

eµ θ 1
s−µ

s2

1−µ s
1

(s−µ)s
1

1−µ s

epθ−eqθ

p−q
1

(s−p)(s−q)
s2

(p− 1
s )(q−

1
s )

1
s

(p−s)(q−s)
1

(p− 1
s )(q−

1
s )

Eα (−µ θα) sα

s(µ+sα)
s2

µ sα+1
sα

s2(µ+sα)
1

(µ sα+1)

1− Eα (−µ θα) µ
s(µ+sα)

sα+2µ
w sα+1

µ
s2(µ+sα)

µ sα

w sα+1

Functions Yang Mohand Natural Shehu

1 s s 1
s

ς
s

µ µ s µ s µ
s

µ ς
s

θ s2 1 ς
s2

ς2

s2

θ2 2 s3 2
s

2 ς2

s3
2 ς3

s3

θn Γ (n+ 1) sn+1 Γ(n+1)
sn−1

Γ(n+1)ςn

sn+1
Γ(n+1)ςn+1

sn+1

sin (µ θ) s2µ
µ2s2+1

s2µ
µ2+s2

µ ς
µ2ς2+s2

µ ς2

µ2ς2+s2

cos (µ θ) s
µ2s2+1

s3

µ2+s2
s

µ2ς2+s2
s ς

µ2ς2+s2

sinh (µ θ) − s2µ
µ2s2−1

s2µ
−µ2+s2

µ ς
−µ2ς2+s2

µ ς2

−µ2ς2+s2

cosh (µ θ) − s
µ2s2−1

s3

−µ2+s2
s

−µ2ς2+s2
s ς

−µ2ς2+s2

eµ θ s2

s−µ − s3

µ s−1
1

s−µ ς
ς

−µ ς+s
epθ−eqθ

p−q
s

(p− 1
s )(q−

1
s )

s2

(p−s)(q−s)
1

(p− s
ς )(q−

s
ς )

ς

(p− s
ς )(q−

s
ς )

Eα (−µ θα) s
µ sα+1

sα+1

µ+sα
sα

s(µ ςα+sα)
sας

s(µ ςα+sα)

1− Eα (−µ θα) µ sα+1

µ sα+1
sµ

µ+sα
µ ςα

s(µ ςα+sα)
µ ςα+1

s(µ ςα+sα)

The table 1 demonstrates some convolutions results against different fractional order
operators.



M. Sohail et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6378 9 of 20

3. Suggested ADM Method by Using Different Fractional Differential
Operators

Let us have a general nonlinear FPDE in the form of

CDα
θ ν (κ, θ) + Fν(κ, θ) +Nν(κ, θ) = f(κ, θ), u(κ, 0) = f(κ). (27)

Where Fν(κ, θ) and Nν(κ, θ) are linear and nonlinear parts, respectively. Taking LT
( 27 ) implies

L[ CDα
θ ν (κ, θ)] = −L

[
Fν(κ, θ) +Nν(κ, θ)− f(κ, θ)

]
. (28)

By using equation ( 16 ), implies

U(κ, s) =
ν(κ, 0)

s
− 1

sα
L
[
Fν(κ, θ) +Nν(κ, θ)− f(κ, θ)

]
.

Now, taking the inverse LT

L−1[U(κ, s)] = ν(κ, θ)

= L−1

[
ν(κ, 0)

s
− 1

sα
L
[
Fν(κ, θ) +Nν(κ, θ)− f(κ, θ)

]]
.

(29)

Nν(κ, θ) will be decomposed by Adomian Polynomial, defined as

Nν(κ, θ) =
∞∑
n=0

Pn,

where

Pn =
1

n!

∂n

∂γn

(
N(

∞∑
i=0

γi νi)

)∣∣∣∣∣
γ=0

n = 0, 1, 2, 3, . . . .

Thus, LADM decompose equation ( 29 ) as

νk+1(κ, θ) = L−1

[
− 1

sα
L
[
Fνk(κ, θ) + Pk

]]
, k ≥ 0. (30)

Where

ν0(κ, θ) = L−1

[
ν(κ, 0)

s
+

1

sα
L
[
f(κ, θ)

]]
. (31)



M. Sohail et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6378 10 of 20

Similarly, ADM iterative terms under different transformations for CFD in equation (32).

νk+1(κ, θ) = L−1

[
− 1

sα L
[
Fνk(κ, θ) + Pk

]]
.

νk+1(κ, θ) = A−1

[
− 1

sα A
[
Fνk(κ, θ) + Pk

]]
.

νk+1(κ, θ) = M−1

[
− 1

sα M
[
Fνk(κ, θ) + Pk

]]
.

νk+1(κ, θ) = Y−1

[
−sα Y

[
Fνk(κ, θ) + Pk

]]
.

νk+1(κ, θ) = E−1

[
−sα E

[
Fνk(κ, θ) + Pk

]]
.

νk+1(κ, θ) = S−1

[
−sα S

[
Fνk(κ, θ) + Pk

]]
.

νk+1(κ, θ) = S−1

[
− ςα

sα S

[
Fνk(κ, θ) + Pk

]]
.

νk+1(κ, θ) = N−1

[
− ςα

sα N
[
Fνk(κ, θ) + Pk

]]
.

(32)

Similarly, using CFFD, iterative terms for ADM under different transformations, are given
in equation (35)

νk+1(κ, θ) = L−1

[
− 1− α

N(α)s
(s+

α

1− α
)L
[
Fνk(κ, θ) + Pk

]]
, k ≥ 0, (33)

and

νk+1(κ, θ) = L−1

[
−sα(1− α) + α

AB(α)sα
L
[
Fνk(κ, θ) + Pk

]]
, k ≥ 0. (34)



νk+1(κ, θ) = L−1

[
− 1−α

N(α)s(s+
α

1−α)L
[
Fνk(κ, θ) + Pk

]]
.

νk+1(κ, θ) = A−1

[
− 1−α

N(α)s(s+
α

1−α)A
[
Fνk(κ, θ) + Pk

]]
.

νk+1(κ, θ) = M−1

[
− 1−α

N(α)s(s+
α

1−α)M
[
Fνk(κ, θ) + Pk

]]
.

νk+1(κ, θ) = Y−1

[
−1+α(s−1)

N(α) Y
[
Fνk(κ, θ) + Pk

]]
.

νk+1(κ, θ) = E−1

[
−1+α(s−1)

N(α) E

[
Fνk(κ, θ) + Pk

]]
.

νk+1(κ, θ) = S−1

[
−1+α(s−1)

N(α) S
[
Fνk(κ, θ) + Pk

]]
.

νk+1(κ, θ) = S−1

[
− (1−α)ς

sN(α) (
s
ς +

α
1−α)S

[
Fνk(κ, θ) + Pk

]]
.

νk+1(κ, θ) = N−1

[
− (1−α)ς

sN(α) (
s
ς +

α
1−α)N

[
Fνk(κ, θ) + Pk

]]
.

(35)



M. Sohail et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6378 11 of 20

Similarly, using ABFD, iterative terms for ADM under different transformations, are given
in equations (36)

νk+1(κ, θ) = L−1

[
− sα(1−α)+α

AB(α)sα L
[
Fνk(κ, θ) + Pk

]]
.

νk+1(κ, θ) = A−1

[
− sα(1−α)+α

AB(α)sα A
[
Fνk(κ, θ) + Pk

]]
.

νk+1(κ, θ) = M−1

[
− sα(1−α)+α

AB(α)sα M
[
Fνk(κ, θ) + Pk

]]
.

νk+1(κ, θ) = Y−1

[
−α(sα−1)+1

AB(α) Y
[
Fνk(κ, θ) + Pk

]]
.

νk+1(κ, θ) = E−1

[
−α(sα−1)+1

AB(α) E

[
Fνk(κ, θ) + Pk

]]
.

νk+1(κ, θ) = S−1

[
−α(sα−1)+1

AB(α) S
[
Fνk(κ, θ) + Pk

]]
.

νk+1(κ, θ) = S−1

[
− sα(1−α)+ α

ςα

AB(α)sα S

[
Fνk(κ, θ) + Pk

]]
.

νk+1(κ, θ) = N−1

[
− sα(1−α)+ α

ςα

AB(α)sα N
[
Fνk(κ, θ) + Pk

]]
.

(36)

If f(κ, θ) = 0 then ν0(κ, θ) is calculated as

ν0(κ, θ) = L−1

[
ν(κ, 0)

s

]
. (37)

Similarly, equation (38) , for ν0(κ, θ) under different transforms.

ν0(κ, θ) = L−1
[ ν(κ,0)

s

]
.

ν0(κ, θ) = E−1
[
s2ν(κ, 0)

]
.

ν0(κ, θ) = A−1
[ ν(κ,0)

s2

]
.

ν0(κ, θ) = S−1
[
ν(κ, 0)

]
.

ν0(κ, θ) = Y−1
[
s ν(κ, 0)

]
.

ν0(κ, θ) = M−1 [ s ν(κ, 0) ] .
ν0(κ, θ) = N−1

[ ν(κ,0)
s

]
.

ν0(κ, θ) = S−1
[
ν(κ, 0) ς

s

]
.

(38)

The term-wise iterative solution of ADM with different transforms for the equations
( 32, 35, and 36 ) are represented as

ν(κ, θ) =
∞∑
n=0

νn(κ, θ). (39)



M. Sohail et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6378 12 of 20

4. Problem Solution

Dα
τ µ(χ, τ)−

d2µ

dχ2
+

dµ

dχ
− µ− µ

(
d2µ

dχ2

)
+ µ2 = 0; µ(χ, 0) = eχ. (40)

Having an exact solution

µ(χ, τ) =eχ+τ . (41)

4.1. Solving under Caputo Fractional Differential Operator

µ0(χ, τ) =eχ. (42)

µ1(χ, τ) =
eχτα

Γ(1 + α)
. (43)

µ2(χ, τ) =
eχ (τα)2

Γ(1 + 2α)
. (44)

µ3(χ, τ) =
eχ (τα)3

Γ(1 + 3α)
. (45)

µ4(χ, τ) =
eχ (τα)4

Γ(1 + 4α)
. (46)

µ(χ, τ) =eχ +
eχτα

Γ(1 + α)
+

eχ (τα)2

Γ(1 + 2α)
+

eχ (τα)3

Γ(1 + 3α)
+

eχ (τα)4

Γ(1 + 4α)
+ · · · . (47)

Taking α = 1, implies

µ(χ, τ) =eχ + eχτ +
eχτ2

2
+

eχτ3

6
+

eχτ4

24
+

eχτ5

120
+ · · · . (48)

Clearly Equation (48) is converging to the exact solution.

4.2. Solving under CF Fractional Differential Operator

µ0(χ, τ) =eχ. (49)

µ1(χ, τ) =
eχ (α (−1 + τ) + 1)

M(α)
. (50)

µ2(χ, τ) =

(
2 + α2

(
τ2 − 4τ + 2

)
+ (4τ − 4)α

)
eχ

2M(α)2
. (51)



M. Sohail et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6378 13 of 20

µ3(χ, τ) =
eχ

6M(α)3

(
α3
(
τ3 − 9τ2 + 18τ − 6

)
+ 9α2

(
τ2 − 4τ + 2

)
+ 18α (−1 + τ) + 6

)
.

(52)

µ4(χ, τ) =
eχ

24M(α)4

(
α4
(
τ4 − 16τ3 + 72τ2 − 96τ + 24

)
+ 16α3

(
τ3 − 9τ2 + 18τ − 6

)
+ 72α2

(
τ2 − 4τ + 2

)
+ 96α (−1 + τ) + 24

)
.

(53)

µ(χ, τ) =eχ +
eχ (α (−1 + τ) + 1)

M(α)
+

(
2 + α2

(
τ2 − 4τ + 2

)
+ (4τ − 4)α

)
eχ

2M(α)2
+ · · · . (54)

By putting α = 1 Equation(54) implies

µ(χ, τ) =eχ + eχτ +
τ2eχ

2
+

eχτ3

6
+

eχτ4

24
+

eχτ5

120
+ · · · . (55)

Clearly Equation (55) is converging to the exact solution.

4.3. Solving under ABC Fractional Differential Operator

µ0(χ, τ) =eχ. (56)

µ1(χ, τ) =−
eχ
(
−1 + α− α τα

Γ(1+α)

)
AB(α)

. (57)

µ2(χ, τ) =−
eχ
(
− α2τ2α

Γ(1+2α) +
2ταα(−1+α)

Γ(1+α) − 1 + 2α− α2
)

AB(α)2
. (58)

µ3(χ, τ) =−
eχ
(
− α3τ3α

Γ(1+3α) −
3ταα(−1+α)2

Γ(1+α) + 3τ2αα2(−1+α)
Γ(1+2α) − 1− 3α2 + 3α+ α3

)
AB(α)3

. (59)

µ4(χ, τ) =−
eχ
(
−1 + 4α3 − α4 − α4τ4α

Γ(1+4α) + 4α− 6α2 + 4ταα(−1+α)3

Γ(1+α) − 6τ2αα2(−1+α)2

Γ(1+2α) + 4τ3αα3(−1+α)
Γ(1+3α)

)
AB(α)4

.

(60)

µ(χ, τ) =eχ −
eχ
(
−1 + α− α τα

Γ(1+α)

)
AB(α)

−
eχ
(
−1 + 2α− α2 − α2τ2α

Γ(1+2α) +
2ταα(−1+α)

Γ(1+α)

)
AB(α)2

+ · · · .

(61)



M. Sohail et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6378 14 of 20

By putting α = 1 Equation(61) implies

µ(χ, τ) =eχ + eχτ +
eχτ2

2
+

eχτ3

6
+

eχτ4

24
+

eχτ5

120
+ · · · . (62)

Clearly Equation (62) is converging to the exact solution.
Here, in figure 1, we present 2D graphs of our solution.

Figure 1: 2D graphs of solutions at different fractional orders.

The error minimization procedure has presented in figure 2.

Figure 2: The error is minimized as the number of solution terms increases.

We have presented graphical illustrations of solutions for various operators in figure 3.



M. Sohail et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6378 15 of 20

(a) (b) (c)
Figure 3: Solution graphs under different differential operators: (a): CFD (b): CFFD (c): ABFD .



M. Sohail et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6378 16 of 20

Fractional Order τ CFD Operator CFFD Operator ABFD Operator

0.10 3.004166021 3.004166021 3.004166021
0.20 3.320116674 3.320116674 3.320116674

1.00 0.30 3.669293792 3.669293792 3.669293792
0.40 4.055183573 4.055183573 4.055183573
0.50 4.481625588 4.481625588 4.481625588

0.10 3.049062427 3.162276707 3.209534249
0.20 3.395391468 3.494836859 3.574059501

0.95 0.30 3.772474744 3.862326090 3.970883434
0.40 4.185951335 4.268359634 4.405859090
0.50 4.640664007 4.716892562 4.883957994

0.10 3.101148306 3.337913899 3.445629081
0.20 3.480848024 3.688814956 3.867192329

0.90 0.30 3.888474328 4.076407078 4.319300743
0.40 4.332160489 4.504383671 4.810635889
0.50 4.817860678 4.976753819 5.347337632

0.10 3.161737693 3.534026565 3.719068724
0.20 3.578250433 3.905096972 4.207468721

0.85 0.30 4.019492351 4.314587569 4.723204464
0.40 4.496450273 4.766204929 5.278274854
0.50 5.016292557 5.263940139 5.879974675

0.10 3.232462725 3.754097180 4.037868933
0.20 3.689797588 4.147155788 4.603938816

0.80 0.30 4.168268841 4.580217667 5.191702098
0.40 4.682091666 5.056930340 5.817070731
0.50 5.239738787 5.581190749 6.488496318

0.10 3.315379658 4.002197630 4.411458682
0.20 3.818266181 4.418910210 5.066371531

0.75 0.30 4.338252697 4.876937276 5.733580680
0.40 4.893167590 5.379793868 6.433753129
0.50 5.492854752 5.931207911 7.176619194

Table 2: Approximate solutions under different differential operators at χ = 1.

In table 2, we have displayed the approximate solutions corresponding to various dif-
ferential operators of fractional orders.

5. Conclusions

This study conducts an analytic investigation of a non-linear FPDE utilizing various
derivative operators and transformations. A clearly stated general methodology is de-
veloped for fractional operators under transformations in the context of solving FPDEs.
The study concludes that the ADM, when applied with various transforms, is an effective



M. Sohail et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6378 17 of 20

tool for solving both linear and non-linear FPDEs, demonstrating high accuracy and re-
duced labor effort. Furthermore, the data presented in the tables and plots indicate that
the solutions to the FPDEs using the Caputo, Caputo-Fabrizio, and Atangana-Baleanu
fractional derivatives at an integer level are equivalent and converge to the exact solu-
tion. Furthermore, Plots indicate that absolute errors diminish as the iterations of ADM
with various transformations increase. 2D & 3D-Plots show that fractional order solu-
tions under different operators are converging to integer order. The ADM series solutions
are the same under different transforms. Our research indicates that ADM with various
transformations offers a straightforward and precise solution to FPDEs involving different
operators.

Acknowledgements

Authors are thankful to Prince Sultan University for APC and support through TAS
research lab.

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