EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 16, No. 2, 2023, 1024-1046
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

The generalization of integral transforms combined with
He’s polynomial

Junaid Idrees Mustafa

Department of Mathematics, College of Education For Pure Sciences, University of Mosul,
Mosul, Iraq

Abstract. Our goal in this paper is to generalize the integral transforms and use it with He’s
polynomial method to find the solution of the nonlinear partial differential equations. All results
of theoretical studies regarding the generalization and its properties are presented. For the He’s
polynomial method, it is used to solve the nonlinear part of the partial differential equation.
It is shown that the importance of my research is the combination of generalization of integral
transforms with He’s polynomial method allows for exact and approximate solutions configurations
to be determined. Furthermore, the generalization of integral transforms has been shown to include
most, or even all, of the integral transforms and be applicable to a variety of equations, making it
a crucial tool in solving them. Finally, the capability of solutions to be obtained quickly and easily
through this combined technique provides an invaluable tool for solving problems.

2020 Mathematics Subject Classifications: 35Q35, 44A05, 41A10, 41A58, 65D25, 65J15,
74H15.

Key Words and Phrases: Generalization of integral transform, He’s polynomial method, non-
linear partial differential equations

1. Introduction

Integral transforms have become an important tool in mathematics in recent years and
have been extensively studied by many researchers [24, 25, 28]. These transforms have
been successfully applied to solve many linear equations, such as ordinary and partial
differential equations (PDEs) and integral equations [17, 19, 20]. The utility of integral
transforms is their ability to simplify the solution process, making it easier to calculate the
solutions in a fraction of time. This has enabled the application of integral transforms to a
wide variety of problems, allowing for the production of new solutions and the refinement
of existing ones [18].

Furthermore, the development of integral transforms have allowed for the efficient com-
putation of PDEs, opening up new possibilities in the production of exact and approximate
solutions of the equations among them, PDEs [5, 23, 32]. Recently, various applications

DOI: https://doi.org/10.29020/nybg.ejpam.v16i2.4747

Email address: j.i.mustafa20@uomosul.edu.iq. (J.I. Mustafa)

https://www.ejpam.com 1024 © 2023 EJPAM All rights reserved.



J.I. Mustafa / Eur. J. Pure Appl. Math, 16 (2) (2023), 1024-1046 1025

of integral transforms have been found in different areas of engineering, mathematics and
physics such as image processing, signal analysis and electric [21, 26, 30]. This is due
to their properties and ability to transform a function from one domain to another do-
main while preserving important features of the function. Their use in these areas makes
them a valuable tool as they provide efficient and accurate results. Consequently, integral
transforms are being used more and more in applied mathematics [13, 15].

It is worth mentioning that the integral transforms can be used in combination with
other methods to address the nonlinear parts of equations. This can provide a much more
efficient solution, and can be used to analyse many different systems with greater accuracy
[7, 29, 32].

One of these methods is He’s polynomial, it is a reliable numerical approach for solving
nonlinear PDEs. This method is based on the idea that a nonlinear PDE can be approxi-
mated by a polynomial in a certain variable. It works by expanding the nonlinear function
into a set of truncated series of polynomials [6, 10]. This set of polynomials is then used
to approximate the solution of the PDE. He’s polynomial method is computationally effi-
cient, since it only requires the evaluation of a few polynomial coefficients rather than a
full numerical solution [9]. It has been used in a wide range of problems in fields such as
fluid dynamics, astrophysics, and acoustics, and it is a powerful tool for solving nonlinear
PDEs [27].

By understanding the underlying principles of integral transforms, it may become
possible to develop more efficient and accurate methods for solving equations [1, 4, 8, 14,
16]. In this paper, the integral transforms are generalized, and then combined it with He’s
polynomial method to solve nonlinear PDEs. As the examples, we apply the combination
of generalization (GN) of integral transforms and He’s polynomial method to solve the
nonlinear PDEs, which include the nonlinear gas dynamic equation, system of coupled
nonlinear Burgers’ equation and the non-homogeneous gas dynamic equation.

There are some scientific applications of these types of nonlinear equations. For
instance, in ideal gas dynamics, several kinds of waves in nonlinear systems are de-
scribed including discontinuities in contact, shock fronts and rare factions. It’s worth
mentioning that many researchers have solved the gas dynamics equations [2, 3]. For
the coupled Burgers’ equation, it falls into the category of integrable equations, such as
nonlinear Schrödinger, Korteweg–De Vries, and Bogoyavlensky-Konopelchenko equations
[11, 12, 22, 31].

The structure of this article is as follows: In Section (2), a general structure for the GN
of the integral transforms is presented. Then, some theorems for the generalized integral
transform of functions are proved in Section (3). As main examples, we prove the constant,
polynomial, trigonometric, and exponential functions. In Section (4), we examine some
useful properties for the GN of integral transform. In Section (5), our mathematical
method is illustrated by using He’s polynomial for PDEs with applying the GN . In Section
(6), based on appropriate GN of integral transforms with He’s polynomial method, we
present three different examples for nonlinear PDEs: the nonlinear gas dynamic equation,
system of coupled nonlinear Burgers’ equation and the non-homogeneous gas dynamic
equation, the effectiveness of the method is evaluated by deriving the solutions of the



J.I. Mustafa / Eur. J. Pure Appl. Math, 16 (2) (2023), 1024-1046 1026

corresponding PDEs.

2. A generalization of integral transforms

Here, we present a GN of integral transforms which includes most integral transforms,
or even all of them in the class of Laplace transforms (see, e.g., [8, 14, 15, 17]). In the next
definition, we define some functions related to the GN , and then formulate the general
form of the GN . After that, based on this GN , the transforms of some functions will be
found by proving some theorems.

Definition 1. Assume that the function F (Ψ(ϑ) t), t ∈ [0,∞) is an integrable. Given the
positive real functions h(ϑ) ̸= 0 and σ(ϑ), the following formula shows the general form
of the GN (GN(ϑ)) of F (Ψ(ϑ) t):

GN [F (t)] =GN(ϑ) = h(ϑ)

∫ ∞

0
F (Ψ(ϑ) t) e−σ(ϑ) tt (1)

= lim
τ→∞

h(ϑ)

∫ τ

0
F (Ψ(ϑ) t) e−σ(ϑ) tt, (2)

then, when the limit in (2) is exist, we get,∣∣∣∣∣h(ϑ)
∫ τ ′

0
F (Ψ(ϑ) t) e−σ(ϑ) tt

∣∣∣∣∣ ≤ h(ϑ)

∫ τ ′

0

∣∣∣F (Ψ(ϑ) t) e−σ(ϑ) t
∣∣∣ t −→ 0, (3)

where, τ → ∞ for all τ ′ > τ . Then GN(F (Ψ(ϑ) t);ϑ) will be convergence.

Definition 2. Let F (t), has an exponential order κ, which means, for any L > 0, ∃M > 0,
where κ, L and M are constants, such that:

|F (t)| ≤M eκt, t > L. (4)

3. Results of the preliminary study

In this section, we will establish the theorems needed that can be used to demonstrate
the GN of integral transforms for solving PDEs.

Theorem 1. Using the GN of integral transforms in the definition 1, we can get the
generalized transforms for bellow functions:

(1.A) When F (t) = 1, then

GN [1] =h(ϑ)

∫ ∞

0
e−σ(ϑ)tt,

=
−h(ϑ)
σ(ϑ)

e−σ(ϑ)t
∣∣∣∣∞
0

=
h(ϑ)

σ(ϑ)
. (5)



J.I. Mustafa / Eur. J. Pure Appl. Math, 16 (2) (2023), 1024-1046 1027

(1.B) When F (t) = t, then

GN [t] = h(ϑ)

∫ ∞

0
ψ(ϑ)t e−σ(ϑ)tt,

by integrating by parts, we obtain

GN [t] =
h(ϑ)ψ(ϑ)

σ2(ϑ)
. (6)

(1.C) When F (t) = tn, then

GN [tn] =
n! h(ϑ)ψn(ϑ)

σn+1(ϑ)
. (7)

(1.D) When F (t) = sin(t), then

GN [sin(t)] = h(ϑ)

∫ ∞

0
sin(ψ(ϑ)t)e−σ(ϑ)tt,

using integrating by part twice, we obtain:

GN [sin(t)] =
h(ϑ)ψ(ϑ)

σ2(ϑ) + ψ2(ϑ)
, (8)

(1.E) When F (t) = et, then

GN [et] =
h(ϑ)

σ(ϑ)− ψ(ϑ)
. (9)

Theorem 2. Suppose F (t) is differentiable, if F (t) and it’s derivatives (F ′(t), F ′′(t), . . . , F (n−1)(t))
are of exponential order κ and are piecewise continuous on [0,∞) and the nth derivative
F (n)(t) is a piecewise continuous on [0,∞), then,

(2.A)

GN [F ′(t)] = − h(ϑ)

Ψ(ϑ)
F (0) +

σ(ϑ)

Ψ(ϑ)
GN [F (t)], (10)

and,

(2.B)

GN [F ′′(t)] = − h(ϑ)

Ψ(ϑ)
F ′(0)− h(ϑ)σ(ϑ)

Ψ2(ϑ)
F (0) +

σ2(ϑ)

Ψ2(ϑ)
GN [F (t)], (11)

then, in general,



J.I. Mustafa / Eur. J. Pure Appl. Math, 16 (2) (2023), 1024-1046 1028

(2.C)

GN [Fn(t)] =
σn(ϑ)

Ψn(ϑ)
GN [F (t)]−

n−1∑
k=0

h(ϑ) σn−k−1(ϑ)

Ψn−k(ϑ)
F (k)(0). (12)

Proof. (2.A) To find GN [F ′(t)], we can integrate as follows:

GN [F ′(t)] = h(ϑ)

∫ ∞

0
F ′(Ψ(ϑ)t) e−σ(ϑ)tt.

Using integrating by part, we get,

GN [F ′(t)] =h(ϑ)

[
1

Ψ(ϑ)
e−σ(ϑ)t F (Ψ(ϑ)t)

∣∣∣∣∞
0

−
∫ ∞

0

−σ(ϑ)
Ψ(ϑ)

e−σ(ϑ)t F (Ψ(ϑ)t)t

]
=
−h(ϑ)
Ψ(ϑ)

F (0) +
σ(ϑ)

Ψ(ϑ)
GN [F (t)]. (13)

Proof. (2.B) To find GN [F ′′(t)]: Let F(t) = F ′(t), then F ′(t) = F ′′(t) . Since we
know,

GN [F ′(t)] =
−h(ϑ)
Ψ(ϑ)

F(0) +
σ(ϑ)

Ψ(ϑ)
GN [F(t)], (14)

substitute F ′(t) = F ′′(t), and F(t) = F ′(t) in (14), we have

GN [F ′′(t)] =
−h(ϑ)
Ψ(ϑ)

F ′(0) +
σ(ϑ)

Ψ(ϑ)
GN [F ′(t)].

Using the relation (2.A), we get,

GN [F ′′(t)] =
−h(ϑ)
Ψ(ϑ)

F ′(0) +
σ(ϑ)

Ψ(ϑ)

[
−h(ϑ)
Ψ(ϑ)

(ϑ) F (0) +
σ(ϑ)

Ψ(ϑ)
GN [F (t)]

]
,

then,

GN [F ′′(t)] =
−h(ϑ)
Ψ(ϑ)

F ′(0)− h(ϑ) σ(ϑ)

Ψ2(ϑ)
F (0) +

σ2(ϑ)

Ψ2(ϑ)
GN [F (t)]. (15)

It is also possible to prove the general form (2.C) in (12) in a similar way.

Theorem 3. Suppose that h(ϑ) and σ(ϑ) are differentiable and σ′(ϑ) ̸= 0, then,

(3.A)

GN [t F (t)] = − h(ϑ)

σ′(ϑ)

(
ϑ

(
GN [F (t)]

h(ϑ)

))
+
h(ϑ)

σ′(ϑ)

∫ ∞

0 ϑ
(F (Ψ(ϑ)t))e−σ(ϑ)tt. (16)



J.I. Mustafa / Eur. J. Pure Appl. Math, 16 (2) (2023), 1024-1046 1029

(3.B)

GN [t2 F (t)] =
h(ϑ)

σ′(ϑ)

(
ϑ

(
1

σ′(ϑ)

(
ϑ

(
GN [F (t)]

h(ϑ)

))))
− h(ϑ)

σ′(ϑ)

(
ϑ

(
1

σ′(ϑ)

(∫ ∞

0 ϑ
(F (Ψ(ϑ)t)) e−σ(ϑ) tt

)))
+
h(ϑ)

σ′(ϑ)

∫ ∞

0
t
ϑ
(F (Ψ(ϑ)t)) e−σ(ϑ) tt. (17)

(3.C) In general,

GN [tn F (t)] = (−1)n
h(ϑ)

σ′(ϑ)

(
ϑ

(
1

σ′(ϑ)

(
ϑ

(
1

σ′(ϑ)
. . .︸ ︷︷ ︸

n−1 times

(
ϑ

(
GN [F (t)]

h(ϑ)

))))))

+
h(ϑ)

σ′(ϑ)

n−1∑
k=0

(−1)k
(
ϑ

(
1

σ′(ϑ)

(
ϑ

(
1

σ′(ϑ)
. . .︸ ︷︷ ︸

k times

×
(∫ ∞

0
(t)n−k−1

ϑ
(F (Ψ(ϑ)t)) e−σ(ϑ) tt

)))))
. (18)

(3.D)

GN [t F ′(t)] = − h(ϑ)

σ′(ϑ) ϑ

(
GN [F ′(t)]

h(ϑ)

)
+
h(ϑ)

σ′(ϑ)

∫ ∞

0 ϑ
(F ′(Ψ(ϑ) t))e−σ(ϑ) tt. (19)

(3.E)

GN [t F ′′(t)] = − h(ϑ)

σ′(ϑ) ϑ

(
GN [F ′(t)]

h(ϑ)

)
+
h(ϑ)

σ′(ϑ)

∫ ∞

0 ϑ
(F ′′(Ψ(ϑ) t))e−σ(ϑ)tt. (20)

(3.F) In general,

GN [t F (n)(t)] = − h(ϑ)

σ′(ϑ) ϑ

(
GN [F (n)(t)]

h(ϑ)

)
+
h(ϑ)

σ′(ϑ)

∫ ∞

0 ϑ
(F (n)(Ψ(ϑ)t))e−σ(ϑ)tt. (21)

Proof. (3.A)
By using the definition of the GN in equation (1),

GN [F (t)] =h(ϑ)

∫ ∞

0
F (Ψ(ϑ) t) e−σ(ϑ) tt,

GN [F (t)]

h(ϑ)
=

∫ ∞

0
F (Ψ(ϑ) t) e−σ(ϑ) tt,



J.I. Mustafa / Eur. J. Pure Appl. Math, 16 (2) (2023), 1024-1046 1030

derive both side with respect to ϑ, gives,

ϑ

(
GN [F (t)]

h(ϑ)

)
=

∫ ∞

0 ϑ

(
F (Ψ(ϑ) t) e−σ(ϑ) t

)
t

=

∫ ∞

0

(
−t F (Ψ(ϑ) t) e−σ(ϑ) tσ′(ϑ) +

F (Ψ(ϑ) t)

ϑ
e−σ(ϑ) t

)
t

=− σ′(ϑ)

h(ϑ)
h(ϑ)

∫ ∞

0
t F (Ψ(ϑ) t) e−σ(ϑ) tt+

∫ ∞

0

F (Ψ(ϑ) t)

ϑ
e−σ(ϑ) tt

=− σ′(ϑ)

h(ϑ)
GN [t F (t)] +

∫ ∞

0

F (Ψ(ϑ) t)

ϑ
e−σ(ϑ) tt,

then,

GN [t F (t)] =− h(ϑ)

σ′(ϑ)

(
ϑ

(
GN [F (t)]

h(ϑ)

))
+
h(ϑ)

σ′(ϑ)

∫ ∞

0 ϑ
F (Ψ(ϑ) t) e−σ(ϑ) tt.

Proof. (3.B)
Using the formula in (3.A) and the definition of the GN in equation (1)

GN [t F (t)] = − h(ϑ)

σ′(ϑ)

(
ϑ

(
GN [F (t)]

h(ϑ)

))
+
h(ϑ)

σ′(ϑ)

∫ ∞

0 ϑ
F (Ψ(ϑ) t) e−σ(ϑ) tt,

h(ϑ)

∫ ∞

0
F (Ψ(ϑ) t) t e−σ(ϑ) tt = − h(ϑ)

σ′(ϑ)

(
ϑ

(
GN [F (t)]

h(ϑ)

))
+
h(ϑ)

σ′(ϑ)

∫ ∞

0 ϑ
F (Ψ(ϑ) t) e−σ(ϑ) tt,

derive both side with respect to ϑ, gives,∫ ∞

0
F (Ψ(ϑ) t) t2(−σ′(ϑ)) e−σ(ϑ) tt+

∫ ∞

0 ϑ
F (Ψ(ϑ) t) t e−σ(ϑ) tt =

−
ϑ

(
1

σ′(ϑ)

(
ϑ

(
GN [F (Ψ(ϑ) t)]

h(ϑ)

)))
+
ϑ

(
1

σ′(ϑ)

(∫ ∞

0 ϑ
F (Ψ(ϑ) t) t e−σ(ϑ) tt

))
,

so,

− σ(ϑ)

h(ϑ)
GN [t2F (t)] +

∫ ∞

0 ϑ
F (Ψ(ϑ) t) t e−σ(ϑ) tt =

−
ϑ

(
1

σ′(ϑ)

(
ϑ

(
GN [F (Ψ(ϑ) t)]

h(ϑ)

)))
+
ϑ

(
1

σ′(ϑ)

(∫ ∞

0 ϑ
F (Ψ(ϑ) t) t e−σ(ϑ) tt

))
,

then,

GN [t2F (t)] =
h(ϑ)

σ′(ϑ)

(
ϑ

(
1

σ′(ϑ)

(
ϑ

(
GN [F (Ψ(ϑ) t)]

h(ϑ)

))))
−

h(ϑ)

σ′(ϑ)

(
ϑ

(
1

σ′(ϑ)

(∫ ∞

0 ϑ
F (Ψ(ϑ) t) e−σ(ϑ) tt

)))
+
h(ϑ)

σ′(ϑ)

(∫ ∞

0 ϑ
F (Ψ(ϑ) t) t e−σ(ϑ) tt

)



J.I. Mustafa / Eur. J. Pure Appl. Math, 16 (2) (2023), 1024-1046 1031

We can prove the general case (3.C) of theorem (3) in a similar way.

Proof. (3.D)
Since,

GN [F ′(t)] =h(ϑ)

∫ ∞

0
F ′(Ψ(ϑ) t) e−σ(ϑ) tt,

GN [F ′(t)]

h(ϑ)
=

∫ ∞

0
F ′(Ψ(ϑ) t) e−σ(ϑ) tt,

derive both side with respect to ϑ, gives,

ϑ

(
GN [F ′(t)]

h(ϑ)

)
=

∫ ∞

0 ϑ

(
F ′(Ψ(ϑ) t) e−σ(ϑ) t

)
t

=

∫ ∞

0

(
−t F ′(Ψ(ϑ) t) e−σ(ϑ) tσ′(ϑ) +

F ′(Ψ(ϑ) t)

ϑ
e−σ(ϑ) t

)
t

=− σ′(ϑ)

h(ϑ)
h(ϑ)

∫ ∞

0
t F ′(Ψ(ϑ) t) e−σ(ϑ) tt+

∫ ∞

0

F ′(Ψ(ϑ) t)

ϑ
e−σ(ϑ) tt

=− σ′(ϑ)

h(ϑ)
GN [t F ′(t)] +

∫ ∞

0

F ′(Ψ(ϑ) t)

ϑ
e−σ(ϑ) tt,

then,

GN [t F ′(t)] = − h(ϑ)

σ′(ϑ)

(
ϑ

(
GN [F ′(t)]

h(ϑ)

))
+
h(ϑ)

σ′(ϑ)

∫ ∞

0

F ′(Ψ(ϑ) t)

ϑ
e−σ(ϑ) tt. (22)

We can prove the forms (3.E) and (3.F) of theorem (3) in similar way.

Theorem 4. Given these functions h(ϑ), σ(ϑ) and F (t) are differentiable (σ′(ϑ) ̸= 0),
then we can prove

GN [t2 F (n)(t)] =
h(ϑ)

σ′(ϑ)

(
ϑ

(
1

σ′(ϑ)

(
ϑ

(
GN [F (n)]

h(ϑ)

))))
+
h(ϑ)

σ′(ϑ)

×

(
ϑ

(
1

σ′(ϑ)

(∫ ∞

0

F (n)(Ψ(ϑ) t)

ϑ
e−σ(ϑ) tt

)))

+
h(ϑ)

σ′(ϑ)

∫ ∞

0
t
F (n)(Ψ(ϑ) t)

ϑ
e−σ(ϑ) tt. (23)

Proof. Recall the general form (3.F) of theorem (3),

GN [t F (n)(t)] = h(ϑ)

∫ ∞

0
t F (n)(Ψ(ϑt))e−σ(ϑ)tt

= − h

σ′(ϑ)

(
ϑ

(
GN [F (n)(t)]

h(ϑ)

))
+
h(ϑ)

σ′(ϑ)

∫ ∞

0 ϑ
(F (n)(Ψ(ϑ)t))e−σ(ϑ)tt, (24)



J.I. Mustafa / Eur. J. Pure Appl. Math, 16 (2) (2023), 1024-1046 1032

derive both sides of (24) with respect to ϑ, to obtain,∫ ∞

0
t2 F (n)(Ψ(ϑ) t) e−σ(ϑ) t(−σ′)t+

∫ ∞

0
t
ϑ
(F (n)(Ψ(ϑ) t)) e−σ(ϑ) tt =

−
ϑ

(
h

σ′(ϑ)

(
ϑ

(
GN [F (n)(t)]

1

)))
+
ϑ

(
1

σ′(ϑ)

(∫ ∞

0 ϑ
(F (n)(Ψ(ϑ)t))e−σ(ϑ)tt

))
,

(25)

Thus, simplifying more, we prove theorem (4) as follows:

σ′(ϑ)

h(ϑ)
GN [t2 F (n)(t)] =

ϑ

(
h

σ′(ϑ)

(
ϑ

(
GN [F (n)(t)]

1

)))

−
ϑ

(
1

σ′(ϑ)

(∫ ∞

0 ϑ
(F (n)(Ψ(ϑ) t)) e−σ(ϑ)tt

))
. (26)

Theorem 5. Let h(ϑ) and σ(ϑ) and F (Ψ(ϑ) t)) are differentiable (σ′(ϑ) ̸= 0), then,

(5.A)

GN [tn F ′(t)] = (−1)n
h(ϑ)

σ′(ϑ)

[
ϑ

(
1

σ′(ϑ)

(
ϑ

(
1

σ′(ϑ)
. . .︸ ︷︷ ︸

n−1 times

(
ϑ

(
GN [F ′(t)]

h(ϑ)

)))))]

+
h(ϑ)

σ′(ϑ)

n−1∑
k=0

(−1)k
[
ϑ

(
1

σ′(ϑ)

(
ϑ

(
1

σ′(ϑ)
. . .︸ ︷︷ ︸

k times

×
(∫ ∞

0
(t)n−k−1

ϑ
(F ′(Ψ(ϑ)t)) e−σ(ϑ) tt

))))]
. (27)

(5.B)

GN [tn F ′′(t)] = (−1)n
h(ϑ)

σ′(ϑ)

[
ϑ

(
1

σ′(ϑ)

(
ϑ

(
1

σ′(ϑ)
. . .︸ ︷︷ ︸

n−1 times

(
ϑ

(
GN [F ′′(t)]

h(ϑ)

)))))]

+
h(ϑ)

σ′(ϑ)

n−1∑
k=0

(−1)k
[
ϑ

(
1

σ′(ϑ)

(
ϑ

(
1

σ′(ϑ)
. . .︸ ︷︷ ︸

k times

×
(∫ ∞

0
(t)n−k−1

ϑ
(F ′′(Ψ(ϑ)t)) e−σ(ϑ) tt

))))]
. (28)



J.I. Mustafa / Eur. J. Pure Appl. Math, 16 (2) (2023), 1024-1046 1033

(5.C)

GN [tn F (m)(t)] =

(−1)n
h(ϑ)

σ′(ϑ)

[
ϑ

(
1

σ′(ϑ)

(
ϑ

(
1

σ′(ϑ)
. . .︸ ︷︷ ︸

n−1 times

(
ϑ

(
GN [F (m)(t)]

h(ϑ)

)))))]

+
h(ϑ)

σ′(ϑ)

n−1∑
k=0

(−1)k
[
ϑ

(
1

σ′(ϑ)

(
ϑ

(
1

σ′(ϑ)
. . .︸ ︷︷ ︸

k times

×
(∫ ∞

0
(t)n−k−1

ϑ
(F (m)(Ψ(ϑ)t)) e−σ(ϑ) tt

))))]
. (29)

where, in the same way as the theorems (2-4), this theorem is proven.

4. Some properties of the generalization of integral transforms

Here, we study some properties for the GN of integral transforms.

Theorem 6. (Linearity property) Let S(t) and P (t) be functions defined for t ≥ 0,
then,

GN{η S(t) + ω P (t)} = η GN{S(t)}+ ω GN{P (t)}, (30)

here, η and ω are scalers.

Proof. Depending on the definition of the GN of integral transforms, we obtain:

GN{η S(t) + ω P (t)} = h(ϑ)

∫ ∞

0
[η S(Ψ(ϑ)t) + ω P (Ψ(ϑ)t)] e−σ(ϑ)tt

= η h(ϑ)

∫ ∞

0
S(Ψ(ϑ)t)e−σ(ϑ)tt+ ω h(ϑ)

∫ ∞

0
P (Ψ(ϑ)t)e−σ(ϑ)tt

= η GN{S(t)}+ ω GN{P (t)}. (31)

Theorem 7. (Convolution property) If S(t) and P (t) are of exponential order and
piecewise continuous on [0, 1), then,

GN{S(t) ∗ P (t)} = GN1{S(t)} GN2{P (t)}. (32)

Proof. Applying the general transform GN on both functions S(t) and P (t), we obtain

GN1{S(t)} =h(ϑ)

∫ ∞

0
S(Ψ(ϑ)ϵ)e−σ(ϑ)ϵϵ,



J.I. Mustafa / Eur. J. Pure Appl. Math, 16 (2) (2023), 1024-1046 1034

GN2{P (t)} =h(ϑ)

∫ ∞

0
P (Ψ(ϑ)ν)e−σ(ϑ)νν,

GN1{S(t)}GN2{P (t)} =

(
h(ϑ)

∫ ∞

0
S(Ψ(ϑ)ϵ)e−σ(ϑ)ϵϵ

)
×
(
h(ϑ)

∫ ∞

0
P (Ψ(ϑ)ν)e−σ(ϑ)νν

)
=h2(ϑ)

(∫ ∞

0

∫ ∞

0
S(Ψ(ϑ)ϵ) P (Ψ(ϑ)ν)e−σ(ϑ)(ϵ+ν)ϵ ν

)
=h2(ϑ)

∫ ∞

0
S(Ψ(ϑ)ν)ν

∫ ∞

0
P (Ψ(ϑ)ϵ)e−σ(ϑ)(ϵ+ν)ϵ. (33)

Let,

t =ϵ+ ν, t = ϵ, (34)

substitute (34) into (33), leads to

GN1{S(t)} GN2{P (t)} =h2(ϑ)

∫ ∞

0
S(Ψ(ϑ)ν)ν

∫ ∞

0
e−σ(ϑ)t P (Ψ(ϑ)(t− ν))t

=h2(ϑ)

∫ ∞

0
e−σ(ϑ)tt

∫ t

0
S(Ψ(ϑ)ν) P (Ψ(ϑ)(t− ν))ν

=h(ϑ)

∫ ∞

0
e−σ(ϑ)t

(
h(ϑ)

∫ t

0
S(Ψ(ϑ)ν)P (Ψ(ϑ)(t− ν))ν

)
t.

So,

GN1{S(t)} GN2{P (t)} =GN{S(t) ∗ P (t)}. (35)

Theorem 8. (Commutativity property) The convolution between two functions is
commutative.

Proof. Let S(t) and P (t), then

S(t) ∗ P (t) = h(ϑ)

∫ ∞

0
e−σ(ϑ)νS(Ψ(ϑ)ν)P (Ψ(ϑ)(t− ν))ν. (36)

Suppose,
t− ν = ω, −ν = ω, (37)

substitute the relation in (37) into (36), gives,

S(t) ∗ P (t) =− h(ϑ)

∫ −∞

t
e−σ(ϑ)(t−ω)S(Ψ(ϑ)(t− ω))P (Ψ(ϑ)ω)ω,

=h(ϑ)

∫ ∞

0
e−σ(ϑ)ωP (Ψ(ϑ)ω) S(Ψ(ϑ)(t− ω))ω.

So,

S(t) ∗ P (t) =P (t) ∗ S(t). (38)



J.I. Mustafa / Eur. J. Pure Appl. Math, 16 (2) (2023), 1024-1046 1035

5. Mathematical Method: The generalization of integral transforms
with He’s polynomial for the PDEs

As a starting point, we consider nonlinear PDEs in the following equation:

L̇[g(x, t)] +R[g(x, t)] + G(x, t) = 0, (39)

here, L̇ is an invertible operator of first order, R consists of linear as well as nonlinear
functions, g(x, 0) is the initial condition (IC) for (39) and g(x, 0) and G(x, t) are both
known functions.
First, the GN to (39) can be applied as:

GN [L̇[g(x, t)]] +GN [R[g(x, t)]] +GN [G(x, t)] = 0. (40)

Now, the IC of (39) and theorem (2) will be used as follows,

− h(ϑ)

ψ(ϑ)
g(x, 0) +

σ(ϑ)

ψ(ϑ)
GN [g(x, t)] +GN [R[g(x, t)]] +GN [G(x, t)] = 0,

σ(ϑ)

ψ(ϑ)
GN [g(x, t)] =

h(ϑ)

ψ(ϑ)
g(x, 0)− (GN [R[g(x, t)]] +GN [G(x, t)]) ,

GN [g(x, t)] =
ψ(ϑ)

σ(ϑ)

(
h(ϑ)

ψ(ϑ)
g(x, 0)− (GN [R[g(x, t)]] +GN [G(x, t)])

)
,

GN [g(x, t)] =
h(ϑ)

σ(ϑ)
g(x, 0)− ψ(ϑ)

σ(ϑ)
(GN [R[g(x, t)]] +GN [G(x, t)]) . (41)

Taking the inverse of the GN of integral transforms (GN−1) to (41), we can find g(x, t),
this yields,

g(x, t) = g(x, 0)−GN−1

[
ψ(ϑ)

σ(ϑ)
GN [R[g(x, t)] + G(x, t)]

]
. (42)

So, the solution can be represented by an infinite series:

g(x, t) =
∞∑
n=0

gn(x, t), (43)

and by using He’s polynomial, we will deal with the nonlinear parts

R[g(x, t)] =

∞∑
n=0

Hn, (44)

where, Hn is defined as fallow

Hn(g0, . . . , gn) =
1

n!

∂n

∂pn

[
R

( ∞∑
n=0

pngn(x, t)

)]
p=0

, (45)



J.I. Mustafa / Eur. J. Pure Appl. Math, 16 (2) (2023), 1024-1046 1036

here, n = 0, 1, 2, . . . , substitute (43) and (44) into (42), gives:

∞∑
n=0

gn(x, t) = g(x, 0)−GN−1

[
ψ(ϑ)

σ(ϑ)
GN

[ ∞∑
n=0

Hm
n +

∞∑
n=0

gn(x, t) + G(x, t)

]]
, (46)

where, m represents the number of nonlinear terms in (39). Compering both sides of (46),
we obtain,

g0(x, t) =g(x, 0),

g1(x, t) =−GN−1

[
ψ(ϑ)

σ(ϑ)
GN [Hm

0 + g0 + G(x, t)]
]
,

g2(x, t) =−GN−1

[
ψ(ϑ)

σ(ϑ)
GN [Hm

1 + g1 + G(x, t)]
]
,

then, the general form will be as,

gn+1(x, t) = −GN−1

[
ψ(ϑ)

σ(ϑ)
GN [Hm

n + gn + G(x, t)]
]
, (47)

here, n ≥ 0. This calculation results a series expression and the convergence of this
expression leads to an approximate solution.

6. Examples: Numerical results for nonlinear PDEs

Here, we apply our mathematical method to three different examples of PDEs.

Example 1. Let us consider the nonlinear equation for gas dynamics,

gt(x, t) =− g(x, t) gx(x, t) + g(x, t)(1− g(x, t)), (48)

g(x, 0) =e−x.

Here, G(x, t) = 0, first, we apply the GN integral transforms of (48):

− h(ϑ)

ψ(ϑ)
g(x, 0) +

σ(ϑ)

ψ(ϑ)
GN [g(x, t)] = GN [−ggx + g − g2(x, t)],

σ(ϑ)

ψ(ϑ)
GN [g(x, t)] =

h(ϑ)

ψ(ϑ)
g(x, 0) +GN [−ggx + g − g2(x, t)],

multiply both side by ψ(ϑ)
σ(ϑ) , gives

GN [g(x, t)] =
h(ϑ)

σ(ϑ)
e−x +

ψ(ϑ)

σ(ϑ)
GN [−ggx + g − g2(x, t)], (49)

take GN−1 to (49), then we obtain:

g(x, t) =GN−1

[
h(ϑ)

σ(ϑ)
e−x +

ψ(ϑ)

σ(ϑ)
GN [−ggx + g − g2(x, t)]

]
,



J.I. Mustafa / Eur. J. Pure Appl. Math, 16 (2) (2023), 1024-1046 1037

g(x, t) =e−x +GN−1

[
ψ(ϑ)

σ(ϑ)
GN [−ggx + g − g2(x, t)]

]
. (50)

Now, we deal with the nonlinear parts g gx and g2 by using He’s polynomial. So, using
the general form (47) will give,

gn+1 = −GN−1

[
ψ(ϑ)

σ(ϑ)
GN

[
H1
n − gn +H2

n

]]
, n ≥ 0, (51)

where, H1 = gg′ and H2 = g2, and g0 = e−x, using the formula (45), we have

H1
0 =g0g

′
0 = −e−2x, (52)

H2
0 =g20 = e−2x. (53)

So,

g1(x, t) =−GN−1

[
ψ(ϑ)

σ(ϑ)
GN

[
H1

0 − g0 +H2
0

]]
=−GN−1

[
ψ(ϑ)

σ(ϑ)
GN

[
−e−2x − e−x + e−2x

]]
=GN−1

[
ψ(ϑ)e−x

σ(ϑ)
GN [1]

]
=GN−1

[
ψ(ϑ)e−x

σ(ϑ)
× h(ϑ)

σ(ϑ)

]
,

then,
g1(x, t) = t e−x. (54)

The next step, we need,

H1
1 =g0g

′
1 + g1g

′
0 = −2te−2x, (55)

H2
1 =2g0g1 = 2te−2x. (56)

So, the second iteration is:

g2(x, t) =−GN−1

[
ψ(ϑ)

σ(ϑ)
GN

[
H1

1 − g1 +H2
1

]]
=−GN−1

[
ψ(ϑ)

σ(ϑ)
GN

[
−2te−2x − te−x + 2te−2x

]]
=GN−1

[
ψ(ϑ)e−x

σ(ϑ)
GN [t]

]
=GN−1

[
ψ(ϑ)e−x

σ(ϑ)
× h(ϑ)ψ(ϑ)

σ2(ϑ)

]
=e−xGN−1

[
h(ϑ)ψ2(ϑ)

σ3(ϑ)

]
,



J.I. Mustafa / Eur. J. Pure Appl. Math, 16 (2) (2023), 1024-1046 1038

Figure 1: The plot of the solution (59)

then,

g2(x, t) =
1

2
t2 e−x, (57)

and so on,

g(x, t) = lim
n→∞

n∑
k=0

gk(x, t) = e−x + te−x +
1

2
t2e−x + . . . . (58)

In such a way, the series solution (58) of (48) represents Taylor’s expansion of the function
g(x, t)

g(x, t) =et−x, (59)

in the variable t (see Figure (1)).

Example 2. Meditate the system of nonlinear Burgers’ equation,

gt(x, t)−gxx(x, t)− 2 g(x, t) gx(x, t) + (g(x, t)v(x, t))x = 0,

vt(x, t)−vxx(x, t)− 2 v(x, t) vx(x, t) + (g(x, t)v(x, t))x = 0, (60)

g(x, 0) = sin(x) = v(x, 0).

Here, G(x, t) = 0. First apply the GN of integral transforms of (60):

−h(ϑ)

ψ(ϑ)
g(x, 0) +

σ(ϑ)

ψ(ϑ)
GN [g(x, t)] = −GN [−gxx(x, t)− 2g(x, t)gx(x, t) + (g(x, t)v(x, t))x],

−h(ϑ)

ψ(ϑ)
g(x, 0) +

σ(ϑ)

ψ(ϑ)
GN [g(x, t)] = −GN [−vxx(x, t)− 2v(x, t)vx(x, t) + (g(x, t)v(x, t))x],

(61)



J.I. Mustafa / Eur. J. Pure Appl. Math, 16 (2) (2023), 1024-1046 1039

apply GN−1 to (61), then we obtain:

g(x, t) =g(x, 0)−GN−1

[
ψ(ϑ)

σ(ϑ)
GN [−gxx(x, t)− 2 g(x, t) gx(x, t) + (g(x, t)v(x, t))x]

]
,

v(x, t) =v(x, 0)−GN−1

[
ψ(ϑ)

σ(ϑ)
GN [−vxx(x, t)− 2 v(x, t) vx(x, t) + (g(x, t)v(x, t))x]

]
.

(62)

Now, we deal with the nonlinear parts g gx, v vx and (g v)x by using He’s polynomial and
the general formula (47), as follows:

gn+1 =−GN−1

[
ψ(ϑ)

σ(ϑ)
GN

[
−gn,xx − 2H1

n +H2
n

]]
,

vn+1 =−GN−1

[
ψ(ϑ)

σ(ϑ)
GN

[
−vn,xx − 2H3

n +H2
n

]]
,

(63)

here, n ≥ 0, H1 = gg′, H3 = vv′ and H2 = (gv)x, and since g0 = v0 = sin(x), H2
n =

(gv)x = g2x or v2x, then using the formula (45) yields,

H1
0 =g0g

′
0 = sin(x) cos(x), g0(x, t) = g(x, 0) = sin(x),

H3
0 =v0v

′
0 = sin(x) cos(x), v0(x, t) = v(x, 0) = sin(x),

H2
0 =(g20)x = 2g0g

′
0 = 2 sin(x) cos(x),

g1 =−GN−1

[
ψ(ϑ)

σ(ϑ)
GN

[
−g0,xx − 2H1

0 +H2
0

]]
,

v0 =−GN−1

[
ψ(ϑ)

σ(ϑ)
GN

[
−v0,xx − 2H3

0 +H2
0

]]
,

g1 = −GN−1

[
ψ(ϑ)

σ(ϑ)
GN [sin(x)− 2 sin(x) cos(x) + 2 sin(x) cos(x)]

]
,

v1 = −GN−1

[
ψ(ϑ)

σ(ϑ)
GN [sin(x)− 2 sin(x) cos(x) + 2 sin(x) cos(x)]

]
,

g1 =−GN−1

[
ψ(ϑ)

σ(ϑ)
sin(x)

h(ϑ)

σ(ϑ)

]
,

v1 =−GN−1

[
ψ(ϑ)

σ(ϑ)
sin(x)

h(ϑ)

σ(ϑ)

]
,

g1 =− sin(x)t,

v1 =− sin(x)t.



J.I. Mustafa / Eur. J. Pure Appl. Math, 16 (2) (2023), 1024-1046 1040

Figure 2: The plot of the solution (65)

So, in the second iteration, we need to find the second He’s polynomial,

H1
1 =g0g

′
1 + g′0 + g1 = 2 sin(x) cos(x)t,

H3
1 =v0v

′
1 + v′0 + v1 = 2 sin(x) cos(x)t,

H2
1 =(2g0g1)x = 4 sin(x) cos(x)t,

(64)

then, substitute (64) into the system (63), yields,

g2 =
1

2
sin(x)t2,

v2 =
1

2
sin(x)t2,

and so on, then, the Taylor’s expansion of g(x, t) and v(x, t) give the following solutions
(see Figure (2)):

g(x, t) = lim
n→∞

n∑
k=0

gk =sin(x)e−t,

v(x, t) = lim
n→∞

n∑
k=0

vk =sin(x)e−t.

(65)

Example 3. The non-homogeneous gas dynamic equation is given below,

gt(x, t) =− g(x, t) gx(x, t) + g(x, t)(1− g(x, t))− et−x, (66)

g(x, 0) =1− e−x.

First take general integral transform of (66) as:

−h(ϑ)

ψ(ϑ)
g(x, 0) +

σ(ϑ)

ψ(ϑ)
GN [g(x, t)] = GN [−ggx + g − g2(x, t)]−GN [et−x],



J.I. Mustafa / Eur. J. Pure Appl. Math, 16 (2) (2023), 1024-1046 1041

σ(ϑ)

ψ(ϑ)
GN [g(x, t)] =

h(ϑ)

ψ(ϑ)
g(x, 0)− e−x

h(ϑ)

σ(ϑ)− ψ(ϑ)
+GN [−ggx + g − g2(x, t)],

then, multiply both side by ψ(ϑ)
σ(ϑ) , gives,

GN [g(x, t)] =
h(ϑ)

σ(ϑ)
(1− e−x)− e−x

ψ(ϑ)

σ(ϑ)

h(ϑ)

σ(ϑ)− ψ(ϑ)
+
ψ(ϑ)

σ(ϑ)
GN [−ggx + g − g2(x, t)],

(67)

take GN−1 to (67), we obtain:

g(x, t) = GN−1

[
ψ(ϑ)

σ(ϑ)
(1− e−x) −e−x

ψ(ϑ)

σ(ϑ)

h(ϑ)

σ(ϑ)− ψ(ϑ)

+
ψ(ϑ)

σ(ϑ)
GN [−ggx + g − g2(x, t)]]

]
,

g(x, t) = 1− e−x − e−x( et − 1) +GN−1

[
ψ(ϑ)

σ(ϑ)
GN [−ggx + g − g2(x, t)]

]
. (68)

After that, we will deal with the nonlinear parts g gx and g2 by using He’s polynomial.
So, using the general form (47) will be as,

gn+1 = −GN−1

[
ψ(ϑ)

σ(ϑ)
GN

[
H1
n − gn +H2

n

]]
, n ≥ 0, (69)

where, H1 = gg′ and H2 = g2, and g0(x, t) = 1− et−x, using the formula (45), we have

H1
0 =g0g

′
0 = et−x − e2(t−x), (70)

H2
0 =g20 = 1− 2et−x + e2(t−x), (71)

so,

g1(x, t) =−GN−1

[
ψ(ϑ)

σ(ϑ)
GN

[
H1

0 − g0 +H2
0

]]
=−GN−1

[
ψ(ϑ)

σ(ϑ)
GN

[
et−x − e2(t−x) − 1 + et−x + 1− 2et−x + e2(t−x)

]]
=GN−1

[
ψ(ϑ)e−x

σ(ϑ)
GN [1]

]
=0,

then,
g1(x, t) = 0. (72)

The next step, we need

H1
1 =g0g

′
1 + g1g

′
0 = 0, (73)



J.I. Mustafa / Eur. J. Pure Appl. Math, 16 (2) (2023), 1024-1046 1042

Figure 3: The plot of the exact solution (76)

H2
1 =2g0g1 = 0. (74)

So,

g2(x, t) =−GN−1

[
ψ(ϑ)

σ(ϑ)
GN

[
H1

1 − g1 +H2
1

]]
=0,

and so on. Thus, the series solution g(x, t)

g(x, t) = lim
n→∞

∞∑
k=0

gk(x, t)

=1− et−x + 0 + . . . , (75)

gives the exact solution of (66) (see Figure (3)):

g(x, t) =1− et−x. (76)

7. Conclusion

We briefly summarized the output of the current paper. First of all, the GN of inte-
gral transforms has been developed by formulating a new form of the generalized integral
transform. The development started with introducing some definitions about the func-
tions which are related to the GN . Then, formulating the theorems of the GN , in which
we proved that the GN of integral transforms of some functions and the derivatives of
unknown functions, which were used in the equations, has been fulfilled. These transforms
have been used as an essential tool in determining the solutions to equations. Additionally,



REFERENCES 1043

the development of the GN of integral transform involved studying some useful proper-
ties. In a second study, a mathematical method has been exhibited by using our newly
developed GN together with He’s polynomial method to solve nonlinear PDEs where the
He’s polynomial method evaluates nonlinear terms. For this combination of our GN and
He’s polynomial method, a convergent series has been obtained. Consequently, a vari-
ety of equations have been successfully solved using this combination, including PDEs in
three types of equations. The nonlinear gas dynamic equation and the system of coupled
nonlinear Burgers’ equation have been solved in the first and second examples, respec-
tively, which determined approximate solution (see Figure (1) and (2)), whereas in the
third example, the non-homogeneous gas dynamic equation has been solved to obtain an
exact solution (see Figure (3)). Finally, the advantages of our method are its efficiency
and accuracy as it led to more accurate results and more efficient calculations, as well as
the fact that it can be used to solve any order of nonlinear equations. This makes it an
invaluable tool for finding the solutions of a various problems in mathematical physics.

Acknowledgements

The author is very grateful to the “College of Education for Pure Sciences at the
University of Mosul, Iraq” for their support of my research.

References

[1] Shams A Ahmed, Ahmad Qazza, and Rania Saadeh. Exact solutions of nonlinear
partial differential equations via the new double integral transform combined with
iterative method. Axioms, 11(6):247, 2022.

[2] Hossein Aminikhah and Ali Jamalian. Numerical approximation for nonlinear gas
dynamic equation. International Journal of Partial Differential Equations, 2013,
2013.

[3] Mohannad Hamid Eljaily Babiker et al. Solution of Partial Differential Equations
with Nonlocal Conditions by Combine Homotopy Perturbation Method and Laplace
Transform. PhD thesis, Sudan University of Science and Technology, 2016.

[4] Benedict Barnes, C Sebil, and A Quaye. A generalization of integral transform. 2018.

[5] Rachid Belgacem, Ahmed Bokhari, Mohamed Kadi, and Djelloul Ziane. Solution
of non-linear partial differential equations by shehu transform and its applications.
Malaya Journal of Matematik, 8(4):1974–1979, 2020.

[6] Changbum Chun. Application of homotopy perturbation method with chebyshev
polynomials to nonlinear problems. Zeitschrift für Naturforschung A, 65(1-2):65–70,
2010.



REFERENCES 1044

[7] Ravi Shankar Dubey, Pranay Goswami, Vinod Gill, et al. A new analytical method
to solve klein-gordon equations by using homotopy perturbation mohand transform
method. Malaya Journal of Matematik, 10(1):1–19, 2022.

[8] AI El-Mesady, YS Hamed, and AM Alsharif. Jafari transformation for solving a
system of ordinary differential equations with medical application. fractal fract. 2021,
5, 130, 2021.

[9] Asghar Ghorbani. Beyond adomian polynomials: he polynomials. Chaos, Solitons &
Fractals, 39(3):1486–1492, 2009.

[10] Asghar Ghorbani and Jafar Saberi-Nadjafi. He’s homotopy perturbation method for
calculating adomian polynomials. International Journal of Nonlinear Sciences and
Numerical Simulation, 8(2):229–232, 2007.

[11] Georgi G Grahovski, Amal J Mohammed, and Hadi Susanto. Nonlocal reductions
of the ablowitz–ladik equation. Theoretical and Mathematical Physics, 197(1):1412–
1429, 2018.

[12] Georgi G Grahovski, Junaid I Mustafa, and Hadi Susanto. Nonlocal reductions of
the multicomponent nonlinear schrödinger equation on symmetric spaces. Theoretical
and Mathematical Physics, 197(1):1430–1450, 2018.

[13] Murat Gubes. A new calculation technique for the laplace and sumudu transforms by
means of the variational iteration method. Mathematical Sciences, 13:21–25, 2019.

[14] Hossein Jafari. A new general integral transform for solving integral equations. Jour-
nal of Advanced Research, 32:133–138, 2021.

[15] C Jesuraj and A Rajkumar. A new modified sumudu transform called raj transform
to solve differential equations and problems in engineering and science. International
Journal on Emerging Technologies, 11(2):958–964, 2020.

[16] Artion Kashuri, Akli Fundo, and Matilda Kreku. Mixture of a new integral trans-
form and homotopy perturbation method for solving nonlinear partial differential
equations. 2013.

[17] Bachir Nour Kharrat. A new integral transform: Kharrat-toma transform and its
properties. World Applied Sciences Journal, 38(5):436–443, 2020.

[18] Adem Kılıçman and Rathinavel Silambarasan. Computing new solutions of algebro-
geometric equation using the discrete inverse sumudu transform. Advances in Differ-
ence Equations, 2018(1):1–17, 2018.

[19] Kevser Köklü. Resolvent, natural, and sumudu transformations: solution of loga-
rithmic kernel integral equations with natural transform. Mathematical Problems in
Engineering, 2020, 2020.



REFERENCES 1045

[20] MAM Mahgoub and Abdelbagy A Alshikh. An application of new transform “mah-
goub transform” to partial differential equations. Mathematical theory and Modeling,
7(1):7–9, 2017.

[21] Amal Jasim Mohammed, Sohaib Talal Al-Ramadhani, and Rabeea Mohammed Hani
Darghoth. The possible solutions for the two kdv-type equations using a semi-
analytical kamal-iteration method. European Journal of Pure and Applied Mathe-
matics, 15(4):1917–1936, 2022.

[22] Amal Jasim Mohammed and Ahmed Farooq Qasim. A new procedure with iteration
methods to solve a nonlinear two dimensional bogoyavlensky-konopelchenko equation.
Journal of Interdisciplinary Mathematics, 25(2):537–552, 2022.

[23] Oludapo Omotola Olubanwo, Olutunde Samuel Odetunde, and Adetoro Temitope
Talabi. Aboodh homotopy perturbation method of solving burgers equation. Asian
Journal of Applied Sciences, 7(2), 2019.

[24] Dinkar Patil. Aboodh and mahgoub transform in boundary value problems of sys-
tem of ordinary differential equations. DP Patil, Aboodh and Mahgoub Transform
in Boundary Value Problems of System of Ordinary Differential Equations, Inter-
national Journal of Advanced Research in Science, Communication and Technology
(IJARSCT), pages 67–75, 2022.

[25] Patarawadee Prasertsang, Supaknaree Sattaso, Kamsing Nonlaopon, and Hwajoon
Kim. Analytical study for certain ordinary differential equations with variable co-
efficients via gα-transform. European Journal of Pure and Applied Mathematics,
14(4):1184–1199, 2021.

[26] LS Sawant. Applications of laplace transform in engineering fields. International
Research Journal of Engineering and Technology, 5(5):3100–3105, 2018.

[27] Dinkar Sharma, Prince Singh, and Shubha Chauhan. Homotopy perturbation trans-
form method with he’s polynomial for solution of coupled nonlinear partial differential
equations. Nonlinear engineering, 5(1):17–23, 2016.

[28] Pilasluck Sornkaew and Kanyarat Phollamat. Solution of partial differential equations
by using mohand transforms. In Journal of Physics: Conference Series, volume 1850.
IOP Publishing, 2021.

[29] Betty Subartini, Ira Sumiati, Riaman Sukono, and Ibrahim Mohammed Sulaiman.
Combined adomian decomposition method with integral transform. 2021.

[30] Janki Vashi and MG Timol. Laplace and sumudu transforms and their application.
Int. J. Innov. Sci., Eng. Technol, 3(8):538–542, 2016.

[31] Yuan Wei, Li Yin, and Xin Long. The coupling integrable couplings of the general-
ized coupled burgers equation hierarchy and its hamiltonian structure. Advances in
Difference Equations, 2019(1):1–17, 2019.



REFERENCES 1046

[32] Djelloul Ziane, Rachid Belgacem, and Ahmed Bokhari. A new modified adomian
decomposition method for nonlinear partial differential equations. Open J. Math.
Anal, 3:81–90, 2019.