Solution of Three-Dimensional Mboctara Equation Via Gamar
Transform
ABDELILAH KAMAL.H.SEDEEG
Department of Mathematics, Faculty of Education, University of Holy Quran and Islamic Sciences,
Omdurman P. O Box 14411, SUDAN.
Department of Mathematics, Faculty of Sciences and Arts-Almikwah, Al-Baha University, Al-Bahah
P. O Box 1988, SAUDI ARABIA.
Department of Physics and Mathematics, College of Sciences and Technology, Merowe University of
Technology- Abdulatif Alhamad, Merowe, SUDAN.
Abstract: - The objective of present paper is to introduce and apply a novel general triple integral transform
named Gamar Transform. First, we outline the basic properties and then establish same important results such
as the existence and triple convolution theorems as well as the derivatives properties. Furthermore, the current
transform is applied to solve a wide range of partial differential equations including homogeneous and
nonhomogeneous Mboctara partial differential equations. Figures are utilized to clarify and exemplify the
solutions.
Key-Words: - Laplace transform; Gamar transform; general triple convolution theorem; partial derivatives;
Mboctara equation.
Received: August 7, 2023. Revised: July 23, 2024. Accepted: August 22, 2024. Published: September 26, 2024.
1 Introduction
Integral transform methods have proved their
value among the most powerful and effective ways
in solving partial differential equations (PDEs).
Their applications cover a wide range of phenomena
in the disciplines of mathematics, physics,
engineering, to mention only a few scientific
specializations [1-13]. Thereby, it is feasible to
transform PDEs in terms of algebraic equations and
thus obtain exact solutions of PDEs.
Many scholars have focused great labor both to
develop and enhance these new methods prior to
applying them to resolve a wide spectrum of
problems in the realm of mathematics. These
methods are best represented by Fourier and Laplace
transforms, to cite only few illustrative examples
[14-25].
In more recent times, triple Laplace transforms
have been extensively used to solve PDEs,
specifically those with unknown function of three
variables, with the ultimate goal of obtaining more
satisfactory solutions accompanying the new method
[26-28]. Furthermore, many enhancements and
extensions have been devised by researchers to the
original triple Laplace transform. These include
triple Sumudu transform [29], triple Elzaki
transform [30], triple Aboodh transform [31] ,triple
Shehu transform [32] , triple Natural transform [33]
,triple Kamal transform [34] and triple Laplace-
ARA -Sumudu transform [35], all of which are
extensions and modifications to the original Laplace
transform.
For his part, Jafri has proposed a new general
integral transform, namely Jafri transform [36],
which is illustrated as follows:
where and are regular complex functions
such that for all belongs to complex
number.
International Journal of Computational and Applied Mathematics & Computer Science
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Recently, Meddahi et.al have proposed a general
double transform [37] defined by:
where
and
are the transform functions
for
and
respectively.
More recently, Abdelilah [38] is introduced a novel
triple general integral transform known as Gamar
transform is defined as:
provided that all integrals exist for some
and , where and are
general transform functions for and ,
respectively.
The inverse Gamar transform is defined by
where and are real constants.
Considered as a special kind of both third-order
homogeneous and nonhomogeneous partial
differential equations, the Mboctara equation, this
equation is mainly utilized to investigate the nature
of collective motion concerning micro-particles in
materials. It is possible to solve this equation by
means of any triple integral transform. Though been
a comparatively new operator, the Gamar transform
has proved its effectiveness in solving various
differential equations [38]. Consequently, it has
been widely adopted in different fields, including
physics, engineering and material science. For
instance, in engineering, it has greatly helped to
model the nature of behavior of certain fluids and to
analyze the dynamics of some kinds of waves.
In this study, we consider the Mboctara partial
differential equations of the following form:
Subject to the boundary and initial conditions
and , where is an unknown
function, is the source term.
A novel concept termed Gamar transform is
introduced in the current study which is specifically
directed to functions with three variables. As a
foundation, we first establish and prove some key
theorems that comprise existence and triple
convolution, among other properties. Subsequently,
we obtain the Gamar transform regarding some
basic functions. Likewise, the Gamar transform of
some partial differential derivatives is established
and obtained. The findings prove that the new
general triple transform indeed implies the original
triple Laplace transform. The effectiveness of
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DOI: 10.37394/232028.2024.4.9
Abdelilah Kamal. H. Sedeeg
E-ISSN: 2769-2477
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Volume 4, 2024
Gamar transforms to solve Mboctara partial
differential equations upon applications is firmly
proved.
2 Some Properties and Theorems of
Gamar transform [38]
In this section, we proceed to prove some basic
properties and theorems such as existence, triple
convolution.
Property 2.1. (Linearity). If
and , then for
any constants and , we have
Proof of Property 2.1. From the definition of
Gamar transform, we obtain
Thus, Gamar transform is linear integral
transformation. Similarly, we can prove that the
inverse Gamar transform is also linear.
Property 2.2.Let
and. Then
where , and are general integral transform
for , and respectively.
Proof of Property 2.2. From the definition of
Gamar transform, we obtain
∞
∞
∞
Definition 2.1. If defined on
satisfies the condition
Y and
Then, is called a function of
exponential orders , and as .
Theorem 2.1.The existence condition of Gamar
transform of the continuous function
defined on is to be of
exponential orders , and , for ,
and .
Proof of Theorem 2.1. From the definition of
Gamar transform, we get
where
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Definition 2.3. The triple convolution of
and is denoted by
defined by
Theorem 2.2. Let .
Then,
(8)
where denotes the unit step function
defined by
Theorem 2.3. (General Triple Convolution Theorem).
If G and G,
then
3. Gamar Transform for Some Basic
Functions
In this section, we introduce the Gamar
transform for some basic functions.
i. Let .Then
From Property 2.2, we have
∞
∞
∞
Thus,
ii. Let and
.Then
∞
∞
∞
From Property 2.2, we obtain
∞
∞
∞
By integrating by parts, we have
Thus, by induction, we prove
Γ
iii. Let
andand and are constants. Then
From Property 2.2, we obtain:
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Thus,
Similarly,
Thus, one can obtain
Using Euler’s formulas:
And the formulas:
Then, we find the Gamar transform of the following
functions:
4. Gamar transform for Partial
Differential Derivatives [38]
In this section, we present some results related to
the Gamar transform of partial derivatives. We
begin by obtaining partial derivatives with respect to
and .
Theorem 4.1. (Derivative properties with respect
to). Let is Gamar transform of
and is general double
transform of , then:
a)
b)
c)
Theorem 4.2. (Derivative properties with respect
to). Let is Gamar transform of
and
is general double
transform of , then:
a)
.
b)
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c)
Theorem 4.3. (Derivative properties with respect
to ). Let is Gamar transform of
and
is general double
transform of , then:
a)
b)
.
c)
Theorem 4.4. (Derivative properties with respect
to ). Let is Gamar transform of
, then
Corollary 4.1.(Gamar transform of integral). Let
and be positive functions and let
the Gamar transform of , then
where for all .
Theorem 4.5. Let is Gamar transform
of , then
a)
b)
c)
5. Applications
In this section, we apply the properties associated
with Gamar transform established above to solve
homogeneous and nonhomogeneous three-
dimensional Mboctara partial differential equations.
All the following figures of the selected examples
were obtained using Mathematica software 13.
Example 5.1
Consider the following homogeneous three-
dimensional Mboctara partial differential equation
Subject to the boundary and initial conditions
Applying Gamar transform on both sides of Eq. (10),
we have
By linearity property and partial derivative
properties of Gamar transform, we get
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Substituting
in Eq. (13) and simplifying, we obtain:
Taking inverse Gamar transform for Eq. (14), we get
Figure 1. The exact solution of Example 5.1.
Example 5.2
Consider the following nonhomogeneous third-
order Mboctara partial differential equation
Subject to the boundary and initial conditions
Applying Gamar transform on both sides of Eq. (15),
we have
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By linearity property and partial derivative
properties of Gamar transform, we get
Substituting
in Eq. (18) and simplifying, we obtain:
Taking inverse Gamar transform for Eq. (19), we get
Figure 2. The exact solution of Example 5.2.
Example 5.3
Consider the following nonhomogeneous third-
order Mboctara partial differential equation
Subject to the boundary and initial conditions
,
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Applying Gamar transform on both sides of Eq. (21),
we have
By linearity property and partial derivative
properties of Gamar transform, we get
Substituting
in Eq. (23) and simplifying, we obtain:
Taking inverse Gamar transform for Eq. (24), we get
Figure 3. The exact solution of Example 5.3.
4 Conclusion
Greatly inspired by the work carried out in the single
integral transform related to one-dimensional spaces
and double integral transform in two-differential
spaces, the new notion of Gamar transform is
introduced. This novel transform is characterized by
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its capacity to both refine and imply its original
model, namely Laplace triple transform in positive
quadrant places. The next goal was to prove some
major properties concerning the proposed Gamar
transform, which include, among other properties,
triple convolution theorem. In order to judge and
evaluate the effectiveness of this transform, it is
utilized to solve a selection of PDEs under standard
conditions. As an extension of the of the present
study, we recommend that scholars pursue
investigations dealing with the feasibility of
applying this transform to solve both differential and
functional differential equations.
Acknowledgement:
I would like to acknowledge my colleagues Holy
Quran and Islamic Sciences University for their
technical advice. The author expresses his gratitude
to the editor and dear unknown reviewers and for
their helpful suggestions, which improved the final
version of this manuscript. The author extends his
sincere thanks to Dr.Nauman Ali and Dr.Ahmed
Elhassan for improving the language of the
manuscript. I also thank Dr. Jamal Derbali for
helping me with the physics aspect, drawing
solutions. Finally, I do not forget to thank my wife,
Elham Suleiman, who provided me with moral and
material support.
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Appendix
Table 1: Here we present a list of the previous results Gamar transform of some special function and general properties:
Γ
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The author contributed in the present research, at all
stages from the formulation of the problem to the
final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The author has no conflict of interest to declare that
is relevant to the content of this article.
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