Conformable Triple Sumudu Transform with Applications
GHARIB M. GHARIB1, MAHA S. ALSAUODI2, MOHAMAD ABU-SEILEEK3
1Department of Mathematics, Faculty of Science,
Zarqa University,
Zarqa,
JORDAN
2Department of Science,
Applied Science Private University,
Amman,
JORDAN
3Department of Physics, Faculty of Science,
Zarqa University,
Zarqa,
JORDAN
Abstract: - One of the important topics in applied mathematics is the topic of integral transformations, due to
their importance in electrical engineering applications, including communications in particular, and other
sciences. In this work, one of the most important transformations in its three dimensions was presented, which
is the triple Sumudu transform, including solving some real-life applications of physics, some of which have
not been solved using such an integral transform before. In this work, we extend the Sumudu transform formula
to the conformable fractional order, as well as other interesting and significant rules. The general analytical
solution of a singular and nonlinear conformable fractional differential equation based on the conformable
fractional Sumudu transform is also presented in this paper. The general solutions of several linear and
nonhomogeneous conformable fractional differential equations can be obtained using the method we've
proposed. As a result, our results reveal that our proposed method is an efficient one that can be used for
solving all conformable fractional differential equations. The relationship between the Sumudu integral
transform and other important and recently proposed integral transforms are also discussed. Finally, the triple
Sumudu transform is used to solve boundary value problems, such as the heat equation with boundary values.
The triple Sumudu integral transform is also used to solve linear partial integro-differential equations. The
transform capability to handle such equations has been proven via its utilization in three applications.
Key-Words:
-
Triple Sumudu Transform; Sumudu Transform, Partial differential equation, Integral equation.
Transform, Double Transform, partial integro-differential equations, fractional Sumudu
transform , fractional equation.
Received: March 6, 2023. Revised: October 5, 2023. Accepted: November 6, 2023. Published: January 26, 2024.
1 Introduction
It is possible to solve some ordinary differential
equations, partial linear ones, and non-linear ones,
using different integral transformations in their
different dimensions. Solutions to the Burgers
equation were presented in, [1] and in [2], the
Navier-Stokes equation and similar equations in,
[3]. More than one method was used to solve
various equations, including the Sumudu transform
in, [4] and [5], as well as Storm-Liouville problems
in, [6], explaining integration coefficients in, [7],
and discussing Laplace applications in, [8]. In, [9]
the generalized Lyapunov type appeared, and
Hermite-Hadamard variants for fractional integrals
were presented in, [10]. In, [11], the transformation
properties were presented and deduced. The results
were discussed and presented in, [12]. The nonlinear
biological model was solved in, [13], and Euler's
results were presented in, [14]. All immunodynamic
results appeared, [15], [16] and show random
behavior at different and random values of the
fractional order, [17]. Conversion of symmetry
disturbance and residual energy chain path using the
analytical method, multidimensional thermal
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equations are solved, [18]. To solve the new types
of equations, the space fractional telegraph equation,
the new mixed diffusion equation Yang-Abdel-Aty-
Cattani, and the gas dynamics equation using the
new fractional symmetry analysis conversion
method, [19], [20], [21], [22]. The authors in, [23],
By applying the first integration method, prepare the
exact solutions to Burger's fractional time equations,
[24]. By generalized two-dimensional differential
transformation (DTM) to solve the Berger equations
paired with fractions of space and time, at present,
new concepts and properties of the corresponding
derivative have been identified for more
information, [25], [26], [27]. Moreover, the authors
in, [28], [29] Using the Laplace Transformation
matching technique, differential equations are
solved. In, [30], the Laplace double congruent
transformation method is used to solve partial
differential equations, to obtain solutions to a
regular and single-dimensional equation for a
partially paired burger provided by the authors in,
[31]. To determine the exact solutions of the burger
fractional equations of time, the first integrated
method was used in, [32].
In this work, a new method called congruent
triple Sumudu will be proposed, and a solution is
done by the decomposition method (CTSDM) to
solve a wonderful mixture type for nonlinear
equations, but the proposed methods are conformity,
analysis method, and triple sumudu conversion
method. This article looks at whether they can apply
SUMU conformance triple structure decoding
methods (CTSDM), [33].
In this article, we will solve regular and single-
dimensional identical burger equations and some
basic concepts and definitions of compatible
derivatives will be published later in this article,
[34].
2 The System of Some NLEEs
For the first time, the Korteweg-de Vries equation
(KdV) equation is used in this article (ISM), [7].
Later, the authors extended it in, [23]. ISM was
initially created in this article for the Schrödinger
nonlinear equation (NLSE) and it was then
deepened in, [6], such that it now contains a
different type of NLEE. These steps are used to
create the AKNS method: (i) construct a suitable, 22
linear scattering (eigenvalue) problem using the
"space" variable, where the NLEE's resolution
serves as the potential; (ii) choose the Eigen
functions' "time" dependence so that the eigenvalues
hold steady while the potential changes in
accordance with the NLEEs; (iii) Select a time for
self-functions such that the eigenvalues are
preserved. We will quickly find a solution to the
direct dispersion issue, and we need to establish
how time affects the dispersion data. Rebuild the
potential using the dispersion data in (iv). This
section focuses on the AKNS method's first phase.
As a result, each DE solution yields a scale on M2
with a constant Gaussian curvature of -1.
Additionally, the aforementioned definition of DE is
equal to saying that DE for you is the problem's
integration condition:
(1)
where denotes exterior dierentiation, is a vector,
and the matrix is traceless:
(2)
The NLPDE is given by Equation (1.2),
(3)
It is the initial NLPDE that needs to be addressed.
The AKNS system provided the following
examples, which we demonstrate here.
(a) Equation 1:
(4)
(5)
where
(b) Equation 2:
(6)
(7)
(c) Equation 3:
(8)
(9)
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(d) Equation 4:
(10)
(11)
(e) Equation 5:
(12)
where is a constant,
(13)
where
.
(f) Equations 6, [33]:
(14)
where satisfies , such that
(15)
(g) Equations 7, [25]:
(16)
where , and
(17)
Keeping in mind that the problem in (1) is
played by the parameter. The equations (1), (2), and
(3) are form invariant under the "gauge"
transformation, are not unique for a particular
NLPDE:
(18)
where is matrix,
(19)
Integrability of (1) is,
(20)
requires the vanishing of the two form:
(21)
3 The System Describe PSS
The, [8], gave the problem and defined it by:
(22)
Examples
(a) Equation 1.
(23)
satisfies equation 10.
(b) Equation 2.
(24)
satisfies equation 6.
(c) Equation 3.
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(25)
satisfies equation 8.
(d) Equation 4.
(26)
satisfies 4.
(e) Equation 5.
(27)
satisfies 12.
(f) Equations 6.
(28)
satisfies the equations I 14.
(g) Equations 7.
(29)
satisfies equations II 16.
4 Method on Bäcklund Transforms
BTs are called classical transformations whose
solutions to the same equation are related to the self-
transformation of B acklund (SBT), that the
solutions related to the sg equation are found other
transformations related to the solutions of specific
equations in, [17], [19], [34], [35].
These conversions are called BTS.
After the classical theory of B acklund has
arisen in the study of pss.
A NLEE's soliton solutions can be obtained
using this method. In the paragraphs that follow, we
demonstrate how some NLEEs that describe pss can
have their BTs determined systematically using the
geometrical features of pss.
Proposition 4.1. On a smooth Riemannian surface
M2, given a coframe _1, _2, and related connection
one-form _3, there exists a new coframe _1', _2',
and new connection one-form _3' that satisfy the
conditions listed below:
(30)
we give a proof of, [28].
Proof.
(31)
satisfying 30.
(32)
if oneforms
and (32)
(33)
is completely integrable for whenever
is a local solution of
[2], [36].
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Proposition 4.2. Let be a
NLEE which describe pss with associated
one‐forms (1.2). Then, for each solution of
, the system of equations for
,
(34)
(35)
using
(36)
where
(37)
then (34) is :
(38)
(39)
Construction u' (x) is the next step.
(40)
(a) BT for equation 4.
In (10) defined by
(41)
Then (38) becomes
(42)
If we choose and as
(43)
then the :
(44)
Equation (41)'s f_11, f_22, and f_32 serve as the
BT for Liouville's equation (4) in equation (44).
For equation 2, use (b) BT.
The functions for any Burgers' equation (6)
solution u(x,t).
(45)
Then (38) becomes, [29]
(46)
If we choose and as
(47)
the :
(48)
(c) Equation 3.
defined by, [30]
(49)
Then (38) becomes [34]
(50)
and as
(51)
we get the :
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(52)
c) BT for an equation 1.
(4) defined by
(53)
Then (38), [32]
(54)
choose and as
(55)
then we get the :
(56)
the values given in (53) for
, , and
For an equation 5, (e) BT.
We take into consideration the u(x,t) functions
defined by for (12);
(57)
Then (38), [33]
(58)
choose and as
(59)
we get the :
(60)
we put
and .
Equation (60) with
, and at (57).
(f) the equations 6.
equations 6 (14), the functions
(61)
Then (38) becomes, [35]
(62)
choose and as
(63)
then we get:
(64)
(g) the equations 7.
defined by
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(65)
Then (38) becomes
(66)
choose and as
(67)
we get the :
(68)
Equation (68) is the BT for the equations (16) with
, and at (65).
5 Conclusions
The congruent triple sumudu conversion of
fractional partial derivatives was also proven and in
this study, the method of placing congruent triple
SUMUDU was presented
The congruent triple Sumudu transformation of
fractional partial derivatives was also proved, and in
this study, the method of placing the congruent
triple SUMUDU was corrected and solved based on
the congruent triple SUMUDU to solve the
congruent two-dimensional BURGERS equations
that are regular and irregular. Moreover, Examples
were given to explain the proposed methods of
solution by applying MATLAB to represent the
solutions
Acknowledgements:
The authors would like to extend their sincere
appreciation to the Deanship of Scientific Research
at Zarqa University for its funding this Research.
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Contribution of Individual Authors to the
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All authors contribute equally to this project.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
This research is funded by the Deanship of
Scientific Research at Zarqa University /Jordan.
Conflict of Interest
The authors declare that they have no Conflicts of
Interest.
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