A New Approach in Solving Regular and Singular Conformable
Fractional Coupled Burger’s Equations
AMJAD E. HAMZA1, ABDELILAH K. SEDEEG2,3, RANIA SAADEH4*, AHMAD QAZZA4,
RAED KHALIL5
1Department of Mathematic, Faculty of Sciences,
University of Ha’il, Ha’il 2440,
SAUDI ARABIA
2Department of Mathematics, Faculty of Education,
Holy Quran and Islamic Sciences University, Sudan,
SUDAN
3Department of Mathematics, Faculty of Sciences and Arts-Almikwah,
Albaha University,
SAUDI ARABIA
4Department of Mathematics, Faculty of Science,
Zarqa University, Zarqa 13110,
JORDAN
5Department of Computer Information Systems, Faculty of Prince Abdullah Bin Ghazi,
Balqaa Applied Univesity,
JORDAN
*Corresponding Author
Abstract: - The conformable double ARA decomposition approach is presented in this current study to solve
one-dimensional regular and singular conformable functional Burger's equations. We investigate the
conformable double ARA transform's definition, existence requirements, and some basic properties. In this
study, we introduce a novel interesting method that combines the double ARA transform with Adomian’s
decomposition method, in order to find the precise solutions of some nonlinear fractional problems. Moreover,
we use the new approach to solve Burgers' equations for both regular and singular conformable fractional
coupled systems. We also provide several instances to demonstrate the usefulness of the current study.
Mathematica software has been used to get numerical results.
Key-Words: - Conformable ARA transform; Conformable double ARA decomposition method; Singular one-
dimensional coupled Burgers’ equation; Conformable partial fractional derivative.
Received: September 9, 2022. Revised: March 29, 2023. Accepted: April 24, 2023. Published: May 9, 2023.
1 Introduction
Fractional partial differential equations have drawn
significant interest from a wide range of specialists
in applied sciences and engineering, including
acoustics, control, and viscoelasticity. In many areas
of mathematics and physics, partial differential
equations are crucial. To examine several time-
fractional partial differential equations, the authors
use a novel strategy termed the "simplest equation
method", [1], [2].
In the context of applied sciences like mathematical
modeling and fluid mechanics, this work focuses on
Burger's equation. Burger's equation was initially
brought up about steady-state solutions, in fact, [3].
Burger later changed the approach to characterize
the viscosity of certain fluid types, [4]. The
conformable double Laplace transform approach,
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which was first proposed in [5], was improved and
used to solve fractional partial differential
equations. This method has been used by a number
of academics to obtain precise and numerical
solutions to this type of equation. To precisely solve
time-fractional Burger's equations, other scholars
used the first integral technique, [6]. Another set of
researchers developed the coupled Burger's equation
solution using the generalized two-dimensional
differential transform approach, [7], [8], [9], [10].
The ARA transform is a revolutionary integral
transform that Saadeh and others introduced in,
[11]. ARA is a new transform; it is not an acronym.
It has novel properties, including the ability to
generate numerous transforms by varying the value
of the index n, a duality with the Laplace transform,
and the capacity to get around the singularity at time
zero. Many researchers have studied the new
approach and implemented it to solve many
problems by merging it with other numerical
methods or other transforms, such as ARA-Sumudu
transform, [12], [13], Laplace-ARA transform, [14],
double ARA transform, [15], [16], ARA residual
power series method, [17], [18].
In this article, we choose to build a unique
combination of Adomian’s decomposition method
and the double ARA transform, so that we obtain
the advantages of these two methods and fully
utilize these two potent techniques. The
conformable double ARA transform method will be
introduced in this research in combination with
Adomian’s decomposition method, [19], to solve
systems of conformable fractional partial
differential equations.
With the help of the conformable double ARA
decomposition approach, this study aims to provide
analytical solutions for the coupled, one-
dimensional, singular, and regular conformable
fractional Burger's equations (CDARADM). The
following space-time fractional order coupled with
Burger's equations were described in [20], and are
given below:
(1)
This article is organized as follows, in the following
section, we present the ARA transform with the
main characteristics. In Section 3, we introduce
some preliminaries about the conformable fractional
derivatives. The conformable ARA transform and
some related results are presented in Section 4. In
Section 5, we introduce some numerical
experiments to prove the efficiency and applicability
of the new method.
2 ARA Integral Transforms, [11]
Definition 1. If is a continuous function on
, then the ARA transform of order
(2)
and the inverse ARA transform is defined as
(3)
where
Theorem 1. Let be piecewise continuous in
every finite interval and satisfies the
condition
where is a positive constant. Then, ARA integral
transform exists for all .
Proof. Using the definition of ARA transform, we
get
Thus, we get
Hence, the integral exists for all , and
exists.
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Now, we present some properties of ARA
transform.
If and and
, then
(4)
(5)
(6)
(7)
(8)
(9)
where is the ARA transform of a
continuous function of order one on ,
and it is given by
(10)
In this research, we denote for .
3 Conformable Fractional Derivatives
(CFD), [21], [22], [23], [24]
Conformable fractional derivatives are investigated
and expanded in [15], [16], respectively. The
following definitions of CFD are utilized in this
study.
Definition 2. The CFD of
order , where
., is given by
(11)
Definition 3. The conformable space fractional
partial derivative of
order , where
, is defined as
(12)
Definition 4. Let
. Then,
the conformable fractional partial derivative of
of order, is given by
(13)
3.1 Conformable Fractional Derivatives of
Some Basic Functions
In the following arguments, we introduce the CFD
for some basic functions.
i. Let
. Then
ii. Let
. Then
iii. Let
. Then
iv. Let
. Then
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v. Let
. Then
Property 1. If is a differentiable function of
order and at the points and , where
. Then
Proof. Using Definition 3 and putting
in Equation (12), we get
Similarly, we can easily prove that
4 Conformable Double ARA
Transform (CDARAT)
In this part of this study, we present the conformable
double ARA transform using the following
definitions.
Definition 5. Assume that is a real-valued
function defined on to , then the
conformable ARA transform of
is given by
(14)
If the integral exists.
Definition 6. Let
be a piecewise
continuous function on the interval
of exponential order, that is considered for some
,
,
.Under these
conditions the conformable double ARA transform
is given by
(15)
where and the integrals
with respect to and respectively, are taken by
conformable fractional derivative.
Property 2. Let
. Then
(16)
Proof. The definition of CDARAT implies
(17)
Substituting
,
, and
in Equation (17) and simplifying, we
obtain
4.1 CDARAT of Some Elementary Functions
In this section, we present the conformable Double
ARA Transform for some basic functions.
i. Let
. Then
From Property 2 and Equation (6), we get
ii. Let
. Then
From Property 2 and Equation (6), we get
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iii. Let
. Then
From Property 2 and Equation (7), we get
iv. Let
. Then
From Property 2 and Equation (8), we get
4.2 Existence Condition for the Conformable
Double ARA Transform
If
is an exponential order and as
, If a constant , such that
for all and
it is easy to get,
as
Theorem 2. Let the function
be
continuous on the region and are of
exponential orders and , then the conformable
double ARA transform of
exists for all
, .
Proof. Using the definition of the CDARAT of
, we have
For .
Theorem 3. Let
,
then
i.
ii.
iii.
iv.
v.
Proof.
Proof of (i). Using the definition of CDARAT of
, we get
(18)
By differentiating both sides with respect to in
Equation (18), we have
Calculating the partial derivative of the second
integral, we can get
(19)
Therefore,
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(20)
Proof of (iii). Differentiating the both sides with
respect to in Equation (19), we have
Thus,
(21)
From Equation (20), we have
Similarly, we can easily prove that:
Proof of (v). Differentiating both sides with respect
to in Equation (19), we have
Therefore,
From (i) and (ii), we have
The proof of (ii) and (iv) can be obtained by similar
arguments of (i) and (iii).
Theorem 4. Let
,
then
i.
ii.
iii.
iv.
Proof.
i. Using the definition of CDARAT for
, we have
(22)
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Applying Property 1,
, then Equation (22) becomes
(23)
Thus, the integral inside bracket is given by
(24)
Substituting Equation (24) into Equation (23), we
obtain
(25)
In the same manner, the CDARAT of
,
and
can
be obtained.
Theorem 5. Let
,
then
i.
ii.
Proof of i. The conformable double ARA
transforms definition for fractional partial
derivatives, implies
(26)
we calculate the partial derivative in the second
integral as follows
(27)
Substituting Equation (27) into Equation (26), we
get
Thus,
Using Theorem 4, we have
Similarly, one can prove that
In the following, we introduce the previous results
in the following table, Table 1, below:
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Table 1. Analysis of the presented results
5 Applications
The CDARADM is used in this section of the study
to solve regular and singular one-dimensional
conformable fractional coupled Burger's equations.
The goal problem is the same as the problem
examined in [1], when and . This is
what we mention here.
Example 1.
Consider the One-dimensional conformable
fractional coupled Burgers’ equation of the form
(28)
subject to
(29)
for . Here,
,
and
are given functions, , and are
arbitrary parameters depending on the Peclet
number, Stokes velocity of particles due to gravity
and Brownian diffusivity, see, [9]. Now, operating
the conformable double ARA transform to Equation
(28) and the single conformable single ARA
transform for Equation (29), to get
(30)
(31)
The CDARADM defines the solution of the target
problem
and
in the form of
infinite series as
(32)
Define the Adomian’s polynomials, and
as
(33)
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We can compute the Adomian polynomials of the
nonlinear terms and by the formulas
Operating the inverse double ARA transform to
Equation (30) and Equation (31), utilizing Equation
(33), we get
(34)
and
.
(35)
Now, we compare both sides of Equation (34) and
Equation (35), to get
(36)
Following that, the recursive relation can be
expressed as
(37)
and
(38)
Herein, we should state that the solutions in (37) and
(38) exist, provided the inverse double ARA
transform exists and .
Putting , and
in Equation (28) and
in Equation (29), we obtain the
one-dimensional homogeneous coupled Burgers
fractional equation is the conformable sense
(39)
with initial condition
(40)
By using Equations (36)–(38) , we have
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and
Therefore, using Equation (32), we can express the
series solution as
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and hence the exact solutions become
By taking and, the fractional solution
of Equation (39) becomes
The behavior of the velocity field of the two-
CDARADM (28) and (29) is depicted in Figure 1
for (a) the approximate and exact solutions of
for Example 1, when , at
, and (b) the approximate and exact
solutions of
, for Example 1, when taking
various values of fractional order (
and .
(a)
(b)
Fig. 1: The figure of (a) The approximate and exact
solutions of for Example 5.1, when
, (b) The approximate and exact solutions of
, for Example 1, when taking various
values of and .
Table 2 below, presents the absolute errors
considering , . We
compare the obtained results by exact solution.
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Table 2. Analysis of error to
for Example
1, on
Exact
Solution
Approximat
e Solution
Example 2.
Consider the singular one-dimensional conformable
fractional coupled Burgers’ equation with the Bessel
operator of the form
(41)
and with initial conditions
(42)
Where the linear terms
is known as a
conformable Bessel operator where , and are
real constants.
Multiplying both sides of Equation (41) by
, we
get
(43)
Applying conformable double ARA transform to
both sides of Equation (43) and single conformable
ARA transform for the initial condition, we get
(44)
Applying Theorem 3 and Theorem 5, we have
(45)
Simplifying Equation (45), we obtain
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(46)
Applying the definite integral
with respect to
to both sides of Equation (46)
Multiplying both sides of the equations by , we get
(47)
Utilizing the CDARADM to present the solution of
and
by infinite series as
(48)
Define the nonlinear operators as
(49)
Operating the double inverse transform to Equation
(47) and making use of Equation (48) and Equation
(49), we have
(50)
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And
(51)
Now, we can express the first few components as
(52)
And
(53)
And
(54)
In addition, we assume that the double inverse
transform in (53) and (54) exist, and substitution
, and
in Equation (41) and
in Equation (42), we obtain the
singular conformable coupled Burgers fractional
equation of one dimensional
(55)
subject to
(56)
By following similar steps, we obtain
(57)
(58)
Using Equations (52) – (54) the components are
given by
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In a similar way, we obtain
.
Following that, one can express the solution as
Therefore, the exact solution is given by
By taking and, the fractional solution
becomes
The behavior of the velocity field of the two-
CDARADM (41) and (42) is depicted in Figure 2
for (a) the approximate and exact solutions of for
Example 2, at , and (b) the
approximate and exact solutions of
, for
Example 2, when taking various values of (
and .
(a)
(b)
Fig. 2: The figure of (a) The approximate and exact
solutions of
for Example 5.2, when
, (b) The approximate and exact solutions of
, for Example 2, when we take different
values of fractional order and
Table 3 below, presents the absolute errors with
respect to , . We
compare the obtained results by exact solution.
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Table 3. Error analysis of
for Example 2
on
Exact
Solution
Approximate
Solution
6 Conclusion
In the current study, we defined and went over some
of the characteristics of the conformable double
ARA transform. The conformable double ARA
decomposition method is a novel approach that we
present for the solution of nonlinear conformable
partial differential equations. We used the proposed
approach, a novel amalgamation of the conformable
double ARA transform and Adomian decomposition
methods, to present solutions to the one-dimensional
regular and singular conformable fractional coupled
Burgers' problem. Additionally, two intriguing
examples were given to demonstrate the
applicability of the novel approach. Different types
of nonlinear time-fractional differential equations
with conformable derivatives can be solved using
this technique. We want to answer more fractional
integral equations and fractional nonlinear problem
classes in the future.
Acknowledgement:
The authors express their gratitude to the dear
referees, who wish to remain anonymous, and the
editor for their helpful suggestions.
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Contribution of Individual Authors to the
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The authors equally contributed to 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 authors have no conflicts of interest to declare.
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