Analytic Computational Method for Solving Fractional Nonlinear
Equations in Magneto-Acoustic Waves
RANIA SAADEH
Department of Mathematics
Zarqa University
Zarqa 13110,
JORDAN
Abstract: - In this article, we employ a useful and intriguing method known as the ARA-homotopy transform
approach to explore the fifth-order Korteweg-de Vries equations that are nonlinear and time-fractional. The
study of capillary gravity water waves, magneto-sound propagation in plasma, and the motion of long waves
under the effect of gravity in shallow water have all been influenced by Korteweg-de Vries equations. We
discuss three instances of the fifth-order time-fractional Korteweg-de Vries equations to demonstrate the
efficacy and applicability of the proposed method. Utilizing, also known as the auxiliary parameter or
convergence control parameter, the ARA-homotopy transform technique which is a combination between ARA
transform and the homotopy analysis method, allows us to modify the convergence range of the series solution.
The obtained results show that the proposed method is very gratifying and examines the complex nonlinear
challenges that arise in science and innovation.
Key-Words: - Nonlinear fractional partial differential equation, ARA-Homotopy transform method, ARA
transform, Caputo derivative.
Received: May 23, 2022. Revised: October 25, 2022. Accepted: December 3, 2022. Published: December 31, 2022.
1 Introduction
Plasma is typically regarded as a unique phase of
matter in physics and lacks a fixed shape or volume.
By heating a neutral gas or exposing it to a desired
electromagnetic field, which causes an ionized
gaseous material to become more electrically
conductive, plasma can be created. Because of the
free electric charges that make the plasma
electrically conductive, the plasma reacts strongly to
electromagnetic fields. Thus, electric and/or
magnetic forces would dominate its characteristics.
The matter in the plasma state is referred to as some
forms of flame, stars, and the Sun's corona. The
realm of intensity production is where plasmas are
most practically utilized. A crucial method for
producing electricity is to use heat sources to turn
water into steam, which drives turbo generators.
Unsettling effects that are compressible propagate
through plasma as magneto-acoustic waves that are
fueled by both gas pressure and attractive force.
Particle nonpartisan effects have a role in the partial
coupling of ionized plasmas, the constituents of
ionized and impartial species. As a result, in the
presence of low magnetic field and low temperature,
the magneto-acoustic wave persists as particle
acoustic waves and Alfvén waves, respectively. The
heating of the solar corona is significantly
influenced by the magneto-acoustic waves [1], [2],
[3], [4].
The phenomenon of nonlinear equations
describes the fundamental physical aspects in nature
ranging from chaotic behaviour in biological
systems [2], plasma physics - plasma containment in
stellarators, and tokamaks to energy generation.
[3],4], quantum mechanics, [5], nonlinear optics,
[6], solid-state physics and up to fibre optical
communication devices, [7], dual wave soliton
solution, [8], unidirectional shallow water waves
[9], analytical wave solutions, [10], unmagnetized
dust plasma, [11], optimal solitons for the nonlinear
dynamics, [12]. The various phenomena of
nonlinear equations are modelled in terms of many
orders of nonlinear partial differential equations,
[13, [14], [15], [16], [17]. Partial differential
equations are largely utilised to represent physical
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systems, but unfortunately, many of them don’t
have the exact solution. Moreover, the accurate
solution to this nonlinear phenomenon is not
available in the literature and hence to solve these
nonlinear systems, there is an essence of studying
the nonlinear phenomena with appropriate and more
efficient methods.
Since the solution of the KdV equation can be
explained exactly and precisely, it is predominantly
distinguished as the archetypal illustration of an
exactly solvable model, [18], [19], [20]. The KdV
equation amalgamates dispersion and nonlinearity
and provides stationary solutions tracing both
periodic and solitary waves. It represents a model
for the interpretation of long waves which are
weakly nonlinear with small dispersion in media.
Subsequently, various kinds of KdV equations
possess many remarkable properties and are being
considered as a model to explain the wide range of
physical phenomena which exist in the connected
branches of mathematics and physics.
There are various analytical and numerical methods
available for handling various forms of fifth-order
KdV-type equations in the literature. Some of them
are the Adomian decomposition technique,
Modified Adomian Decomposition Method, Laplace
decomposition approach, Hyperbolic and
exponential ansatz methods, Multiple Exp-function
method, and others [30-33].
In the present investigation, we consider the time-
fractional fifth-order KdV equations with initial
conditions as follows, [18]:
(1)
with the initial condition
(2)
with initial conditions.
(3)
with initial conditions
Here, Eqs. (3) and (4) are called fifth-
order KdV equations and Equation (5) is called the
Kawahara equation [38].
The KdV equations (3) and (4) are crucial for
explaining how long waves move in shallow water
when there is gravity. In order to study the
propagation of oscillatory solitary waves in a
dispersive medium, Kuwahara first applied the
Kawahara equation (5) in 1972, [38]. The above
equations describe the interaction between
nonlinearity and dispersion in the theoretically
simplest terms possible. The higher order nonlinear
factors that are present in the equations under
consideration express higher amplitude internal
waves.
Now, the solutions for the above-mentioned
equations have been investigated by employing a
new computational technique, known as ARA-
homotopy transform method (or briefly, ARA-
HTM). The considered technique is a graceful
unification of the homotopy algorithm and ARA
transform [34], [35], [36], [38].
The proposed technique gives a great degree of
freedom in picking initial approximations and
auxiliary linear operators; as a result, the complexity
of the problem can be reduced by transforming it
into an infinitely countable number of easier, linear
subproblems, helping in reducing the time of
computational work.
Following that, the article's remaining portion is
decorated as: The fundamental and standard
definitions of the fractional derivatives, and the
basic idea of ARA transform of Caputo fractional
derivative is presented in section 2. The
methodology of the considered analytical technique
for nonlinear fractional partial differential equations
can be seen in Section 3. The investigation of the
considered problem along with the incorporation of
their graphical results using projected technique is
done is section 4. Section 5 cites the description
about the obtained results. Section 6 is decorated
with the concluding remarks followed by the
references.
2 Basic Facts and Theorems
The basic definitions of fractional operators and the
ARA transform, which are related to this study, are
presented in this section of this research.
Definition 1. The Caputo fractional derivative of
order of
is defined as
(4)
Definition 2. [37] If is a continuous function
on the interval ,then ARA transform of order
of is defined by
(5)
We define the inverse ARA transform as follows
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where and
Now, we introduce some basic properties of ARA
transform that are important in our study.
Assume that and are two continuous
functions defined on the interval such that
ARA transform exists, then for , we have
where and are two
constants.
(6)
(7)
(8)
Remark 1. In this study, we focus the proposed
method on ARA transform of order 1, to simplify
the notation, we use instead of to denote ARA
transform.
3 The basic idea of the ARA-HTM
To illustrate the main idea of ARA-HTM, let us
consider the following time fractional PDE
,
(9)
where cites the Caputo fractional
derivative of the function , denotes the
source term, linear and nonlinear differential
operators are represented by and respectively.
Now, hiring ARA transform with respect to the
variable , to Eq.(9) and using the properties of
ARA to the fractional derivative, we conclude
(10)
Simplifying Eq.(10), we have
(11)
The nonlinear operator with HAM can be
expressed as
,
(12)
where is a real valued function of
and and
.
The zeroth order deformation equation involving the
auxiliary function is as follows:
,
(13)
where is the convergence control parameter,
is the embedding parameter,
is an initial guess of , is a
function to be determined. The below equations
justify for and
.
,
(14)
respectively. As we move from to
, the solution
converges from to the solution
. After operating, the Taylor theorem for the
function around leads to
,
(15)
where
.
(16)
On choosing the appropriate the auxiliary parameter
, the initial guess and , the
auxiliary linear operator, the series (15) converges at
, which leads to one of the solutions of the
original nonlinear equation of the form
.
(17)
Next, the order deformation equation obtained
by differentiating the zeroth order deformation
equation m-times followed by dividing the resulting
equation by at leads to
,
(18)
and the vector
is demonstrated as
(19)
The following recursive is obtained by hiring the
inverse ARA transform to Eq.(18) as
,
(20)
where
,
(21)
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and
(22)
Lastly, the terms of the ARA-HTM series solution
are attained by evaluating Eq.(20).
4 Numerical Simulations
The investigation of the following examples proves
the efficiency and applicability of the presented
scheme.
Problem 4.1.
The fifth-order time-fractional KdV equation
defined in Equation (3)
(23)
with the initial condition
(24)
Introduce ARA transform in Eq.(23) along with the
starting solution in Eq.(24), leads to
(25)
The nonlinear operator is defined as follows
(26)
The order deformation equation is
,
(27)
where
(28)
On implementing inverse ARA transform on
Eq.(24), we get
(29)
Solving the above equations consistently gives
,
Finally, after getting further iterative terms, the
essential series solution of Eq.(23) is presented by
(30)
By taking and , then the
obtained solution
, converges
to the exact solution
of the Eq.(23), as
.
(a)
(b)
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(c)
Figure 1. (a) 3D plot for ARA-HTM solution, (b)
surface of exact solution, (c) approximate solution
surface at and for Problem
4.1.
Figure 2. with versus for Problem 4.1
when and for distinct values
of .
(i)
(ii)
Figure 3. -curve for the acquired solution
versus for the considered Problem 4.1 when (i)
and (ii) when for
distinct values of .
In the following table, (Table 1) we present comparisons of the 3rd and 6th order ARA-HTM solutions with
HPTM [30] in terms of absolute error values for Problem 4.1 at and.
[14]
0
2.44582
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Example 4.2.
Consider the nonlinear time-fractional fifth-order
KdV equation cited in Eq.(4)
(31)
with initial conditions
(32)
Introduce ARA transform in Eq.(31) along with the
starting solution in Eq.(32), leads to
(33)
The nonlinear operator is defined as follows
.
(34)
The order deformation equation is
,
(35)
where
.
(36)
When treated with inverse ARA transform with
Eq.(36), we get
(37)
Solving the above equations consistently gives
,
Finally, after getting further iterative terms, the
essential series solution of Eq.(31) is presented by
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Taking and then the
solution we get is in the form
, converges to the
exact solution of the
Eq.(31) as
(38)
(a)
(b)
(c)
Figure 4. 3D plots of solution surfaces indicating
ARA-HTM solution, the exact solution, and an
approximate error solution, respectively ((a), (b) &
(c)) at and for Problem 4.2.
Figure 5. versus for Problem 4.2 at
and for distinct values of .
(i)
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(ii)
Figure 6. -curve for acquired solution for
Problem 4.2 when (i) and (ii) when
and for distinct values of .
In the following table, (Table 2) we present the 3rd to 6th-order approximations of the obtained ARA-HTM
series solution for Problem 4.2 at and .
Example 4.3. Consider the time-fractional fifth-
order KdV equation
(39)
with the initial condition
(40)
By performing ARA transform on Equation (39)
and then considering Eq.(40), we get
(41)
The nonlinear operator is defined as follows
.
(42)
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The order deformation equation is
,
(43)
where
.
(44)
By enforcing the inverse ARA transform with
Eq.(43), we get
.
(45)
Solving the above equations consistently gives
,
,
Finally, after getting further iterative terms, the
essential series solution of Equation (39) is
presented by
(46)
If we take and, then the
secured solution
, converges to
the exact solution
of the Eq.(39), as
(a)
(b)
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(c)
Figure7. (a) 3D plot for ARA-HTM solution, (b)
surface of exact solution, (c) approximate error
solution surface, at ℏ = −1, = 4, = 1 and = 1.
Figure8. versus for the contemplated
Problem 4.3 at and
for distinct values of .
(i)
(i)
(ii)
Figure 9. -curve for acquired solution for
Problem 4.3 when (i) and (ii) when
and for distinct values of
.
In the following table, (Table 3) we present a numerical study of the achieved results in terms of absolute error
for Problem 4.3 at and and different values of and .
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5 Numerical results and discussion
This portion of the article provides an incorporation
of numerical simulations of the investigated
problem that show the validity and effectiveness of
the considered scheme q-HATM. Moreover,
incorporated a detailed description of the graphical
solutions that were found. The secured results are
very satisfying and in good fit with the exact
solutions to the contemplated problem. The
comparison of 3D surface plots of the obtained
approximate solution and the exact solution along
with their absolute error solutions is presented in
Figure 1. We can see the accuracy of the obtained
approximated solution of Problem 4.1 in Figure 1(c)
with the least error values. The nature of the
obtained solutions for different fractional order as
we move along time is cited in figure 2. We can
see the variation in the solution affected by different
fractional orders. Figure 3 cites the plot of solution
curves for distinct fractional orders, which gives the
precise range of convergence control parameter to
achieve the convergence. The figure shows that we
can choose values between to for the
faster convergence of the approximated solution
towards the exact solution. The convergence of the
obtained solution is achieved by considering
in this work. Table 1 depicts the comparison of
secured results with the homotopy perturbation
transform method (HPTM) in terms of absolute
error values with and . The
calculations of table 1 are carried out by taking
and with the time interval . Surface
plots of the q-HATM solution, the exact solution,
and the approximated error solution for Example 4.2
are cited in Figure 4. The 2D plot of the obtained
solution of Problem 4.2 with respect to time for
different fractional order is cited in figure 5. We
can observe that the solution curf leads to different
consequences for different fractional orders . The
performance of with ℏ in an accomplished
outcome of the provided method is shown in figure
6. The solution of Problem 4.2 in terms of an
absolute error from 3rd-order to 6th-order
approximations is given in Table 2. That shows we
can achieve better results by increasing the number
of iterations. Figures 7(a) and 7(b) explore the 3D
surfaces of the ARA-HTM solution and the exact
solution of Problem 4.3. The approximated absolute
error solution of Problem 4.3 is cited in Figure 7(c).
The fractional behaviour of the considered nonlinear
time-fractional fifth-order KdV equation over time
for distinct fractional order is plotted in figure 8.
To attain the precise range of convergence control
parameters to have a faster rate of convergence to
the exact solution, we have plotted Figure 9. Table 3
cites the approximated absolute error values of
Problem 4.3 for different values of and .
The 3D plots presented in Figures (1), (4) and (7)
describe the wavy nature of the considered
nonlinear KdV equations. For the purpose of
accuracy we can consider the plots for the
fractional/classical order , there we can see
the close association of the ARA-HTM solution
with the exact solution. The physical interpretation
of the considered fractional problems are well
described by the fractional orders and
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due to their feature of memory effect. From the
2D plots one can see the considered value for the
convergence control parameter works for
all the fractional orders.
From all figures, we can observe that the hired
fractional operator in the considered model
exemplifies some interesting consequences and it
authorizes the model to noticeably defend time and
history behaviour. Moreover, the illustrated
numerical simulations confirm the applicability as
well as the accuracy of the considered solution
procedure, and, we prove that we go close to the
exact solution as we increase the number of
iterations. Some relevant study can be found in [39].
6 Conclusion
In the present work, the investigation of the time-
fractional nonlinear fifth-order KdV equation is
carried out using an analytical algorithm called
ARA-HTM. We discussed three nonlinear problems
to testify to the ability of the projected method to
handle complex nonlinear problems. The results are
highly pleasing and attest to the effectiveness of the
strategy under consideration. The fractional operator
considered in the present framework gives more
degrees of freedom and incorporates the nonlocal
effect in the projected model. The innovative aspect
of this approach is its straightforward process,
which enables us to arrive at a solution quickly and
identifies a substantial region of convergence. The
rate of convergence of the obtained series solution
to the exact solution is accelerated with the help of
optimal values of the convergence control parameter
ℏ. The obtained numerical simulations guarantee
the results with higher accuracy. Tables provide
great satisfactory results when comparing with
homotopy perturbation transform method (HPTM).
As a future research direction, readers can use the
hybrid methodologies merging with our projected
scheme to achieve better consequences. Finally, we
claim that our proposed technique is incredibly
dependable and can be applied to large study
classifications relating to fractional-order nonlinear
scientific methods.
Acknowledgement:
The author expresses her gratitude to the dear
unknown referees and the editor for their helpful
suggestions, which improved the final version of
this paper.
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Volume 1, 2021, DOI:
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WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2022.17.22
Rania Saadeh
E-ISSN: 2224-347X
254
Volume 17, 2022
