A Generalized Hybrid Method for Handling Fractional Caputo Partial
Differential Equations via Homotopy Perturbed Analysis
RANIA SAADEH1, AHMAD QAZZA1,*, ABDELILAH KAMAL SEDEEG2,3
1Faculty of Science,
Zarqa University,
Zarqa 13110,
JORDAN
2Department of Mathematics, Faculty of Education,
Holy Quran and Islamic Sciences University,
SUDAN
3Department of Mathematics, Faculty of Sciences and Arts-Almikwah,
Albaha University,
SAUDI ARABIA
*Corresponding Author
Abstract: - This article describes a novel hybrid technique known as the Sawi transform homotopy perturbation
method for solving Caputo fractional partial differential equations. Combining the Sawi transform and the
homotopy perturbation method, this innovative technique approximates series solutions for fractional partial
differential equations. The Sawi transform is a recently developed integral transform that may successfully
manage recurrence relations and integro-differential equations. Using a homotopy parameter, the homotopy
perturbation method is a potent semi-analytical tool for constructing approximate solutions to nonlinear
problems. The suggested method offers various advantages over existing methods, including high precision,
rapid convergence, minimal computing expense, and broad applicability. The new method is used to solve the
convection–reaction–diffusion problem using fractional Caputo derivatives.
Key-Words: - Sawi transform, Homotopy perturbation method, Fractional partial differential equations, Caputo
fractional derivative, Series solution, Nonlinear partial differential equations.
Received: April 12, 2023. Revised: November 26, 2023. Accepted: December 13, 2023. Published: December 31, 2023.
1 Introduction
Various phenomena in physics, engineering,
biology, and other disciplines are typically modeled
using fractional partial differential equations
(FPDEs), [1], [2], [3]. Due to the nonlocal and
singular nature of fractional derivatives, however,
accurate or numerical solutions to FPDEs are
frequently difficult to discover. The fractional
power series method, the fast convolution algorithm,
the fractional differential transform method, the
finite difference method, and the fixed point and
upper and lower solution methods, [4], [5], [6], [7],
[8] are examples of analytical and numerical
methods that have been developed to solve FPDEs.
In this paper, the Sawi transform homotopy
perturbation method (STHPM) is introduced as a
novel hybrid strategy for solving FPDEs with
Caputo fractional derivatives. The Caputo fractional
derivative, which is widely recognized as one of the
most significant definitions of fractional derivatives,
offers the distinct advantage of maintaining the
beginning conditions in the classical sense, as
supported by references [9], [10] and [11].
Approximate series solutions for FPDEs are
generated using the Sawi transform-homotopy
perturbation method (STHPM) combination, [12],
[13], [14], [15], [16].
In this study, homotopy perturbation methods
are applied to solve fractional Caputo partial
differential equations (PDEs) in a novel manner, as
noted in references [17] and [18].
According to references [19] and [20], the
STHPM produces extremely precise results and
saves a significant amount of calculation time when
compared to other techniques like the variational
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Rania Saadeh, Ahmad Qazza,
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iteration approach and the Adomian decomposition
method.
Additionally, a variety of complex and non-
linear partial differential equations (PDEs) can be
solved quickly and effectively with STHPM's
flexibility, something that is challenging to achieve
with conventional numerical approaches, [21], [22]
and [23]. Scholars and professionals alike will find
the STHPM to be a beneficial tool as it provides a
novel perspective on the analysis and solution of
fractional Caputo PDEs. The solution of the
convection-reaction-diffusion equation shows the
practicality and efficacy of the STHPM. We
compare our results with the accuracy,
computational expense, and convergence of known
numerical methods or solutions, as documented in
references [24], [25] and [26]. As shown in
references [27], [28] and [29], we also go over
potential STHPM extensions and uses to handle
other types of FPDEs.
It is crucial to acknowledge that the method has
limits, including the need for meticulous selection of
homotopy parameters. These constraints will be
thoroughly described in the subsequent portions of
this study, as referenced by [30], [31], [32], [33],
[34].
The Sawi transform homotopy perturbation
method is a significant improvement in the field of
fractional differential equations, addressing a critical
gap in the current literature and providing a more
generic, efficient, and accurate method for solving
fractional Caputo PDEs, [35].
2 Basic Concepts of Sawi Transform
This section is concerned with the presentation of
the Sawi transform. We out line some basic
properties regarding the existence conditions,
linearity and the inverse of this transform.
Moreover, some essential properties and results
regarding Sawi transform are discussed. We
introduce the Sawi convolution theorem and the
derivative properties. For more details about Sawi
transform see, [17], [18].
Definition 2.1. If is a function defined over a
positive domain. Then, Sawi transform of ,
denoted by , is given by
(1)
The inverse Sawi transformation is provided as
(2)
Theorem 2.1. If is continuous function
defined for and of exponential order . Then
exists for and satisfies
where , then Sawi transformation exists for
Suppose that and
and , then the following properties hold:
Theorem 2.2. Let . Then,
(3)
where denotes the unit step function defined
by
otherwise
Theorem 2.3. (Sawi Convolution Theorem). If
and , then
(4)
where
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3 Fundamental Facts of Fractional
Calculus
In this section, some definitions and properties of
fractional calculus that will be used in this work are
presented.
Definition 3.1. [35], The three - parameters Mittag-
Leffler function is defined as:
(5)
where is the Pochhammer symbol.
Putting 1 in Eq. (5), we have the new
function turns into the two - parameters Mittag-
Leffler function:
(6)
Putting 1 in Eq. (6), we have the new
function turns into the classical Mittag-Leffler
function:
(7)
we note that
Definition 3.2. [36], Let be a continuous
function, and . Then the Caputo
fractional derivative of the function with
respect to of the order is defined as:
(8)
Theorem 3.1. Let be the Sawi transform of
.Then the Sawi transform of Caputo fractional
derivative of is expressed as:
(9)
Proof. By the definition of convolution integral, we
have:
Therefore,
Using the properties of Sawi transform, we have
Thus,
Corollary3.1. Let be the Sawi transform of
and .Then the Sawi transform of
Caputo fractional derivative of is expressed
as
(10)
4 Analysis of Sawi Transform
Homotopy Perturbation Method
In this part of the paper, we present the fundamental
idea of Sawi transform homotopy perturbation
method for solving fractional Caputo partial
differential equations. In order to show the
fundamental plan of the STHPM, we consider the
following general partial differential equation
and
(11)
subject to the conditions
(12)
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where are linear and nonlinear differential
operators, denotes the Caputo fractional
derivative with respect to the variable , is
the unknown function and is a given
function.
Applying the Sawi transform for Eq. (11), with
respect to we obtain
(13)
Thus, the homotopy parameter is defined as
(14)
The nonlinear terms in Eq.(11) can be written as
(15)
Where are He’s polynomials, which can be
calculated by using the following formula
(16)
We carry out the component of the Caputo
operator result by substituting Eqs. (14) and (15)
into Eq. (13).
(17)
Appling the inverse Sawi transform to Eq. (17), we
have:
(18)
Thus, Eq. (18) , when solved with respect to , are
defined as
(19)
when is applied, suppose that Eq .(19) is the
approximated solution to Eq. (11) , and the solution
is
(20)
5 Applications
In this section of this paper, we present some
examples to show the efficiency of the presented
method.
Application 5.1. Consider the following
convection–reaction–diffusion equation
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(21)
subject to the conditions
(22)
Applying Sawi transform homotopy
perturbation method for Eq. (21), we obtain:
(23)
Taking the inverse Sawi transform to Eq. (23) ,we
get
(24)
Note that, the first few terms of in this case
is given by:
(25)
The function of the Caputo derivative result is
achieved by calculating the powers of
(26)
(27)
Putting into Eq. (27) , we get
Putting into Eq. (27), we get
in the same way, we get
Therefore, the solution of Eq.(21) is given by
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(28)
at , then the exact solution is
Below, we sketch the graph of the exact solution
in Figure 1, and the approximate
solution in Eq. (28) with different values of ,
in Figure 2.
Fig. 1: The exact solution convection–reaction–
diffusion equation (21)
Fig. 2: The approximate solution in Eq. (28) with
different values of
In Figure 3, the 3D plots showing the absolute
error between the solution of Eq.(21) and the exact
solution for each specified value of .
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Fig. 3: The absolute error of Application 5.1
The above plots show the difference between
the approximate solution and the exact solution
for different values of . This difference can
be interpreted as the deviation of the exact function
from a simple exponential function .
As decreases, the difference becomes more
pronounced, especially for larg values of and .
This suggests that the approximate solution deviates
more from as decreases. The complexity of
the surface increases as decreases, indicating that
the function becomes more sensitive to changes in
and .
Application 5.2. Consider the following
convection–reaction–diffusion equation:
(29)
subject to the conditions:
(30)
Applying Sawi transform homotopy perturbation
method for Eq. (29), we obtain:
(31)
Taking inverse Sawi transform to Eq.(31) ,we get:
Thus, the function of the Caputo derivative result is
achieved by calculating the powers of:
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(32)
(33)
Putting into Eq. (33), we get:
Putting into Eq. (33), we get:
in the same way, we get:
Therefore, the solution of Eq. (29) is given by:
(34)
At , the exact solution is
In Figure 4, we sketch the exact solution of
Application 5.2, that is
Fig. 4: The exact solution convection–reaction–
diffusion Eq. (29)
In Figure 5, we plot the approximate solution in
Eq. (34) with different values of .
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Fig. 5: The approximate solution in Eq. (34) with
different values of
Application 5.3. Consider the following
convection–reaction–diffusion equation
(35)
subject to the conditions
(36)
Applying Sawi transform homotopy
perturbation method for Eq. (35), we obtain
(37)
Taking inverse Sawi transform to Eq. (37) ,we get
Note that, the first few terms of in this case
is given by:
(38)
The function of the Caputo derivative result is
achieved by calculating the powers of :
(39)
(40)
Putting into Eq.(40) , we get
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in the same way, we get:
Therefore, the solution of Eq. (35) is given by:
at , then the exact solution is
Here are the 3D plots Figure 6 showing the
absolute error between the solution of Eq. (35) and
the exact solution for each specified value of
:
Fig. 6: The absolute error of Application 5.3
6 Conclusion
This paper provided a thorough analysis of the Sawi
transform homotopy method perturbation, a novel
and effective technique for solving fractional
Caputo PDEs. In terms of computational efficiency
and solution precision, the STHPM has
demonstrated significant gains over traditional
techniques like the Variational iteration method and
the Adomian decomposition method. Through the
integration of homotopy techniques and the Sawi
transform, we effectively resolved a number of the
most formidable challenges associated with the
solution of nonlinear PDEs.
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One benefit of this undertaking is that it is possible
to decrease the processing time while maintaining
the precision of the solutions. For this reason,
researchers and professionals who need to solve
fractional Caputo PDEs rapidly and precisely will
find the STHPM to be an extremely useful
instrument. There are numerous potential paths for
further investigation in the future. We feel that the
STHPM has the potential to transform how
fractional Caputo PDEs are treated and solved, and
we are hopeful about its future contributions to
academia and industry.
Acknowledgments:
The authors express their 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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Conflict of Interest
The authors have no conflicts of interest to declare.
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