A New Computation Approach: ARA Decomposition Method
AHMAD QAZZA
Department of Mathematics,
Zarqa University,
Zarqa 13110,
JORDAN
Abstract: - In this study, we present a novel combination between the ARA transformation and the
decomposition method, termed the ARA decomposition approach. We present the method in a simple algorithm
and use it to solve nonlinear integro-differential equations. To test the efficiency of the new approach, we solve
some examples and calculate the absolute errors and sketch the approximate and exact solutions.
Key-Words: - Decomposition method; ARA transform; ARA decomposition method; Volterra integro–
differential equations
Received: July 28, 2022. Revised: February 23, 2023. Accepted: March 19, 2023. Published: April 26, 2023.
1 Introduction
Integral equations are presented in numerous areas
of engineering, physics, and mathematics, used in
initial and boundary value problems, and
transferred to Fredholm and Volterra integro–
differential equations (VIE), e.g. Dirichlet problems
in astrophysics, conformal mapping, mathematics,
physical models, diffusion problems, water, [1].
Nonlinear integral equations are used in many
fields of study, e.g. queuing theory, chemical
kinetics, fluid dynamics, etc., [2], [3], [4], are also
used in numerical solution by various methods such
as Galerkin, decomposition, quadrature, cubic
spline polynomials, etc., [5], [6], [7]. One of the
useful and important methods that have received a
lot of attention is the Adomian decomposition
method (ADM). In this method, more emphasis is
placed on finding reliable and efficient solution
methods in various fields of science and
technology, [8], [9], [10], [11], [12].
The ARA transformation is introduced in 2020,
[13]. It is defined by the improper integral.
This transformation has attracted a lot of attention
from researchers due to its ability to produce
multiple transformations of index , and it could
also easily overcome the challenges of having
singular points in differential equations. Despite all
these merits, it could be used to solve different
types of problems. In this work, we use the first-
order ARA transform , which we denote
by for the sake of simplicity.
This work aims to develop a combined form of the
ARA transformation method with the ADM, called
the ARA-decomposition method (ARA-DM), to
obtain exact solutions or high-precision
approximations for the nonlinear VIE. The
advantage of this method is its ability to combine
the two powerful methods for obtaining exact
solutions to nonlinear integral equations.
In this study, we investigate the solutions of the
nonlinear VIE of the second kind is
where the kernel and are real-valued
functions, and is a nonlinear function of
, such as , , .
The rest of the paper is constructed as follows.
Section 2 defines the basic definitions of the ARA
transform and ADM. Section 3 introduces the
concept of applying the ARA transform in
combination with the ADM to solve the second
type of nonlinear VIE. By solving significant
examples in Section 4, the effectiveness and
efficiency of the proposed method are illustrated.
Finally, in Section 5, the conclusion of the work is
presented.
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2 Preliminaries
In this section, we present the basic definitions and
properties of the ARA transform. In addition, the
basic idea of the ADM method is presented.
2.1. ARA Integral Transform, [13]
Definition 1. Let be a piecewise continuous
function defined on . Then ARA transforms
for denoted and defined by
The inverse ARA transform for denoted
and defined by
Theorem 1.(Existence Condition) If is a
piecewise continuous function on and
satisfies the condition
, for some .
Then, ARA transform exists
for .
Proof. Using the definition of ARA transform, we
obtain
Hence, ARA integral transform exists for
. □
Now, we mention some properties of ARA
transform to the basic functions. Suppose that
and and
, then
Now the following table (Table 1) introduces
some values of ARA transform to some elementary
functions.
Table 1. ARA transform for some functions.
2.2 Adomian Decomposition Method, [2]
In this section, we introduce the main idea of
ADM, which is a powerful technique used to
handle a large class of nonlinear ordinary
differential equations and partial differential
equations.
The ADM is a very powerful approach used to
solve broad classes of nonlinear partial and
ordinary differential equations. It has wide
applications in engineering, physics, and applied
mathematics.
The ADM depends on decomposing the unknown
equation into the sum of some components to be
determined. The sum of these components
represents the solution with high accuracy. ADM's
algorithm is illustrated in the following steps:
Assume that the target problem has the
following series solution represented as
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Establish a recursive relation of the nonlinear
term in the target problem and substitute the
value of the series solution depending on the
relation
3 Solving Nonlinear VIE by ARA-
DM
In this section, we apply the ARA transform in
combination with the ADM to solve the nonlinear
VIE of the second type. Also, we assume that the
given kernel is of a different kind, which could be
expressed in the form , such as
, ,
Now let us consider the nonlinear VIE equation of
the form
(1)
subject to the initial conditions (ICs)
(2)
where is a nonlinear function on .
To obtain the solution of Equation (1) by ARA-
DM, we firstly apply ARA transform to both sides
of Equation (1)
Applying the differential and the convolution
properties of the ARA transform, we can rewrite
Equation (1) as
(3)
Thus, substituting the ICs (2) and simplifying
Equation (3), we obtain
(4)
Now, utilizing the ADM to handle the nonlinear
term , we need to express as an
infinite series with components as
(5)
The components , can be obtained
from a recurrence relation, and the nonlinear term
can be presented as
(6)
where are defined as
(7)
where 's are called the Adomian polynomials for
the nonlinear function , the Adomian
polynomial can be determined by
(8)
Thus, by substituting Equations (5) and (6) in
Equation (4), we get
(9)
The recursive relation from ADM implies
(10)
From Equation (9), one can get
(11)
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Remark 1. A necessary condition for Equation (11)
to be well-defined is that
Operating the inverse ARA transform to the
equations in (11) recursively, we can obtain the
values of the components .
The solution of the VID Equation (1) is
The proposed method is efficient in finding
approximate solutions of nonlinear VIEs. To
measure the accuracy of the method, we solve some
problems and use the maximum absolute error,
given as
which is given in some intervals.
4 Numerical Applications
In this section, we apply ARA-DM to solve some
applications of VIEs, and we use absolute error to
determine the efficiency of our results.
Problem 1.
Consider the following nonlinear VIE of the form
(12)
Solution. The exact solution of Equation (12) is
.
To get the solution by the proposed method, we
again apply ARA transform to Equation (12), to get
(13)
For the nonlinear term , it can be
decomposed using the formula in Equation (7), one
can obtain the following components
(14)
Making comparisons in the iterative form of
Equation (7) and applying the inverse ARA
transform, to obtain
Thus, the approximate solution can be expressed as
Table 2 below presents the values of the exact and
ARA-DM solutions of Problem 1, and to test the
efficiency we compute the absolute error as
follows.
Table 2. The exact and ARA-DM solution of
Problem 1, and the absolute error.
Exact
Solution
ARA-DM
Solution
Absolute Error
In the following figure below, we sketch the exact
and approximate solutions in Figure 1 below. Also,
we sketch the absolute error of Problem 1 in Figure
2.
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Fig. 1: The exact and approximate solutions of the
nonlinear VIE in Problem 1.
Fig. 2: The absolute error of the exact and
approximate solutions of Problem 1.
Problem 2. Consider the following nonlinear VIE
of the form
(15)
Solution. The exact solution of Equation (15) is
.
Applying ARA transform to Equation (15), we get
(16)
Thus by similar arguments to Problem 1 one can
obtain
Thus, the approximate solution can be expressed as
Table 3 below presents the values of the exact and
ARA-DM solutions of Problem 2 and tests the
efficiency we compute the absolute error.
Table 3 The exact and ARA-DM solutions of
Problem 2, and the absolute error.
Exact Solution
ARA-DM
Solution
Absolute Error
In the following figures below, we sketch the exact
and approximate solutions in Figure 3 below, and
we sketch the absolute error in Figure 4.
Fig. 3: The exact and approximate solutions of the
nonlinear VIE in Problem 2.
Fig. 4: The absolute error of the exact and
approximate solutions of Problem 2.
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Problem 3. Consider the following nonlinear VIE
of the form
(17)
(18)
Solution. Applying ARA transform to Equation
(17), we get
Now, we have
(19)
The Adomian polynomials of , can be
determined as
Taking the inverse ARA to transform to the
functions (19) and using the given recursive
relation, one can obtain
Hence, the approximate series solution of Problem
3 is
which converges to the exact solution .
Table 4 below, presents the values of the exact and
ARA-DM solutions of Problem 3, and to test the
efficiency we compute the absolute error.
Table 4. The exact and ARA-DM solution of
Problem 3, and the absolute error.
Exact Solution
ARA-DM
Solution
Absolute Error
In the following figure below, we sketch the exact
and approximate solutions in Figure 5 below. We
also sketch the absolute error of the exact and
approximate solutions of Problem 3 in Figure 6.
Fig. 5: The exact and approximate solutions of the
nonlinear VIE in Problem 3.
Fig. 6: The absolute error of the exact and
approximate solutions of Problem 3.
Problem 4. Consider the following nonlinear VIE
of the form
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(20)
(21)
Solution. Applying the same procedure from the
previous problems, we can obtain
Thus, the approximate solution of (20) and (21) can
be expressed as
which converge to the exact solution
Table 5 below presents the values of the exact and
ARA-DM solutions of Problem 4, and to test the
efficiency we compute the absolute error.
Table 5. The exact and ARA-DM solutions of
Problem 4, and the absolute error.
In the following figure below, we sketch the exact
and approximate solutions in Figure 7 below.
Lastly, the absolute error of the exact and
approximate solutions of Problem 4 is presented in
Figure 8.
Fig. 7: The exact and approximate solutions of the
nonlinear VIE in Problem 4.
Fig. 8: The absolute error of the exact and
approximate solutions of Problem 4.
5 Discussion and Conclusion
The main goal of this research is to develop an
effective approach to solving nonlinear VIE. We
obtain an approximate series solution of a specific
family of nonlinear VIE problems using a new
approach, that combines a combination of the ARA
transform and the decomposition method, called
ARA decomposition approach. The given problems
are first simplified using the ARA transform, and
then the results are treated by applying the
Adomian decomposition method. The solutions to
VIE problems are examined and found to best
represent the true dynamics of the problem.
To demonstrate the validity of the proposed
method, the results are presented graphically and
tabulated. The main advantage of the proposed
method is the rapid convergence of the series form
solutions to the precise ones. It turns out that the
presented method for solving nonlinear integro-
differential equations is both simple and effective,
and thus can be applied to other scientific
problems.
The method provides a useful way to develop an
analytical treatment for these equations. In future
Exact Solution
ARA-DM
Solution
Absolute Error
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work, we will use the proposed scheme to solve
other nonlinear equations and fractional differential
equations.
The ARA-DM is used in this research to solve
nonlinear integro-differential equations. We solved
some numerical examples and sketched the
solutions. From the problems discussed, one can
see the efficiency of the proposed method. From
the previous figure, we can see the agreement
between the exact and approximate solution. We
also made comparisons and calculated the absolute
errors.
Acknowledgement:
The author expresses his 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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at all stages from problem formulation to final
results and solution.
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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.
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