Solving Nonlinear Volterra Integral Equations by Mohanad
Decomposition Method
RANIA SAADEH1,*, RAED KHALIL2
1Department of Mathematics, Faculty of Science,
Zarqa University, Zarqa 13110,
JORDAN
2Department of Computer Information Systems, Faculty of Prince Abdullah Bin Ghazi,
Balqaa Applied University,
JORDAN
*Corresponding Author
Abstract: - In this research article, we introduce the Mohanad transform-decomposition method, which is a new
analytical approach. The basic characteristics and facts of the proposed method are presented and analyzed.
This new method is a simple method that combines the Mohanad transform with the decomposition method.
This new approach is utilized to handle nonlinear integro-differential equations, the results obtained from this
method are expressed in the form of an infinite series that converges rapidly to the exact ones. The maximum
absolute error is computed for the proposed examples, and some figures are presented to show the accuracy of
the obtained results. All the numerical results and computations in this study are gained by using Mathematica
software.
Key-Words: - Integral transform; Mohanad transform; Adomian Decomposition method; Error analysis;
Convolution theory; Integral equations; Nonlinear problems.
Received: April 7, 2023. Revised: December 8, 2023. Accepted: February 11, 2024. Published: March 22, 2024.
1 Introduction
One of the most important tools for solving
problems in science and engineering is the integral
equation. Both Volterra and Fredholm integral
equations are widely used, and offer useful solutions
for a variety of initial and boundary value issues.
Integral equations have advanced greatly as a result
of advancements in potential theory, [1]. Integral
equations have many applications in different fields
of science and engineering, because of these
applications, they got great interest from authors and
specially mathematicians, they appeared in quantum
mechanics astrophysics, conformal mapping,
scattering, and water waves, [2], [3], [4] and [5].
The importance of studying nonlinear integral
differential has been increased due to the different
branches of applications that could be handled.
Queuing theory and chemical kinetics, and it is
expanded now in many other scientific disciplines.
[6], [7], [8], [9] and [10]. So, researchers,
established many methods to solve these problems,
such as the homotopy analysis approach, [11], the
variation method, the iteration method, [12], the
least squares method, [13] and Adomian’s method,
[14]. Utilizing these methods, we can overcome the
difficulty in the process of solving nonlinear integral
equations.
By offering useful approximate analytical series
solutions for such complex problems, the
decomposition method has established itself as one
of the most effective strategies for solving nonlinear
differential and integral equations. The method of
Adomian decomposition, first presented by [15] and
[16], was created especially for solving integral
equations. The authors in [17], [18], [19] and [20],
improved this research to resolve the Volterra
integral differential equation successfully. The
strategy has been used later to address a variety of
issues in several different domains in response to
time, as stated in [21], [22], [23], [24] and [25].
Integral equations have become more than an
essential tool for solving integral equations, but also
crucial in deriving solutions for intricate situations.
The Laplace transform is one of these
transformations that has shown to be quite helpful,
[26]. To further expand the scope of solving integral
equations and improve the overall effectiveness of
the decomposition method in addressing various
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scientific and engineering challenges, other
transforms, such as the ARA transform, [27], a lot
of scientific transforms such as formable transform,
and others, [28], [29] and [30], have also been
checked and verified.
Once introduced in 2013, [31], the Mohanad
transform is a significant literary transformation
with a huge number of mathematical issues. Given
by the following integral formula is the Mohanad
transform:
The transform can answer a wide range of
problems, academics are paying close attention to it.
To overcome the difficulty in nonlinear cases, we
can easily merge the transform with one of the
numerical methods, such as, [32], [33] and [34].
This will introduce a new hybrid method called: The
Mohanad-decomposition method (MDM), it merges
the two powerful techniques, the Mohanad
transform and decomposition method which is the
main objective of this article.
This article investigates solving nonlinear
Volterra IDE of the form:
where is the difference kernel of the
equation, is piecewise continuous function,
and is a given analytic function of the
unknown , that could be ,
The paper follows the following structure: In
Section 2, we introduce the definition of Mohanad
transform along with its basic properties, and we
also explain the core concept of the Adomian
decomposition method. Section 3 presents the
application of the Mohanad decomposition method
(MDM) for handling nonlinear Volterra integral
differential equations (IDEs). To demonstrate the
method's effectiveness, we solve several numerical
examples of IDEs. Lastly, in Section 5, we provide
the concluding remarks for this article.
2 Basic Facts
In this section, the needed properties and theorems
of the Mohanad transform and the decomposition
method are presented.
2.1 Mohanad Transform
In this section of the article, we introduce some
definitions and properties of Mohanad integral
transform.
Definition 1. Assume that is a continuous
function with domain subset of , then
Mohanad integral transform of is given by the
formula
The Mohanad transform inverse of a function
is defined as:
Theorem 1. Assume that is a piecewise
continuous function with domain and
assume that the following condition holds:
, for a real number . Then,
Mohanad integral transform is well
defined for .
Proof. The formula of Mohanad transform implies:
Thus, Mohanad transform is well defined and
exists for .
We present some properties and the values of
Mohanad transform to some elementary functions.
Assume that and
and , then
Table 1, presents some quantities of Mohanad
integral transform to the standard basic functions.
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Table 1. Mohanad integral transform
2.2 Iterative Decomposition Technique
The Adomian decomposition method is well-known
for its efficiency in solving several types of
nonlinear differential equations, ordinary or partial.
It is a commonly used technique in the domains of
engineering, physics, and applied mathematics. The
Adomian decomposition method's main idea is to
divide the equation's nonlinear term into a number
of elements. When these elements are added
together, the result is represented astonishingly
accurately. To use the method, we assume the series
representation for the solution of the required
equation, given by:
The nonlinear term in the considered problem is
then given in the form of a recursive formula. The
series answer is then entered into the equation. We
proceed to solve the resultant equation recursively
to identify the series components after
simplifying it. We may approximate the solution
more precisely with each iteration, which produces
results that are highly accurate in real-world
applications.
3 Nonlinear Volterra IDEs
In this section, we apply the Mohanad integral
transform to the required IDE, and then we operate
the decomposition technique, that considered the
basic part of the MDM. In addition, the given
kernels in the equation are assumed to be in the
difference form.
Now, let us construct the solution of the following
IDE of the form:
(1)
associated with the initial conditions (ICs)
(2)
To solve the problem (1) by MDM, firstly, apply
Mohanad transform to it, to get:
Using the convolution and the differential properties
from Table 1 of Mohanad transform, we can
simplify the equation to:
(3)
Following that, we substituting the ICs (2) in
Equation (3) to obtain:
(4)
Now, using the decomposition method to handle the
analytic function , express the analytic
function in a form of infinite series of the
form:
(5)
where , can be calculated and
expand the function in the form:
(6)
noting that are given by
(7)
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The Adomian polynomials are the components 's,
are used to handle the nonlinear function
as:
(8)
Thus, substituting Equation (5) and (6) into
Equation (4), to get:
(10)
(3.9)
The relation of Adomian decomposition technique
gives us:
(11)
Equation (9), implies:
(12)
Remark 1. Equation (12) is well defined if the
condition
is satisfied. Applying the inverse Mohanad
transform to Equations (11) and (12) respectively,
we get the values of .
The solution of the required equation Volterra
IDE (1) can be expressed in the form:
The proposed method is effective in expressing
approximate numerical results of Volterra IDEs
(linear and nonlinear). To show accuracy of the
proposed technique, we present some numerical
examples and solve them by MDM. Moreover, we
calculate the absolute error, given by the formula:
defined on some interval.
4 Numerical Examples
This section presents some examples of integral
equations: (IE)s and IDEs that are solved by MDM,
the maximum absolute error is computed to each
example to verify the efficiency of the proposed
results.
Example 4.1
Take the nonlinear Volterra IE:
(13)
(4.1)
Solution. The accurate solution of problem (4.1) is
. To solve Equation (13) by MDM, the
Mohanad transform is operated to Equation (13) to
get:
(14)
(4.2)
The substitution of the series and the usage of
the Adomian polynomials for , give:
The nonlinear term is decomposed by
Equation (7), to get the components as follows:
(15)
Comparing the terms obtained in equation (7) and
operating the inverse Mohanad transform, to get:
(9)
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Thus, the numerical approximation series solution is
given by:
Table 2 proposes comparisons between the
accurate solution and obtained numerical
approximated solution to Example 4.1. To prove the
accuracy of the MDM, we calculate the absolute
error.
Table 2. Approximate and exact solutions of
Example (4.1)
Nodes
Exact
Solution
Approximate
Solution
Absolute
Error
Figure 1 below, presents the graph of
approximate and exact solutions. The absolute error
is presented in Figure 2, of Example 4.1.
Fig. 1: The exact and approximate solutions of
Example 4.1
Fig. 2: The graph of absolute error of Example 4.1
Example 4.2. Take the nonlinear Volterra IE:
(19)
(20)
Solution. Operating Mohanad integral transform to
Equation (19), we obtain:
which can be simplified to:
(18)
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Also, we get:
The Adomian components of the polynomials
of , can be obtained by:
Applying the inverse Mohanad transform to the
functions in (19) and using the recursive formula,
we have:
Thus, the approximate solution of Example 4.2 is:
that converges directly to the exact solution
.
In Table 3, we introduce the accurate and
approximate solutions of Example 4.2, and to prove
the accuracy of the method, we calculate the
absolute error.
Table 3. The exact and approximate solutions of
Example 4.2, and the absolute error
In Figure 3, we sketch the exact and
approximate solutions. We also sketch the absolute
error of the exact and approximate solutions of
Example 4.2 in Figure 4.
Fig. 3: The approximate and exact solutions of the
Example 4.2
Fig. 4: The graph of absolute error of Example 4.2.
5 Conclusion
The main objective of this study was to introduce an
innovative and efficient approach for solving
nonlinear Volterra integral differential equations
(IDEs). We achieved this by presenting approximate
solutions for a family of nonlinear IDEs in the form
of infinite series solutions, employing the MDM
(Mohanad transform combined to Adomian’s
decomposition method). Several examples of
Volterra IDEs were examined to validate and
demonstrate the simplicity and efficiency of the
proposed method. The findings of this research
article indicate that MDM is a straightforward and
effective method for handling nonlinear IDEs. The
accuracy and efficiency in providing approximate
solutions proposed in this article offer promising
prospects for solving a wide range of challenging
problems. In future research, we plan to further
enhance and refine the method to tackle nonlinear
fractional integral equations. This extension would
broaden the scope of its applicability and could
potentially open up new opportunities for solving
more complex mathematical models.
Nodes
Exact
Solution
Approximate
Solution
Absolute
Error
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Acknowledgment:
This work was funded by the deanship of Zarqa
University.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
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
This work was funded by the deanship of Zarqa
University.
Conflict of Interest
The authors have no conflicts of interest to declare.
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(Attribution 4.0 International, CC BY 4.0)
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