Analytical Approaches for Computing Exact Solutions to System of
Volterra Integro-Differential Equations
NIDAL ANAKIRA1,2,*, ADEL ALMALKI3, M. J. MOHAMMED4, SAFWAT HAMAD4,
OSAMA OQILAT5, ALA AMOURAH1
1Department of Mathematics, Faculty of Education and Arts,
Sohar University,Sohar 3111,
SULTANATE OF OMAN
2Jadara University Research Center,
Jadara University,
JORDAN
3 Department of Mathematics, Al-Qunfudhah University College,
Umm Al-Qura University,
Mecca,
SAUDI ARBIA
4Department of Mathematics, College of Science,
University of Anbar Ramadi,
IRAQ
5Department of Basic Sciences, Faculty of Arts and Science,
Al-Ahliyya Amman University,
Amman 19328,
JORDAN
*Corresponding Author
Abstract: - This paper presents a modified technique that utilizes the homotopy-perturbation method (HPM) to
solve a system of integro-differential equation of Volterra kind. By providing practical examples and
conducting numerical simulations, we showcase the effectiveness and efficiency of this modification in solving
these systems encountered in various scientific fields. Furthermore, we compare the performance of the HPM
with the exact solution, emphasizing its advantages in terms of accuracy, convergence, and computational
efficiency.
Key-Words: - Volterra integro-differential equations; Numerical approximation; HPM; Series solution; Laplace
transformation; Padé approximants.
Received: August 29, 2023. Revised: April 11, 2024. Accepted: May 8, 2024. Published: June 27, 2024.
1 Introduction
Integro-differential equations are commonly
encountered in engineering and scientific
disciplines. Scientific and engineering disciplines
frequently use integral-differential equations. A
wide range of physical phenomena, including wind
ripples in deserts, dropwise consideration in
nonhydrodynamics, and the formation of glass, are
explained by them, [1], [2] and [3]. In addition,
many equations equation have important
applications in theoretical physics and as
mathematical representations of viscoelasticity.
Numerical solutions are essential for
comprehending complicated dynamical systems in
physics, biology, and economics, among other
disciplines.Volterra integro-differential equations,
which incorporate integrals and derivatives, describe
these systems. These equations are difficult to find
analytical solutions for, which is why approximating
their solutions numerically is a common practice.
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Different numerical techniques have been developed
by many scholars to solve systems of linear integro-
differential equations. Numerous scholars have
devised numerous numerical techniques to resolve
linear integro-differential equation systems.
Specifically, systems of Volterra integro-differential
equations are solved using the reconstruction of the
variational iteration approach, [4]. This method
produces results that are more accurate than the
homotopy perturbation method, according to
comparisons between the two. Researchers in their
work, [5], used the homotopy perturbation method
to estimate the solution of Volterra integro-
differential equation systems. The optimal
homotopy asymptotic method was employed to find
the solutions of a system of Volterra integro-
differential equations, [6].
Moreover, authors in their research, [7], used
the Sinc collocation and Chebyshev wavelet
methods to solve linear Volterra integro-differential
equation systems. Chebyshev polynomials [8], the
single-term Walsh series technique, [9], the
differential transform method, [10], the power series
method, [11], the homotopy perturbation method,
[12], the homotopy analysis method, [13], and the
modified Adomian decomposition method, [14], are
other numerical techniques that are frequently used
to solve such systems. However, the presence of
integrals requires special attention to efficiently
handle the integral terms. Additionally, in general,
the stability and convergence of numerical schemes,
[15], [16], [17], [18], [19], [20], [21], [22], [23],
[24], [25], [26], are crucial factors in obtaining
accurate results.
In this work, we introduce a new improvement to
the solution of the HPM procedure, specifically for
solving systems of Volterra integro-differential
equations. Our method applies to any given
problem, offering accurate approximate solutions
that approach the exact solution as the number of
approximation terms increases. It is important to
note that the accuracy of our method depends on the
order of the approximation used, which may require
additional computational effort and time, especially
for nonlinear problems. Therefore, researchers are
continuously striving to develop or modify
numerical techniques to achieve higher accuracy or
exact solutions.
The main objective of this paper is to enhance the
accuracy of the HPM by employing an alternative
approach. This approach involves modifying the
series solution of the HPM by applying the Laplace
transformation to the truncated HPM solution.
Subsequently, the transformed series is converted
into a meromorphic function using Padé
approximants. Finally, we apply the inverse Laplace
transformation to obtain the desired solution for the
given problem. This method is straightforward and
yields precise results with high performance,
without requiring significant effort. The structure of
this paper is organized as: Section 2 introduces the
fundamental concept of the HPM, along with a brief
explanation of the Pade approximants. In Section 3,
numerical examples are presented to demonstrate
the effectiveness of the discussed procedure in
obtaining the analytic solution of systems of
Volterra integro differential equations. The results
highlight that accurate solutions can be obtained
with only a few terms. The final section summarizes
the conclusions of this work.
Fundamental Idea of HPM
Procedure
To demonstrate the fundamental concept of the
HPM procedure, [27], [28], [29] and [30], consider:
Given that is a general integral operator, is
a boundary operator, is a known analytic
function, and is the boundary of the domain .
The operator can be divided into two parts: is
linear, while is nonlinear. Thus, the Eq. can
be rewritten as follows:
Now, we construct R which satisfies
or
where , that is parameter, and
is an initial approximation of Eq. Hence
and the process of changing from to to and
from to
which is called deformation in topology, this,
where and are called
homotopic. Since considered a small
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parameter, we assume the solution of Eqs. or
expressed in the following:
when , Eq becomes the approximate
solution of Eq. i.e.,
Padѐ Approximation
The function is defined by the Padé
approximation, [31], [32], [33] and [34].
where the highest degree polynomial for is
and the highest degree polynomial for is
Regarding the formal power series
We can find the coefficients of the polynomials
by using the following equation:
and .
When the denominator and numerator's
functions
is multiplied by a constant that is
not zero, the fractional values stay the same, such
that we can set up the normalization requirement
as
It should be noted that the polynomial for
functions and has no public factors. If
the coefficients of the polynomial functions
and are expressed as:
We can derive the following linear systems of
coefficients by multiplying Eq. (8) by to be:
These equations will be solved using Eq. (11),
which represents a set of linear formulas for the
unidentified variables. Once the ' are identified,
we can derive an explicit formula for the unknown
p's, which will provide the solution to the problem.
4 Applications of HPM
The purpose is to demonstrate the effectiveness and
reliability of our modified procedure.
Example Given the system of integro-differential
equations of Volterra type, [34], [35], [36], and [37].
,
Subject to and exact
solutions
Based on the algorithm presented in Section 2, we
will now proceed to construct the following
homotopy equation.
[
[
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The zeroth-order problem is expressed in Eqs.
as follows:
which has the solution
Based on Eqs. (15) the first-order problem is given
in the form of
under the conditions
Therefore, it has the following solution.
The following formula defines the second-order
problem.
using the conditions We
have the following solution
Based on the HPM procedure, we have the 5th -order
HPM approximate solution
This leads to
as
. Table and
depicts numerical results for HPM
Procedure and the exact one. The HPM is well-
known for its simplicity and versatility. It allows the
calculation of approximate solutions to differential
equations without requiring linearization or
restrictive assumptions. To enhance the accuracy of
the HPM solution, we will use the MHPM that
builds upon the capabilities of the HPM by
addressing its limitations in terms of the number of
terms, convergence rate, and computational
complexity. We achieve this by employing Pade
approximation, Laplace transformation, and
ultimately the inverse Laplace transformation, as
described below:
Use
, leads to
The Pade approximates of order
in term of
, gives
.
The modified approximation solution
is obtained by
applying the inverse Laplace transform to the
Pade approximate.
Example Considering the system of differential
equations of Volterra integro type, [38], [39] and
[40].
, (27)
Subject to
and exact solutions
Based on the algorithm presented in Section 2,
we will now proceed to construct the following
homotopy equations.
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[
[
(29)
Following the same process in example one, we
have the 5th -order HPM approximate solution
This leads to the exact solution
Table 3 and Table 4 depicts numerical results
for HPM procedure and the exact solutions. We
observed that accuracy depends on the order of the
approximations. To obtain more accurate results, we
will modify the HPM solutions. We will achieve
this by employing the Laplace transformation on the
initial terms of the HPM series solutions, using the
Pade approximants, and finally applying the inverse
Laplace transformation as depicted below.
Use
, leads to
Using of
, Then, Pade approximates of order
, yield to
The exact solutions
are obtained by applying the
inverse Laplace transform to the
Pade
approximate.
Table Numerical result of example
Exact Solution
Approximate
Solution
HPM
Absolute
Error
2.47×
3.05×
Table Numerical result of example
Exact Solution
Approximate
Solution
HPM Absolute
Error
1.75×
1.09×
Table Numerical result of example
Exact Solution
Approximate
Solution
HPM Absolute
Error
4.44×
0
1.05×
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Table Numerical result of example
Exact Solution
Approximate
Solution
HPM
Absolute Error
9.99×
2.06×
5 Conclusion
In this research study, we propose a new procedure
based on the HPM for solving a system of Volterra
integro-differential equations. This procedure is not
only effective and reliable, but it also offers a
distinct advantage over other methods. Its ability to
provide accurate solutions for challenging systems
highlights its potential as a valuable tool for
researchers and practitioners seeking to understand
and analyze dynamic phenomena governed by these
systems. Through illustrative examples and
comparisons with numerical results reported in the
literature, we observed that this procedure can
achieve the exact analytical solution by utilizing
only a few terms of the truncated series solution
derived from the HPM solutions. Consequently, we
conclude that this procedure represents a potent
approach and a promising tool for resolving not only
this particular class of differential equations but also
various other types of differential equations.
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