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A Novel Approach to Nonlinear Volterra-Fredholm Integral
Equations Using Abaoub Shkheam Decomposition Method
Ali E. Abaoub
1
, Abejela S. Shkheam
2
, Khireya A. Alkeweldy
3
1,2
Mathematical Department, Essential School of Science, Libyan Academy of Postgraduate, Tripoli – Libya
3
Mathematical Department, Faculty of Science, Zintan University, Zintan-Libya
doi.org/10.51583/IJLTEMAS.2024.131021https://DOI :
Received: 24 October 2024; Accepted: 03 November 2024; Published: 17 November 2024
Abstract: In this study, we introduce a novel approach to the solution of a nonlinear Volterra -Fredholm integral equations by
applying the Adomian decomposition method under the effect of the Abaoub- Shkheam transform. We demonstrate the existence
and uniqueness of the solution in Banach space and illustrate this idea with an example.
Keywords: Abaoub Shkheam-Transform, Volterra- Integral Equations, Fredholm, Integral Equations
I. Introduction
We examine a Volterra Fredholm integral equations of second kind that is nonlinear, given by
where is the unknown function that will be determined,
and the function are given real-valued
functions. The functions
are given nonlinear functions of u, λ is the parameter, a and b are constants.
In 2020, Abaoub and Shkheam introduced a novel integral transform known as the Abaoub Shkheam transform [14], which they
utilized to address linear Volterra integral equations [15]. The following year, Shkheam and collaborators used the Abaoub
Shkheam trans form to solve the Linear Volterra Integro-Differential Equation of the First Kind [16]. In 2022, Asmaa Mubayrash
applied this transform to solve partial differential equations [19], and in 2022 Asma Mubayrash and her colleagues using this
transform for Solving Partial Differential Equations [17]. Building on this work, in 2023, Suad Zali employed the transform to
tackle linear partial integro-differential equations [20], while Nagah Elbhilil and colleagues used it to solve Volterra Integral and
Volterra Integro-Differential Equations [18].
The Adomian Decomposition Method (ADM) is a brand-new, extremely powerful method that Adomian [2],[21] first presented
in the early 1980s for solving a wide range of equations, including integral, differential, partial differential, and linear and non-
linear algebraic equations [22]-[31]. The solution series has shown to rapidly converge using this strategy. The non-linear term is
broken down into a set of specialized polynomials known as Adomian’s polynomials in order for it to function. Our main goal in
this paper is to solve non-linear Volterra Fredholm integral equations by using the Combined Aboub Shkheam Transform-
Adomian Decomposition Method.
The Abaoub Shkheam Decomposition Method
The nonlinear Volterra-Fredholm integral equation with difference kernels is expressed as follows:
Abaoub Shkheam transform method can solves the nonlinear Volterra-Fredholm integro differential equation (2). We utilize the
Abaoub Shkheam transform on bothe sides of (2) we get
using convolution theorem of the Q-Transform in (3), we obtain
The Adomian decomposition method, along with Adomian polynomials, can be employed to tackle equation (4) and address the
nonlinear term
. Initially, we represent the linear term
on the left side as an infinite series of components,
given by:
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where the components
will be determined recursively. However, the nonlinear terms
and
on the
right side of equation (4) will be expressed as infinite series involving Adomian polynomials
and
, respectively, in the
following form:
where
, are defined by
where the Adomian polynomials
can be computed for various forms of nonlinearity. Specifically, for a given nonlinear
function
, the Adomian polynomials are defined as follows:
Similarly, We can evaluated the Adomian polynomials
of the nonlinear function
as following
Substituting equation (5) and equation (6) into equation (4) leads to:
The recursive relation is presented by using the Adomian decomposition method
When the first part of equation (7) is applied with the inverse Abaoub-Shkheam transform,
is obtained. Determine
and
yields
is used to evaluate
and so on. This leads to the complete determination of the
components of
,n ≥ 0 upon applying the second part of Eq. (7). The series solution follows immediately after applying Eq.
(5). The obtained series solution may converge to an exact solution, if such a solution exists.
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Existence and Uniqueness Analysis of Solutions
We shall introduce and prove the results regarding the existence and uniqueness of solution to Equation (2). In order to prove
these results, we first present the suitable hypotheses.
in a neighborhood of , and for any n (the derivatives of
at are bounded in norm);
There exists a constant such that, for any
in Banach space
).
We suppose that for all the kernels
are satisfies the conditions:
The functions
satisfying the Lipschitz condition:
Theorem 3.1 [ 32]
Under the previous hypotheses
and
, the series
is absolutely convergent and, furthermore
where
is the minimal of
.
Theorem 3.2 (Existence and Uniqueness)
Suppose that
hold. If where
and
Then there exists a unique solution
to Eq.(2).
Proof:
By using the Adomian decomposition method, we get
λ
λ
taking the norm of equation (8), yields
λ
λ
utilizing the Cauchy- Schwarz inequality, we get
λ
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Now by using hypotheses (
), (
), (
) and the theorem (3.1), we get
Since then
Under the aforementioned condition (9), the bound ensures that the infinite series
converges uniformly. Consequently,
the function
can be expressed as
:
Since each
is continuous,
inherits this property, demonstrating that
is continuous and convergent. This confirms
the existence of a solution to Equation (2).
Now, we shall prove the uniqueness solution, suppose that Eq. (2) has two solutions
and . Applying the norm on both
sides of Eq.(2), we obtain
By Cauchy-Schwarz inequality and Using hypotheses (
), (
) and (
), we have
We apply the Lipschitz condition
hence
Since this implies that
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Theorem 3.3 (Convergence).
Assume that (
), and (
), then the series solution (5) of the equation (2) converges to the exact solution provided
and 0 1.
Proof
Let
) denote the Banach space of all continuous real-valued functions defined on
. Consider the sequence of
partial sums
defined by:
which represents the partial sums of the series solution (5).Since
So,
Let
and
be arbitrary partial sums with, then
from (10), we have
Let , then
and since
Since 0 then
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But
so, as then
We conclude that
is a Cauchy sequence in Banach space, the convergence of the sequence is equivalent to the convergence
of the series.
4. Applications:
An examination of the following example demonstrates the Abaoub Shkheam decomposition method for solving the nonlinear
Volterra Fredholm equations.
Example. Consider the
the nonlinear Volterra-Fredholm integro differential equation of the second kind
Taking the -ttransform of both sides of the equation (14) gives
Utilized the Adomian decomposition method to both sides of the equation (15), gives:
The Adomian decomposition method presents the recursive relation:
where
and
are the Adomian polynomials for the nonlinear term
and
, respectively. The Adomian
polynomials for
and
are given by:
Substituting in Eq.(16), we obtain
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.
that converges to the exact solution
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