In recent papers the solution of nonlinear Fredholm integral equations was discussed using Adomian decomposition method (ADM). For case in which the integrals are analytically impossible, ADM can not be applied. In this paper a discretized version of the ADM is introduced and the proposed version will be called discrete Adomian decomposition method (DADM). An accelerated formula of Adomian polynomials is used in calculations. Based on this formula, a new convergence approach of ADM is introduced. Convergence approach is reliable enough to obtain an explicit formula for the maximum absolute truncated error of the Adomian’s series solution. Also, we prove that the solution of nonlinear Fredholm integral equation by DADM converges to ADM solution. Finally, some numerical examples were introduced.
Integral equations provide an important tool for modeling a numerous phenomena and processes and also for solving boundary value problems for both ordinary and partial differential equations. Their historical development is closely related to the solution of boundary value problems in potential theory. Progress in the theory of integral equations also had a great impact on the development of functional analysis. Reciprocally, the main results of the theory of compact operators have taken the leading part to the foundation of the existence theory for integral equations of the second kind [1-4]. Therefore, many different methods are used to obtain the solution of the linear and nonlinear integral equations. Among these methods ADM which has gained a great interest in the analytical solutions of linear and nonlinear Fredholm integral equations [5-9]. This is due to many advantages such as simplicity and high accuracy [5,6]. The Adomian solution is obtained as an infinite series which converges to exact solution [10], under some mild conditions. In this work, the nonlinear the Fredholm integral equation
is considered where is known continuous function on and the kernel is continuous on the square and bounded such that where, M is the upper bound on the square E. The nonlinear term is Lipschitz continuous with L is Lipschitz constant and has Adomian polynomials representation
where the traditional formula of is
The author in [11,12] deduced a new formula to the Adomian’s polynomials which can be written in the form
where the partial sum and
Formula (4) is called an accelerated Adomian polynomials and it was used successfully in [13] for solving a class of nonlinear fractional differential equations and in [14] for solving a class of nonlinear partial differential equations. Formula (4) has the advantage of absence of any derivative terms in the recursion, thereby allowing for ease of computation. In this work, it will be used directly in convergence analysis (see Theorem 2) and all calculations concerning the numerical examples. Application of ADM on (1) yields:
where the components are computed using the following recursive relations
The computation of each component requires the computation of integral in Equation (7). If the evaluation of that integral analytically is possible, ADM can be applied in a simple manner. In case where the evaluation of the integral in (7) is analytically impossible, ADM can not be directly applied. In order to overcome this obstacle, please see the details of Sections 2 and 3. In Section 2, a problem is solved in a special case where the kernel is separable [15]. In Section 3, a problem is solved in a more general case where the kernel is not separable and we introduce a discretized modified version of the ADM which is called DADM. In Section 4, convergence of DADM is discussed and the maximum absolute truncated error is estimated. Finally, to verify the theoretical results, some numerical examples are presented in Section 5.
2. Numerical Implementation of ADM
For the sake of making this paper self-contained, a brief summary of numerical implementation of ADM will be introduced in this section (for more details see [15]). Let the kernel function be separable of the form
then Equation (7) becomes
Consider any numerical integration scheme to approximate definite integral by the following formula [16-18]
where is continuous function on, are the nodes of the quadrature rule, and are the weight functions. Applying formula (10) on Equation (9) to obtain
Now, the approximate solution of Equation (1) is the sum of all the components in Equation (11) and the first component in Equation (6).
3. Discrete Adomian Decomposition Method
In case the kernel function, is not separable, the integral in (1) can not be computed and hence the ADM will not be able to continue in order to obtain solution. Therefore, we suggest DADM to overcome this obstacle. The idea is to discretize the independent variable; t, just before applying the quadrature rule. This gives an opportunity to evaluate the integral in Equation (7) numerically but, of course, at the discretization points of the independent variable. Thus, the discrete version of Equations (6) and (7) may take the form
and are the weight functions of any numerical integration scheme. The approximate solution of Equation (1) using DADM can be computed as
Rewriting Equations (12)-(14) in matrix form
where are all vectors of dimension and B is matrix such that
The main advantage of the proposed DADM is that the matrix B is unchanged during the computation of components and the computation of the solution need not to solve linear algebraic system of equations like Nystrom method and projection methods. Also, this method can be used for solving Equation (1) with nonseparable kernel. Thus DADM is more general than the numerical implementation of ADM introduced in [15].
4. Convergence Approach of DADM
Convergence of the Adomian series solution was studied for different problems and by many authors. In [19,20] convergence was investigated when the method applied to a general functional equations and to specific type of equations in [21,22]. In convergence analysis, Adomian’s polynomials play a very important role however, these polynomials cannot utilize all the information concerning the obtained successive terms of the series solution, which could affect and directly the accuracy as well as the convergence region and the convergence rate. In the present analysis we suggest an alternative approach for proving the convergence. This approach depends mainly on El-Kalla accelerated Adomian polynomial formula (4). As a result to this approach, the maximum absolute truncated error of the series solution is estimated. Define a mapping where, is the Banach space of all continuous functions on D with the norm
4.1. Uniqueness Theorem
Theorem 1. Problem (1) has a unique solution whenever
where,
Proof. Define the mapping to be:
. and let x and
be two different solutions to (1) then
Under the condition the mapping F is contraction therefore, by the Banach fixed-point theorem for contraction [23], there exist a unique solution to problem (1) and this completes the proof.
4.2. Convergence Theorem
Theorem 2. The series solution (5) of problem (1) using ADM converges if: and
Proof. Let and be arbitrary partial sums with We are going to prove that is a Cauchy sequence in Banach space B
From Formula (4) we have
so
Let, then
From the triangle inequality we have
Since so, then
But so, as then We conclude that is a Cauchy sequence in so, the series converges and the proof is complete.
4.3. Error Estimate
Theorem 3. The maximum absolute truncation error of the series solution (5) to problem (1) is estimated to be:
where
Proof. From Theorem 2 inequality (18) we have
As then and
so,
Finally, the maximum absolute truncation error in the interval D is:
This completes the proof.
4.4. Equivalence between DADM and ADM
Let D be a closed bounded set in and define operator such that
where is a compact operator on to and is bounded on to since
Now, Equation (1) can be written as
let be the solution obtained by using ADM, where
and Define numerical integral operator as
where is linear finite rank bounded operator on to since
With the operator, Equation (1) may be written as
where here is the solution obtained by using DADMand and
Theorem 4. Since as where [18]. Then, the solution of Equation (1), using DADM converges to the solution of the same equation when using ADM, i.e.
Proof. Since
and.
Starting with
Since
and (25)
Then, by induction and substituting from Equation (25) and Equation (26) into inequality (24), this completes the proof.
5. Numerical Experiments
Consider the following linear Fredholm integral equation
whose exact solution is. In this example the ADM can not be applied because the evaluation of
is conditioned to compute.
Since, the kernel is separable, the numerical implementation of ADM introduced in [15] can be used as well as DADM.
The solution by numerical implementation of ADM introduced in [15]
and the computation of needs Equation (11) and Simpson’s rule [16-18] with number of subintervals and step size to obtain
and so on. The approximate solution by this method is
and the maximum error is
Using Equations (15)-(17) and Simpson’s rule with number of sub-intervals n and step size the results of DADM can be tabulated in Table 1. Table 1 shows the effect of n and m in the maximum absolute error
Example (2) consider the following nonlinear Fredholm integral equation
whose exact solution is. In this example the ADM can not be applied because the integral
has no analytical solution. The numerical implementation of ADM introduced by [15] can be used, because the kernel is separable. Also, DADM can be used to obtain solution. Table 2 shows the effect of n and m in the maximum absolute error
Example (3) consider the following nonlinear Fredholm integral equation
whose exact solution is. In this example the ADM can not be applied because the integral
has no analytical solution. Also, the numerical implementation of ADM introduced by [15] can not be used, because the kernel function is not separable. Here, DADM is the suitable method to obtain solution. Table 3 shows the effect of n and m in
Table 1. The effect of n and m in the maximum absolute error (example 1).
Table 2. The effect of n and m in the maximum absolute error (example 2).
Table 3. the effect of n and m in the maximum absolute error (example 3).
the maximum absolute error
6. Conclusion
Based on the accelerated Adomian polynomials formula (4) and the well known contraction mapping principles, convergence of DADM is discussed. Convergence approach is reliable enough to obtain an explicit formula for the maximum absolute truncated error of the Adomian’s series solution. The proposed DADM is more general method than that in [15] because it is capable to solve linear and nonlinear Fredholm integral equation with separable as well as non-separable kernel functions. DADM is recommended to solve linear and nonlinear Fredholm integral equation due to many advantages such as the matrix B is unchanged during the computation of the components, the solution need not to solve linear algebraic system of equations like Nystrom method and projection methods. Another advantage when applying DADM to solve linear Fredholm integral equation with symmetric kernel is the matrix B will be symmetric matrix as in example (1).
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is known continuous function on 
and the kernel 
is continuous on the square 
and bounded such that 
where, M is the upper bound on the square E. The nonlinear term 
is Lipschitz continuous with 
L is Lipschitz constant and has Adomian polynomials representation
is
and 
are computed using the following recursive relations
requires the computation of integral in Equation (7). If the evaluation of that integral analytically is possible, ADM can be applied in a simple manner. In case where the evaluation of the integral in (7) is analytically impossible, ADM can not be directly applied. In order to overcome this obstacle, please see the details of Sections 2 and 3. In Section 2, a problem is solved in a special case where the kernel 
is separable [
is not separable and we introduce a discretized modified version of the ADM which is called DADM. In Section 4, convergence of DADM is discussed and the maximum absolute truncated error is estimated. Finally, to verify the theoretical results, some numerical examples are presented in Section 5.
is continuous function on
, 
are the nodes of the quadrature rule, 
and 
are the weight functions. Applying formula (10) on Equation (9) to obtain
in Equation (11) and the first component in Equation (6).
, is not separable, the integral in (1) can not be computed and hence the ADM will not be able to continue in order to obtain solution. Therefore, we suggest DADM to overcome this obstacle. The idea is to discretize the independent variable; t, just before applying the quadrature rule. This gives an opportunity to evaluate the integral in Equation (7) numerically but, of course, at the discretization points of the independent variable. Thus, the discrete version of Equations (6) and (7) may take the form
and 
are the weight functions of any numerical integration scheme. The approximate solution of Equation (1) using DADM can be computed as
are all vectors of dimension 
and B is 
matrix such that
and the computation of the solution need not to solve linear algebraic system of equations like Nystrom method and projection methods. Also, this method can be used for solving Equation (1) with nonseparable kernel. Thus DADM is more general than the numerical implementation of ADM introduced in [
where, 
is the Banach space of all continuous functions on D with the norm 
where, 
. and let x and 
the mapping F is contraction therefore, by the Banach fixed-point theorem for contraction [
and 
and 
be arbitrary partial sums with 
We are going to prove that 
is a Cauchy sequence in Banach space B
then
so, 
then
so, as 
then 
We conclude that 
is a Cauchy sequence in 
so, the series converges and the proof is complete.
where 
then 
and
and define operator 
such that
is a compact operator on 
to 
and is bounded on 
to 
since
be the solution obtained by using ADM, where
and 
Define numerical integral operator 
as
is linear finite rank bounded operator on 
to 
since
, Equation (1) may be written as
here is the solution obtained by using DADMand 
and 
as 
where 
[
and
.
and (25)
. In this example the ADM can not be applied because the evaluation of
is conditioned to compute
.
needs Equation (11) and Simpson’s rule [16-18] with number of subintervals 
and step size 
to obtain
the results of DADM can be tabulated in 
. In this example the ADM can not be applied because the integral
has no analytical solution. The numerical implementation of ADM introduced by [
. In this example the ADM can not be applied because the integral
has no analytical solution. Also, the numerical implementation of ADM introduced by [
is the matrix B will be symmetric matrix as in example (1).