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 [
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 [
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 [
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 [
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).
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 [
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
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 [
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.
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.
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 [
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.
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 [
The solution by numerical implementation of ADM introduced in [
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
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 [
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 [
the maximum absolute error
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 [