Development of the Adomian's Method for Solving non-linear Fredholm-Fredholm Integral Equations

Abstract: A new and effective direct method to determine the numerical solution of nonlinear Fredholm-Fredholm integral equations is proposed. The method is based on a series solution for the unknown quantity by Adomian's method. We obtained very good results compared with classic Adomian's method Some numerical examples are provided to illustrate accuracy and computational efficiency of the method.


Introduction
Over several decades, numerical method in electromagnetic have been the subject of extensive researches.many problems in electromagnetic can be modeled by integral and integro-differential equation for example, electric field integral equation (EFIE) and magnetic field integral equation (MFIE) [3].Some papers studied the problem of existence and uniqueness of solution of non-linear Volterra-Fredholm integral equations see [1,6,7].Some authors use Adomian's method for solving this equations [4] .
In this article we present Adomian's method solution solving nonlinear integral equation of Fredholm-Fredholm type:-

Analysis:
In this section, we first describe the algorithm of Adomian's method as it applies to a general non-linear equation of the form [2]: where N is a non-linear operator on a Hilbert space H and f is a known element of H. we assume that for a given f .It is well known that (AM) considers ) (x where n A are the so-called Adomian polynomial's.Substituting (2.3) and (2.5) into the functional equation (2.2) yields: If the series in (2.6) is convergent, then (2.6) holds upon setting: Thus, one can recursively determine every term of the series   o n n  .The convergence of this series has been established.The two hypotheses necessary for proving convergence of the Adomian's method as given in [5] These hypotheses, for proving convergence, are generally satisfied in physical problems [5].
The modified Adomian method [4] may be roughly described as a reassignment of the initial approximates , then we may rewrite (2.7) as: The choice of how to assign 1   and o is experimental, yet it leads to less computational and does accelerate the convergence.
To compute Adomian polynomials we as a new method mentioned in [4].Consider the equation (1.1), to solve (1.1) by AM, we write and substituted the series (2.3) , (2.9) and (2.10 ) in to (1.1) giving If the series is convergent, then we can determined each term of the series The algorithm in (2.13) Go on this course, we will get n A

Numerical Examples:
Here we list the results of approximating some problems solved by Adomian's method and modified Adomian's method.
Example 1: we apply the standard Adomian's method and solved by two methods Adomian's and modification method, and the results will be compared.

Figure (1.1) comparison of convergence rate for classic Adomian's method and exact solution
We can see also from Figure (1.2) that this modified is very good.x .This method was applied at the 6 th iteration.
A comparison of the approximate solution between classic and the modified Adomian's method with the exact solution

Example 2:
We apply the standard Adomian's method      A comparison of the approximate solution between classic and the modified Adomian's method with the exact solution

Figure( 1 . 5 ).
Figure(1.5)comparison of convergence rate for modify Adomian's method and exact solution are as follows.
determines the nA as follows: