A Numerical Study for Solving the Systems of Fuzzy Fredholm Integral Equations of the Second Kind Using the Adomian Decomposition Method

In this paper, the Adomian decomposition method (ADM) is successfully applied to find the approximate solutions for the system of fuzzy Fredholm integral equations (SFFIEs) and we also study the convergence of the technique. A consistent way to reduce the size of the computation is given to reach the exact solution. One of the best methods adopted to determine the behavior of the approximate solutions. Finally, the problems that have been addressed confirm the validity of the method applied in this research using a comparison by combining numerical methods such as the Trapezoidal rule and Simpson rule with ADM

Most scientific problems and phenomena are in nonlinear forms, such as the flow of fluids, it is not easy to find a linear formula to solve, so finding the approximate or analytical solution is very complex, so we resort to developing non-linear formulas and finding the analytical or approximate solution to this type of problems, among the appropriate and most effective methods to finding approximate solutions are smooth in dealing with linear and non-linear problems is the Adomian decomposition method (ADM) [3].
In the literature, solving linear Fredholm fuzzy integral equations of the second kind is based on two m-sets of triangular functions [1]. Solve the non-linear ordinary differential equations by variational iteration method (VIM), Homotopy perturbation method (HPM), and ADM are introduced and applied to solve the steady three-dimensional flow of Walter's B fluid in a vertical channel. Authors discussed the convergence of the Adomian method when applied to a class of non-linear Volterra integral equations [4]. The Homotopy analysis method (HAM) for solving linear and non-linear integral equations of the second kind is applied in [5]. The collocation method solved systems of non-linear Fredholm integral equations in terms of continuous Legendre multi-wavelets on the interval [0, 1) [6].
Two main goals are worked in this research, the first goal is to study the convergence of the fuzzy ADM and to treat the sufficient condition for convergence. For the second goal, we use the standard ADM to solve non-linear system of fuzzy Fredholm integral equations (SFFIEs). Also, we display a comparison of the numerical results applying the ADM with the numerical solution for the iterations of the given SFFIEs with the Trapezoidal rule (ADM-TRAP) and the Simpson rule (ADM-SIMP) obtained with the minimum amount of computation are compared with the exact solutions to show the efficiency of the ADM. The two goals are successfully achieved.

Basic Concepts
Fuzzy numbers are classic generalized real numbers. We can define them as an ambiguous subset of the real line, in the sense that it refers not to a single value but to a continuous set of possible values, where each possible value weights 0 and 1. This weight is called the membership function. Thus, the fuzzy number is a special case of the convex set of the real line. The concept of the fuzzy number is essential for fuzzy analysis and fuzzy integral equations, as well as it is a useful tool in a variety of applications of the fuzzy set. The basic definitions of fuzzy numbers are given as follows: Definition 1 [7,8]: In a fuzzy set, an element can belong to a certain extent of the fuzzy set = {( , ( ) ), ∈ } where ( ) is the membership function of fuzzy set is defined by 1], and the value of ( ) is called the membership degree . Definition 2 [9,10]  In Banach space, we represent a crisp number x by ( ( ), ( )) = ( , ), 0 ≤ ≤ 1. By appropriate definitions, the fuzzy number space { ( ) ≤ ( )} becomes convex if it is isometric and isomorphic. Let ̃= ( ( ), ( )), ̃= ( ( ), ( )) , 0 ≤ ≤ 1, and ∈ ℝ. Then 1) ̃= ̃ ( ) = ( ), ( ) = ( ).  Definition 5 [19]: Let = ( ( ), ( )) , = ( ( ), ( )) , 0 ≤ ≤ 1 be two any arbitrary fuzzy numbers, and is scalar, we define the operation of a fuzzy number by the following

Systems of Fuzzy Fredholm Integral Equations
We consider the following system of fuzzy Fredholm integral equations (SFFIEs) of the second kind and system [12,21,23]: (2)

Applied ADM to SFFIEs
The ADM includes decomposing the unknown function ̃( , )for any equation to the sum of the infinite numbers of the basic elements known as the series of analysis, it usually converges to the closed-form solution. The unknown functions ̃( , ) = [ ( , ), ( , )] given by [17,24] as follows: the above inequality is a sufficient condition to get the solution of linear SFFIEs (1). Note: This theorem is true where the unknown functions inside the integral sign ( ( )) is non-linear functional because we take the condition 1 = {̃1 ,0 }, 2 = {̃2 ,0 }. Which completes the proof. Theorem 2. The speed of the convergence of the solution SFFIEs (1) on [a,b] has the following cases: , then the SFFIEs (1) is linear convergence.

Applications and Numerical Results
In this section, we apply fuzzy ADM to obtain an approximate solution for linear and nonlinear SFFIEs are displayed in the following two problems. To show the high accuracy of the solution results compared with the exact solution, the maximum errors are defined as: The fuzzy exact solutions of the SFFIEs (11) are The parametric form of the given SFFIEs (10) can be written as Operating by the same way proceeding Equations (2)-(6) as above on the SFFIEs (12), and applying the fuzzy ADM, the lower iterations (L) are then determined in the following recursive way: , by Theorem (2) the approximate solution fuzzy ADM (̃, 5 ( , )) of the SFFIEs (11) is Q-super linearly convergence with the fuzzy exact solutions (̃( , )) for all 0 < ≤ 1 as in Tables 1-9.         In the following Figures 1-4, we present the contour plot in 2 on the ( , ) − plane for the exact solutions (̃( , )) and the ADM (̃, 5 ( , )). We represent the exact solutions with a continuous line and the ADM with the symbol ∘.   ) of the SFFIEs (11) is linearly convergent with the fuzzy exact solutions (̃( , )) for all 0 < ≤ 1 as in Tables 10-18.
The fuzzy exact solutions of the SFFIEs (14) are In the same way, we proceed equations (2)-(6) as above, and applying the fuzzy ADM, the lower iterations (L) are then determined in the following recursive way: In this problem, if we take 1 = 2 = 1 100 , then by Theorem (2), the approximate solution fuzzy ADM (̃, 5 ( , )) of the system (14) is Q-superlinear convergence with the exact solutions (̃( , )) for all 0 < ≤ 1 as in Tables 19-26. Tables 19-22          In the following Figures 5-8, we present plot of the exact solutions (̃( , )) and the ADM (̃, 5 ( , )) when = 0.3. We represent the exact solutions with a continuous lines and the ADM with the symbol ∘.

Conclusion and discussion
Two main goals were worked in this research, firstly, the convergence of the fuzzy ADM and to treat the sufficient condition for convergence are studied. Second, we use the fuzzy ADM to obtain the approximate solutions for the non-linear SFFIEs. Also, a comparison of the numerical results applies the ADM with the numerical solution for the iterations of the given SFFIEs with the Trapezoidal rule (ADM-TRAP) and the Simpson rule (ADM-SIMP) obtained with the minimum amount of computation are compared with the exact solutions to show the efficiency of the ADM. From the tables of the numerical results, the ADM-SIMP is converge to the exact solutions and the ADM is better than the ADM-TRAP.