An Accurate and Efficient Technique for Approximating Fuzzy Fredholm Integral Equations of the Second Kind Using Triangular Functions

In this work an accurate and efficient method is suggested to solve the Fredholm fuzzy integral equations of the second kind. The orthogonal triangular function (TF) based method is first applied to transform the fuzzy Fredholm integral equations to a coupled system of matrix algebraic equations. An iterative algorithm of finite nature is then applied to solve the coupled system to obtain the coefficients used to obtain the form of approximate solution of the unknown functions of the integral problems. Finally, an algorithm is presented to solve the fuzzy integral equation by using the trapezoidal rule. This algorithm is implemented on some numerical examples by using software MATLAB. The obtained numerical results are compared with other numerical method and the exact solutions. The main purpose of this paper is to approximate the solution of linear dimensional fuzzy Fredholm integral equations of the second kind (1D-FFIE-2). We use fuzzy triangular functions (1D-TFs) to replace the Fredholm fuzzy integral with a coupled system of matrix algebraic equations. An iterative algorithm of finite nature is then applied to solve the coupled system to obtain the coefficients used to obtain the form of approximate solution of the unknown functions of the integral problems. Moreover, we prove the convergence of the method. Finally we illustrate this method with some numerical examples to demonstrate the validity and applicability of the technique. In this work an accurate and efficient hybrid technique is suggested to solve the Fredholm fuzzy integral equations of the second kind. First, a two msets of orthogonal triangular basis functions (TFs) method is first used to replace the Fredholm fuzzy integral with a coupled system of matrix algebraic equations. An iterative algorithm of finite nature is then applied to solve the coupled system to obtain the coefficients used to obtain the form of approximate solution of the unknown functions of the integral problems. To illustrate the accuracy and the efficiency of the proposed method, set of numerical examples are solved where obtained numerical results are compared with other numerical method and the exact solutions.


Introduction
The importance of fuzzy integral equations appears in studying and solving a large proportion of the problems for different topics in applied mathematics, in particular in relation to biology, physics, medical and geographic. Usually in many applications some of the parameters in our problems are represented by fuzzy number rather than crisp, and hence it is important to develop mathematical models and numerical procedures that would appropriately treat general fuzzy integral equations and solve them. Numerous techniques have been recently implemented for solving integral equations. Many different methods have been used to approximate the solution of integral equation systems [18][19][20]. Many basic and fundamentals functions are recently used to approximate the solution of integral equations like wavelet basis orthogonal bases, see Maleknejad et al. [13], Rationalized Haar functions are developed by Maleknejad and Mirzaee [4] to approximate the solutions of the linear Fredholm integral equations system. A general method by Jahantigh et al. [9] for solving fuzzy Fredholm integral equation of the second kind is introduced. Triangular functions direct method for solving Fredholm integral equations of second kind are proposed in [6], A Direct Method for Numerically Solving Integral Equations System Using Orthogonal Triangular Functions is introduced in [21]. [22], Introduce Application of Triangular Functions to Numerical Solution of Stochastic Volterra Integral Equations. Fredholm integral equations of second kind are solved by using triangular functions method hybrid with iterative algorithm [1]. Also, Fredholm fuzzy integral equations of the second kind is solved via direct method using triangular functions [2] and numerical solution of linear Fredholm fuzzy equation of the second kind by block-pulse functions is considered in [5]. A numerical method for solving the fuzzy Fredholm integral equation of second kind is presented Barkhordary et al. [8] where the trapezoidal rule is used to compute the integrals. Maleknejad et al. [3] proposed a numerical solution of integral equation system of the second kind by block pulse functions, and Babolian et al. [6] proposed a method for solving Fredholm integral equations. Numerical solution of two-dimensional fuzzy Fredholm integral equations of the second kind is presented via direct method using triangular functions [14].H. Nouriani et al. [15] is proposed a quadrature iterative method for numerical solution of two-dimensional fuzzy Fredholm integral equations. R.Ezzati et al. [16] is given numerical solution of two-dimensional fuzzy Fredholm integral equations using bivariate bernstein polynomials. In [17], Modified homotopy perturbation method is solving two-dimensional fuzzy Fredholm integral equation. In this paper we are going to use a kind of these bases that is orthogonal triangular functions. In many applications some of the parameters in our problems are usually represented by fuzzy number rather than crisp state, and thus developing mathematical models and numerical procedures that would appropriately treat general fuzzy integral equations and solve them is important. The paper is organized as follows. In Section 2, some definitions and properties of the orthogonal triangular functions (TFs) are presented. Also, expanding two variable functions by TFs and fuzzy numbers is given. In section 3, to solve coupled system of matrix equations a finite iterative algorithm is presented. In section 4, the suggested method is introduced. In section 5, we solve some numerical examples to illustrate the applicability and the accuracy of the proposed technique.
2 Brief review of triangular functions (TFs)

Triangular Functions (TFs)
Definition 1. Two m-sets of triangular functions (TFs) are defined over the interval [0, T) as: where i = 0, 1, . . . , m − 1 and m has a positive integer value. Also, consider h = T m and T1 i as the ith left-handed triangular function and T2 i as the ith right-handed triangular function. In this paper, it is assumed that T = 1, so TFs are defined over [0 , 1 ) and h = T m . From the definition of TFs, it is clear that triangular functions are disjoint, orthogonal and complete [4]. We can write Now, write the first m terms of the left-hand triangular functions and the first m terms of the right-hand triangular functions as m-vectors: We call T 1(t) and T 2 (t) as left-handed triangular functions (LHTF) vector and right-handed triangular functions (RHTF) vector, respectively. The product of two TFs vectors are presented by: and where 0 is the zero m × m matrix. Also, In which Iis an m × m identity matrix.
The expansion of function f (t) over [0, 1) with respect to TFs, may be compactly written as where we may put c i = f (ih) and d i = f ((i + 1)h) for i=0, 1. . . . . . m-1.

Expanding two variables function by TFs [2]
Each function f (t, s) ∈ L 2 ([0, 1) × [0 , 1)) can be expanded by two TFs vectors with m 1 and m 2 components, respectively. For convenience, take m 1 = m 2 = m . To get desired results, first fix the independent variables. Then, expand f (t, s) by TFs with respect to independent variable t as follows: Now, each of the functions f (ih, s) , s for i = 0, 1, . . . . . . , m − 1 is expanded by TFs with respect to independent variable s. Thus, the expansion of f (t, s) takes the form: In which, Let T (t) be a 2m-vector defined as: The two vector functions T 1(t) and T 2(t) defined in Eqs. (4) and (5). Now, suppose that f (t, s) is a function of two variables. Thus, we can expand it with respect to TFs as follows: where T (s) and T(t) are 2m 1 and 2m 2 dimensional TFs and F a 2m 1 × 2m 2 is TFs coefficient matrix. For convenience, we put m 1 = m 2 = m , so matrix F can be written as where F11, F12, F21 and F22 in above-stated Equation, are previously defined in Eqs. (14): (17).

Fuzzy functions
In this subsection, two definitions that are needed in this work are stated.

Definition 2.
A fuzzy number is a fuzzy set u: R 1 → [0, 1] where the following conditions are to be hold: Definition 3. A fuzzy number u is a pair(u(r), u(r)) of functions u(r) and u(r) , 0 ≤ r ≤ 1, satisfying the following requirement: (a) u(r)is bounded and monotonic increasing as well as left continuous function, (b) u (r) is bounded ,monotonic decreasing and left continuous function, For arbitrary u = (u(r), u(r)) , v = (v(r), v(r)) for k > 0, addition (u + v) and multiplication by k are defined as: 3 Solving coupled system of matrix equations using finite iterative algorithm [1] There are many variant forms of finite iterative algorithms for solving matrix equations, see for example [1,[10][11][12]. We are concerned with iterative solutions to coupled system of similar forms of the Sylvester matrix equations [1].
and second algorithm for solving coupled system of Sylvester matrix equations To solve the matrix equation (21) a finite iterative algorithm is constructed and used as follows, A finite iterative algorithm is constructed to coupled system of Sylvester matrix equations (22) 1-Input A 1 ,B 1 ,A 2 ,B 2 ,C 1 ,C 2 2-Choose arbitrary matrices Y 1 1 ∈ C n×p and Y 2 1 ∈ C r×p 3-Set Proposed hybrid iterative technique for solving linear fuzzy Fredholm integral equation

Converting Fredholm integral equations of second kind to two crisp coupled systems
In this subsection, a TFs method is presented to transform the fuzzy Fredholm integral equation of second kind linear (FFIE-2) to two crisp coupled systems. First consider the following equation: where k(x,t) is an arbitrary kernel function over the square 0 ≤ x, t ≤ 1 and u(x) is a fuzzy real valued function.
The main task is to determine TFs coefficients of u(x) in the interval [0, 1) from the know functions f (x) and kernel k(x,t).
Therefore, we rewrite system (23) in the following form Let us expand u(x, r) , f (x, r) and k(s,t) by TFs (LHTF and RHTF) as follows: with the equation We have Set, which lead to the following two coupled crisp linear systems and Similarly, we expand u(x, r) and f (x, r) by TFs (LHTF and RHTF) and substituting in Eq. (25) two coupled crisp linear systems, similar to (26) and (27)  u(x, r) approx. = T T (x) U T (r) .
The following efficient finite iterative algorithm is proposed which is a generalization of algorithm 2.

Proposed iterative algorithm for solving the two coupled systems
A proposed iterative algorithm is presented in this subsection to solve the two coupled systems (26) and (27) as a modification to algorithm 2.

Algorithm 3.
In this algorithm we modified and generalized algorithm 2 to work out for systems (26) and (27) as follows. First for the coupled system (26) 4-if R K = 0 then stop and U11, U21 is the solution else let K = K + 1 go to step 5.

Numerical results and discussions
To demonstrate the accuracy and effectiveness of our proposed hybrid method, T Fsand an iterative algorithm, some examples are considered. The solution of each example is obtained for different values of r , x and m and is compared with the exact solution and the direct method presented by Mirzaee et al. [ 2 ] and with Ghanbari et al. [5].
Example 51 Consider the following FFIE-2 with and k (x,t) = x 2 (1 + 2t) , 0 < x, t < 1 and λ = 1. The exact solution in this case is given by From the obtained numerical results of the first test example, we can see that our proposed hybrid iterative method gives the same accuracy compared with the direct method. Also, It is worth noting that the number of iterations to execute the algorithm taking tolerance criteria is residual > e −4 was k = 5 which means that the technique is quite efficient. The accuracy can be further improved by increasing the stopping tolerance.
Example 52 Consider the following FFIE-2 with and k (x,t) = (2t − 1) 2 (1 − 2x) , 0 < x, t < 1 and λ = 1 . The exact solution in this case is given by The problem in example 2 is solved by proposed method and the results are given in Table 5 and the numerical results are which compared with the obtained results using the direct method [2]. The last column in this table shows the absolute for the system (4.4) and system (4.5) respectively in our suggested iterative algorithm to obtain the coefficient matrices in (4.4) and (4.5). The number of iterations is k = 5. Moreover, the presented method is compared with the block-pulse function method proposed by Ghanbari et al. [5]. As we can see from the numerical results in Tables 6 and 7, the proposed method is of high accuracy as it is also highly efficient.
Example 53 Consider the following FFIE-2 with The exact solution in this case is given by The results are shown in Tables 8 and 9. TheThe problem in example 3 is also solved by the proposed method and the results are given in tables 8, 9 which are compared with the one obtained using the direct method [2]. From the forth and last column, we can see that both have almost the same accuracy.

Conclusion
In this paper, an approximate numerical solution for linear FFIE-2 is considered. The original Fredholm integral equations of second kind are first transformed to two crisp coupled systems. Then, we use the two msets of TFs to approximate of the unique solution of FFIE-2. Here, a hybrid method of a triangular functions and an iterative algorithm are considered. By examining this hybrid method, the numerical results obtained for three examples show that: the proposed method produces results almost similar to that obtained using the direct method and other numerical method based on block-pulse function method with acceptable percentage error and the absolute error is reduced with the reduction of the solution tolerance. The advances of this method are the number of calculations is very low as well as a good accuracy is mentioned.