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A Highly Efficient and Accurate Finite Iterative Method for Solving Linear Two-Dimensional Fredholm Fuzzy Integral Equations of the Second Kind Using Triangular Functions
This work introduces a computational method for solving the linear two-dimensional fuzzy Fredholm integral equation of the second form (2D-FFIE-2) based on triangular basis functions. We have used the parametric form of fuzzy functions and transformed a 2D-FFIE-2 with three variables in crisp case to a linear Fredholm integral equation of the second kind. First, a method based on the use of two m-sets of orthogonal functions of triangular form is implemented on the integral equation under study to be changed to coupled algebraic equation system. In order to solve these two schemes, a finite iterative algorithm is then applied to evaluate the coefficients that provided the approximate solution of the integral problems. Three examples are given to clarify the efficiency and accuracy of the method. The obtained numerical results are compared with other direct and exact solutions.
Several methods have been developed to estimate the solution of integral equation systems [1–3]. Many simple functions are used to approximate the solution of integral equations, such as orthogonal bases dependent on wavelets . In addition, Maleknejad and Mirzaee developed the rationalized Haar functions  to approximate the solutions of the Fredholm linear integral equation method. In addition, second-type Fredholm integral equations are solved using direct triangular functions method as seen in  and using iterative algorithm-hybrid triangular functions method presented by Ramadan and Ali  where this hybrid method treats Fredholm integral equation of one dimension. More recently, Ramadan et al.  implemented such hybrid method to tackle system of two linear Fredholm integral equations of one dimension.
Furthermore, Maleknejad et al.  suggested by block pulse functions a numerical solution of the integral second-type equation.
It is explained using a series of orthogonal triangular functions, derived from the series of block pulses. Nevertheless, the fuzzy integral equations (FIEs) are required to solve and research a wide number of problems in various applied mathematics subjects, such as connection to physics, spatial, medical, and biology. FIEs therefore require approximate numerical solutions, as they are typically difficult to analytically solve. This thesis introduces a methodology used by the triangular functions (TFs) to solve the fuzzy linear FIE method of the second kind. In various implementation problems, certain parameters are typically represented by a fuzzy number rather than a crisp state, which involves the creation of mathematical models and computational algorithms to handle and solve the general fuzzy integral equations. A general method for solving the fuzzy Fredholm second-type integral equation is proposed in . Recently, numerical methods have been developed to solve linear fuzzy Fredholm integral equation of the second kind in one-dimensional space (1D-FFIE-2) and two-dimensional space (2D-FFIE-2). Also, Fredholm fuzzy integral equations of the second kind are solved using the triangular functions , and numerical solution of linear Fredholm fuzzy equation of the second kind by block pulse functions is considered in . Barkhordary et al. and Ramadan et al. [13, 14] presented a numerical technique for solving the fuzzy Fredholm integral equation of second kind. Numerical solution of two-dimensional fuzzy Fredholm integral equations of the second kind is presented via direct method using triangular functions . Nouriani et al.  proposed a quadrature iterative method for solving the two-dimensional fuzzy Fredholm integral equations. Ezzati and Ziari , Hengamian Asl and Saberi-Nadjafi , and Bica and Popescu  illustrated a solution of the two-dimensional fuzzy Fredholm integral equations. A modified homotopy perturbation method for solving the two-dimensional fuzzy Fredholm integral equation is detailed in . A two-dimensional nonlinear Volterra–Fredholm fuzzy integral equation is solved by using the Adomian decomposition method  and fuzzy bivariate triangular functions .
Section 2 presents some definitions and properties of the orthogonal triangular functions (TFs) (1D-TFs and 2D-TFs). Also, it expands functions by TFs. In Section 3, the definitions and properties of fuzzy function are given while a finite iterative algorithm is presented to solve coupled system of matrix equations in Section 4. The two-dimensional fuzzy integral equation is demonstrated and explained in Section 5 while the suggested method and the proposed iterative algorithm are detailed in Section 6. The illustrative examples and numerical results obtained are presented and discussed in Section 8.
2. Review of Triangular Functions (TFs)
2.1. Triangular Functions (TFs) of One Dimension
Definition 1. Two m-sets of triangular functions (TFs) are defined over the interval [0, T) :where has a positive integer value; ; is the ith left-handed triangular function; and is the ith right-handed triangular function. Assuming , the TFs are defined over and . Based on this definition, it is clear that TFs are disjoint, orthogonal, and complete . Therefore, one may writeThe first terms in the left-hand triangular functions and in the right-hand triangular functions can be written concisely in -vectors format aswhere and are called left-handed triangular function (LHTF) vector and right-handed triangular function (RHTF) vector, respectively. The product of two TF vectors yields the following properties:where 0 is the zero matrix. Also,in which is an identity matrix.
2.2. Two-Dimensional Triangular Functions and Their Properties 
An -set of 2D-TFs on the region is defined bywhere: ; ; ; and and are arbitrary positive integers. Therefore,
Furthermore,where is the th block pulse function defined on and as
Each of the sets and is obviously disjoint:
For , and .
Also, the 2D-TFs are orthogonal, that is,where denotes the Kronecker delta function and
On the other hand, ifthen , the 2D-TF vector, can be defined as
These relations are also satisfied for , similarly. Hence,
Finally by the orthogonality of , we havewhere denotes the Kronecker product defined for two arbitrary matrices P and Q as
The same equations are implied for , , and , by similar computations. Hence, we can carry out double integration of :where is matrix as follows:where and .
2.3. Function Expansion with 1D-TFs and 2D-TFs
The expansion of functions using triangular functions occurs in four situations.(1)The expansion of function over with respect to 1D-TFs is compactly written as where we may put and for i = 0, 1, …, m − 1.(2)The expansion of the function defined over Ω by 2D-TFs is as follows: where F is a vector given by and is defined in equation (21). The 2D-TF coefficients in can be computed by sampling the function at grid points and such that and , for various and . So, we have where and , . The vector F is called the 2D-TF coefficient vector.(3)The expansion of the function of three variables on with respect to 2D-TFs and 1D-TFs is as follows: where and are 2D-TF vector and 1D-TF vector of dimension and , respectively, and is a 2D-TF coefficient matrix. This matrix can be represented as where each block of F is an -matrix that can be computed by sampling the function at grid points such that Let ; then,(4)The expansion of the function of four variables on with respect to 2D-TFs is as follows: where and are 2D-TF vectors of dimension and respectively, and is a 2D-TF coefficient matrix. This matrix can be represented as where each block of is an matrix that can be computed by sampling the function at grid points such that Let and ; then,
In this paper, we suppose that for convergence.
3. Fuzzy Functions
We now remember through the paper some definitions that are required.
Definition 2. A fuzzy number is a fuzzy set u: R1 ⟶ [0, 1] that conforms to the following condition :(a) is upper semicontinuous(b) outside some interval [c, d](c)There are real numbers a and b, c ≤ a ≤ b ≤ d, for which(i) is increasing in monotonic manner on [c, a](ii) is decreasing in monotonic manner on [b, d](iii) for a ≤ x ≤ b
Definition 3. A fuzzy number u is a pair of functions and , 0 ≤ r ≤ 1, satisfying the following requirement :(a) is bounded monotonic increasing left continuous function(b) is bounded monotonic decreasing left continuous function(c), 0 ≤ r ≤ 1For arbitrary , , and k > 0, we define addition and multiplication by k as
4. Solving Coupled System of Matrix Equations Using Finite Iterative Algorithm 
Matrix equations can be solved using various forms of the finite iterative algorithms example [1–3, 5]. We consider iterative solutions to coupled system similar to the forms of Sylvester matrix equations .and second algorithm to solve coupled system of Sylvester matrix equations:
Algorithm 1 (see ). A finite iterative algorithm is developed to solve equation (35) as follows:(1)Input A, B, C.(2)Pick arbitrary matrices and .(3)Set(4)If , then stop and and are the final solutions; else, let and go to step 5.(5)Calculate
Algorithm 2 (see ). The following finite iterative algorithm is proposed to solve coupled system of Sylvester matrix equation (36):(1)Input matrices: A1; B1; A2; B2; C1; C2.(2)Pick arbitrary matrices and .(3)Set(4)If , then stop and and are the solutions; else set and then go to step 5.(5)Calculate
5. Two-Dimensional Fuzzy Fredholm Integral Equation
Two-dimensional FIE of the second kind is defined as follows :
The linear (2D-FFIE-2) is defined aswhere and are fuzzy real functions on , is an arbitrary kernel function over , and is unknown on .
Throughout this paper, we consider 2D-FFIE-2 with , , and .
Now, introduce parametric form of a 2D-FFIE-2 with respect to Definition 3. Let and , be parametric form of and , respectively. Then parametric form of 2D-FFIE-2 is as follows:for each and . We can see that equations (43) and (44) are system of Fredholm integral equation of the second kind with three variables in crisp case.
6. Proposed Hybrid Iterative Technique
6.1. Converting Linear Two-Dimensional FIEs of Second Kind to Two Crisp Coupled Systems
This section presents an efficient method for soling a 2D-FFIE-2 by using 2D-TFs.
First, consider the following equation:
Now, the problem is to find the TF coefficients of from the known functions and kernel . 2D-TFs are applied for equations
To describe the approach of equation (47), first expand , and by 2D-TFs as follows:where and are defined in equations (3) and (21), respectively, and F are matrix of 2D-TF coefficients of and , respectively, and is -matrix 2D-TF coefficients of .
Using equation (22), we haveand thenwhere and are -matrix and is - matrix, so is -matrix, where is unknown.
Then, we havewith