A Highly Efficient and Accurate Finite Iterative Method for Solving Linear Two-Dimensional Fredholm Fuzzy Integral Equations of the Second Kind Using Triangular Functions

%is 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 twom-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. %ree examples are given to clarify the efficiency and accuracy of the method. %e obtained numerical results are compared with other direct and exact solutions.


Introduction
Several methods have been developed to estimate the solution of integral equation systems [1][2][3]. Many simple functions are used to approximate the solution of integral equations, such as orthogonal bases dependent on wavelets [4]. In addition, Maleknejad and Mirzaee developed the rationalized Haar functions [5] 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 [6] and using iterative algorithm-hybrid triangular functions method presented by Ramadan and Ali [7] where this hybrid method treats Fredholm integral equation of one dimension. More recently, Ramadan et al. [8] implemented such hybrid method to tackle system of two linear Fredholm integral equations of one dimension.
Furthermore, Maleknejad et al. [9] suggested by block pulse functions a numerical solution of the integral secondtype 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. is 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 [10]. 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 [11], and numerical solution of linear Fredholm fuzzy equation of the second kind by block pulse functions is considered in [12]. 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 [15]. Nouriani et al. [16] proposed a quadrature iterative method for solving the two-dimensional fuzzy Fredholm integral equations. Ezzati and Ziari [17], Hengamian Asl and Saberi-Nadjafi [18], and Bica and Popescu [19] 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 [20]. A two-dimensional nonlinear Volterra-Fredholm fuzzy integral equation is solved by using the Adomian decomposition method [21] and fuzzy bivariate triangular functions [22]. e aim of paper is to generalize the work proposed in [7] and [8] of these basis orthogonal triangular functions on (0, 1) to solve two-dimensional fuzzy Fredholm integral equations.

y, s, t) u(s, t)ds dt. (1)
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. e 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. e illustrative examples and numerical results obtained are presented and discussed in Section 8.

Triangular Functions (TFs) of One Dimension
Definition 1. Two m-sets of triangular functions (TFs) are defined over the interval [0, T) [5]: where i � 0, 1, . . . , m − 1; m has a positive integer value; h � (T/m); T1 i is the ith left-handed triangular function; and T2 i is the ith right-handed triangular function.
Assuming T � 1, the TFs are defined over [0, 1) and h � (1/m). Based on this definition, it is clear that TFs are disjoint, orthogonal, and complete [5]. erefore, one may write e first m terms in the left-hand triangular functions and in the right-hand triangular functions can be written concisely in m-vectors format as where T1(t) and T2(t) are called left-handed triangular function (LHTF) vector and right-handed triangular function (RHTF) vector, respectively. e product of two TF vectors yields the following properties: where 0 is the zero m × m matrix. Also, in which I is an m × m identity matrix.

Two-Dimensional Triangular Functions and eir
Properties [15]. An (m 1 × m 2 )-set of 2D-TFs on the region where: ; and m 1 and m 2 are arbitrary positive integers. erefore, Furthermore, where φ i,j (s, t) is the im 2 + j + 1 th block pulse function defined on ih 1 ≤ s ≤ (i + 1)h 1 and jh 2 ≤ t ≤ (j + 1)h 2 as Each of the sets For Also, the 2D-TFs are orthogonal, that is, where δ denotes the Kronecker delta function and On the other hand, if then T(s, t), the 2D-TF vector, can be defined as ese relations are also satisfied for T12(s, t), T21(s, t), T22(s, t), similarly. Hence, Finally by the orthogonality of T11, we have where ⊗ denotes the Kronecker product defined for two arbitrary matrices P and Q as e same equations are implied for T12(s, t), T21(s, t), and T22(s, t), by similar computations. Hence, we can carry out double integration of T(s, t): where D is 4m 1 m 2 × 4m 1 m 2 matrix as follows: where I 1 � I m 1 ×m 1 and I 2 � I m 2 ×m 2 .

Function Expansion with 1D-TFs and 2D-TFs.
e expansion of functions using triangular functions occurs in four situations.
(1) e expansion of function f(t) over [0, 1) with respect to 1D-TFs is compactly written as where we may put

Mathematical Problems in Engineering
(2) e expansion of the function f(s, t) defined over Ω ([0, 1) × [0, 1)) by 2D-TFs is as follows: where F is a 4m 1 m 2 vector given by and T(s, t) is defined in equation (21). e 2D-TF coefficients in C, D, E, and L can be computed by sampling the function f(s, t) at grid points s i and t j such that s i � ih 1 and t j � jh 2 , for various i and j. So, we have where k � im 2 + j and i � 0, 1, 2, . . . , m 1 − 1, j � 0, 1, 2, . . . , m 2 − 1. e 4m 1 m 2 vector F is called the 2D-TF coefficient vector. (3) e expansion of the function f(s, t, r) of three variables on (Ω × [0, 1]) with respect to 2D-TFs and 1D-TFs is as follows: where T(s, t) and T(r) are 2D-TF vector and 1D-TF vector of dimension 4m 1 m 2 and 2m 3 , respectively, and F is a (4m where each block of F is an (m 1 m 2 × m 3 )-matrix that can be computed by sampling the function f(s, t, r) at grid points (s i , t j , r k ) such that (28) (4) e expansion of the function k(s, t, x, y) of four variables on (Ω × Ω) with respect to 2D-TFs is as follows: where T(S, T) and T(x, y) are 2D-TF vectors of dimension 4m 1 m 2 and 4m 3 m 4 , respectively, and K is a (4m 1 m 2 × 4m 3 m 4 ) 2D-TF coefficient matrix. is matrix can be represented as Mathematical Problems in Engineering where each block of K is an (m 1 m 2 × m 3 m 4 ) matrix that can be computed by sampling the function k(s, t, x, y) at grid points (s i 1 , t j 1 , x i 2 , y j 2 ) such that In this paper, we suppose that m 1 � m 2 � m 3 � m 4 � M for convergence.

Fuzzy Functions
We now remember through the paper some definitions that are required.

Definition 2.
A fuzzy number is a fuzzy set u: R 1 ⟶ [0, 1] that conforms to the following condition [23]: 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 [5]: , v(r)), and k > 0, we define addition (u + v) and multiplication by k as

Solving Coupled System of Matrix Equations Using Finite Iterative Algorithm [5]
Matrix equations can be solved using various forms of the finite iterative algorithms example [1][2][3]5]. We consider iterative solutions to coupled system similar to the forms of Sylvester matrix equations [5].
and second algorithm to solve coupled system of Sylvester matrix equations: Algorithm 1 (see [5]). A finite iterative algorithm is developed to solve equation (35) as follows: (1) Input A, B, C.
(2) Pick arbitrary matrices V ∈ R n×p and W 1 ∈ R r×p .
(3) Set 6 Mathematical Problems in Engineering (4) If R K � 0, then stop and V K and W K are the final solutions; else, let K � K + 1 and go to step 5.
Algorithm 2 (see [5]). e following finite iterative algorithm is proposed to solve coupled system of Sylvester matrix equation (36): (1) Input matrices: (4) If R K � 0, then stop and Y 1 K and Y 2 K are the solutions; else set K � K + 1 and then go to step 5.

Converting Linear Two-Dimensional FIEs of Second Kind to Two Crisp Coupled Systems.
is 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 u(x, y) from the known functions f(x, y) and kernel k(x, y, s, t). 2D-TFs are applied for equations To describe the approach of equation (47), first expand u(x, y, r),f(x, y, r), and k(x, y, s, t) by 2D-TFs as follows: u(x, y, r) ≃ T11 T (x, y)U11T1(r) + T12 T (x, y)U21T1(r) + T21 T (x, y)U31T1(r) where T(x, y) and T(r) are defined in equations (3) and (21), respectively, U and F are (4M 2 × 2M) matrix of 2D-TF coefficients of u(x, y, r) and f(x, y, r), respectively, and K is (4M 2 × 4M 2 )-matrix 2D-TF coefficients of (x, y, s, t).

Mathematical Problems in Engineering
To obtain the solution of equation (47)

T T (x, y)KT(s, t)T T (x, y)UT(r)ds dt,
Using equation (22), we have and then where U and F are 4M 2 × 2M-matrix and KD is (58)

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Set which lead to the following two crisp linear systems: and Similarly, we expand u(x, y, r) and f(x, y, r) by 2D-TFs, and by substituting them into equation (42) us, we need to obtain the coefficient matrices U11, U12, U21, U22, U31, U32, U41, and U42 in order to get the approximate numerical solution of the form: (68) (61) and (65). An iterative algorithm is proposed here to solve the two coupled systems (61) and (65) as a generalization of Algorithm 2.

Convergence Analysis of the Proposed Method
In this section, we obtain error estimate for the numerical method proposed in previous section.
Also, we have lim M⟶∞ u approx (x, y) � u exact (x, y), so ‖u exact (x, y) − u approx (x, y)‖ ⟶ 0 as M ⟶ ∞ and since S is bounded. us, so the proof of the theorem is completed.
Remark 1. In our theoretical investigation for the proposed method, we take m1 � m2 � M.

Numerical Results and Discussion
is section demonstrates the effectiveness and the accuracy of our proposed hybrid method, 2D-TFs and an iterative algorithm, on some examples. e solution of each example is obtained for different values of x, y, r, and M and is compared with the exact solution, the direct method, and the presented method when the tolerance criteria residual is >e − 4 and >e − 8 .
(78) e number of iterations for solving the two coupled matrix equations to obtain the coefficient matrices taken by our proposed iterative algorithm is k � 3 when the tolerance criteria residual is >e − 4 which indicates that the hybrid proposed method is quite efficient and has good accuracy as seen from Tables 1 and 2.

Remark 2.
e numerical results for the approximate solution using the direct method in Tables 1 and 2 are taken  from Table 3 in [15], while the numerical results using the direct and the proposed iterative methods in Tables 3 and 4 are obtained using our own program written using MAT-LAB R2018b. Also, the number of iterations for solving the two coupled matrix equations to obtain the coefficient matrices taken by our proposed iterative algorithm is k � 3 when the tolerance criteria residual is >e − 4 which indicates that the hybrid proposed method is quite efficient and has good accuracy as seen from Tables 3 and 4.

(79)
In this case, the exact solution is given by Remark 3. e number of iterations for solving the two coupled matrix equations to obtain the coefficient matrices taken by our proposed iterative algorithm is k � 3 when the tolerance criteria residual is >e − 4 which indicates that the hybrid proposed method is quite efficient and has good accuracy as seen from Tables 5 and 6.

(81)
In this case, the exact solution is given by u(x, r) � (2 − r)x 2 y 2 .
(82)       Table 7, the number of iterations for this example by our proposed iterative algorithm is k � 3 when the tolerance criteria residual is >e − 4 which indicates that the hybrid proposed method is quite efficient. Moreover, we can see that our method has good accuracy which can be further improved by increasing the residual.

Conclusion
Fuzzy control applications and a large proportion of applied mathematical topics require the solution of the fuzzy integral equations. e paper introduced the 2D-TFs method for approximating the solution of linear 2D-FFIE-2, which is a hybrid of triangular functions and an iterative algorithm. e method is simple, efficient, and accurate and is based on converting the original equation into two crisp systems (2D-FFIE-2). e efficiency and simplicity of the proposed method are demonstrated via numerical examples with known exact solutions, and the results are given. Furthermore, the exceptional value of the proposed method is low cost of the equation setting with no need for any projection method or integration.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.

Authors' Contributions
e authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the manuscript.