Solving two-dimensional fuzzy Fredholm integral equations via sinc collocation method

*Correspondence: ma_yan_ying@126.com 1School of Mathematics and Physics, Anhui Jianzhu University, Hefei, P.R. China Full list of author information is available at the end of the article Abstract In this paper, we present a numerical method to solve two-dimensional fuzzy Fredholm integral equations (2D-FFIE) by combing the sinc method with double exponential (DE) transformation. Using this method, the fuzzy Fredholm integral equations are converted into dual fuzzy linear systems. Convergence analysis is performed in terms of the modulus of continuity. Numerical experiments demonstrate the efficiency of the proposed method.

In [22], the sinc method was proposed to solve one-dimensional fuzzy integral equations. However, the convergence analysis was not given. The primary aim of this paper is to extend the application of the sinc method together with DE transformation to find the approximate solution of 2D-FFIE. In addition, we prove the convergence of the proposed method. The proposed method has its advantages such as simple structure, easy to pro-gramming, and high accuracy. In addition, the proposed method does not need iterative operation, and the calculation cost is reduced The outline of this paper is as follows. In Sect. 2, we introduce some preliminaries and basic definitions. In Sect. 3, we present the proposed algorithm. Section 4 is focused on discussing the convergence of the proposed method. To demonstrate the effectiveness of the proposed method, we will show some numerical results on several tests in Sect. 5. Finally, some conclusion remarks are given in Sect. 6.

The basic concepts of fuzzy equations
First, we review several necessary basic definitions and relevant results about fuzzy numbers and fuzzy-number-valued functions. Definition 2.1 (see [31]) A fuzzy number is a function u : R → [0, 1] satisfying the following properties: (1) u is upper semicontinuous over R, The set of all fuzzy numbers is denoted by R . Note that every α ∈ R can be seen as a fuzzy number α = χ {α} , and so R ⊂ R .
Definition 2.2 (see [33]) For all u = (u(r), u(r)) and v = (v(r), v(r)), the quantity It is proved that (R , D) is a complete metric space, which possesses the following properties: Definition 2.3 (see [34]) Suppose f , g : [a, b] → R are fuzzy real-valued functions. The uniform distance between f , g is defined by Definition 2.4 (see [35]) If for every ε > 0, there exists δ > 0 such that  [29] Definition 2.6 (see [29]) Suppose that f : [a, b] × [c, d] → R , and n x : a = x 0 < x 1 < · · · < x n = b and n y : c = y 0 < y 1 < · · · < y n = d are two segmentations of the intervals [a, b] and [c, d], respectively. Consider any intermediate points and P y = ([y j-1 , y j ]; η j ) (i, j = 1, . . . , n) are abbreviated to P x = ( n , ξ ) and P y = ( n , η), which are called as δ-fine and σ -fine, respectively, if The function f is said to be two-dimensional Henstock integrable to I ∈ R if for each

Lemma 2.1 If f and g are Henstock-integrable mappings on
( 1 ) Definition 2.7 (see [36]) The fuzzy linear system where A = (a ij ) m×m and B = (b ij ) m×m are crisp coefficient matrices, and Y = (y 1 , . . . , y m ) T is a fuzzy vector, is called a dual fuzzy linear system, and X = (x 1 , . . . , x m ) T is called a solution of the fuzzy linear system (2).
Definition 2.8 (see [36]) A fuzzy number vector (x 1 , x 2 , . . . , x m ) T given by Next, we introduce the method of [36] for solving Eq. (2). To solve Eq. (2), we write it in the form where the elements of S = (s ij ) 2m×2m and T = (t ij ) 2m×2m are determined as follows: assuming that the remaining s ij and t ij are zeros. Moreover, the right-hand side vector and the unknowns are Y = (y 1 , y 2 , . . . , y m , -y 1 , -y 2 , . . . , -y m ) T , respectively. The structures of S and T suggest that s ij ≥ 0, t ij ≥ 0 (i, j = 1, . . . , 2m) and where B 1 and B 2 include the positive entries of A and B, respectively, and C 1 and C 2 include the absolute values of negative entries of A and B, respectively. It is clear that

Sinc method
Definition 2.9 (see [37]) Let f be a function on R with step size h > 0. Its Whittaker cardinal is defined by the series whenever this series converges; is known as the jth sinc function. Definition 2.10 (see [37]) A function g(t) is said to decay double exponentially if there exist constants α and C such that Equivalently, a function g(t) is said to decay double exponentially with respect to conformal map φ if there exist constants α and C such that We describe the following sinc quadrature rule by means of DE transformation, which has been fully discussed in [38]: where , t ∈ R.
By [39], if f (x, y) is decaying double exponentially with respect to conformal maps φ 1 (s) = b-a 2 tanh( π 2 sinh(s)) + b+a 2 and φ 2 (t) = d-c 2 tanh( π 2 sinh(t)) + d+c 2 , then the function f (x, y) can be expanded in series of sinc function as follows: where e(s, t, h 1 , h 2 ) is the remainder term, which depends on the variables s, t and the mesh sizes Integrating this expression with respect to x and y, we

Convergence analysis
In this section, we discuss the convergence of the proposed approach in terms of the modulus of continuity.

Theorem 4.5 Assume that K(x, y, s, t) is analytic and positive on E × E and that the exact solution u(x, y) is continuous on E.
If Q * = λLL 1 < 1, then we have the following error bound: Proof Since K(x, y, s, t) is analytic on the compact set of E × E, we obtain that K(x, y, s, t) is uniformly continuous. Therefore for every ε > 0, there exists δ > 0 such that for all (s 1 , t 1 ), (s 2 , t 2 ) ∈ E, K(x, y, s 1 , t 1 ) -K(x, y, s 2 , t 2 ) < ε whenever (s 2s 1 ) 2 + (t 2t 1 ) 2 < δ.
Therefore K(x, y, s, t) is uniformly bounded, that is, there is L > 0 such that |K(x, y, s, t)| ≤ L for all (x, y, s, t). Let s i = φ 1 (ih 1 ) and t j = φ 2 (jh 2 ). According to Eq. (10), Eq. (18) and the properties of (R , D), we have On the other hand, based on the differential mean value theorem, we have s i+1 - Therefore we can conclude that Thus the proof of the theorem is completed.

Numerical experiments
In this section, the performance of our numerical approach was tested on From Table 1 and Fig. 1 we can find that the numerical solutions are more and more close to the exact solutions as N and M increase. In Table 2, comparing the proposed method with the triangular function method [27] and block pulse function method [26], we see that the proposed method has a higher accuracy and much smaller error with less collocation points. The condition number of the matrix S -T is uniformly bounded with infinity norm in Fig. 2. This shows that the current method is stable. Figure 3 shows that the numerical solutions are in good agreement with exact solution.    It is evident from the Table 3 and Fig. 4 that if we increase the collocation points, then the absolute error decreases. Comparing the presented method and the triangular function method [27] in Table 4, we see that the former is more accurate than the latter. Figure 5       The results of Table 5 and Fig. 7 confirm the theoretical results. Figure 8 shows that the condition number of the matrix S -T is bounded. Figure 9 shows the numerical and exact solutions for some values of x, y, and r. We applied the proposed method to Example 5.3 and compared to the best results obtained in [41,42] in Table 6. From Table 5 we see that our method needs a very small number of collocation points to achieve high accuracy in comparison with the iterative method [41] and the Bernstein polynomials method [42]. Furthermore, the proposed method is easy to implement, in contrast to the iterative method [41], which needs complicated iterations costing too much time.

Conclusion
In this paper, we introduce a numerical scheme based on the sinc method together with DE transformation to solve 2DFFIE. By solving dual fuzzy linear systems, we obtain approximate solutions. Moreover, we give an error analysis of the proposed method. We provide numerical results to illustrate the effectiveness and accuracy of the presented method. We also compared the proposed scheme to the other numerical methods, which confirms its superiority and the importance of employing the sinc collocation method. In future work, we will study the stability of the current method and utilize the proposed method to deal with other kinds of fuzzy integral equations.