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Numerical Solution of Two-Dimensional Linear Fuzzy Fredholm Integral Equations by the Fuzzy Lagrange Interpolation.

1. Introduction

Recently many authors proposed various numerical methods for solving one-dimensional fuzzy integral equations [19]. Also, two-dimensional fuzzy integral equations have been noticed by a lot of researchers because of their broad applications in engineering sciences. Some of the most important papers in this area are trapezoidal quadrature rule and iterative method [10-12], triangular functions [13], quadrature iterative [14], Bernstein polynomials [15], collocation fuzzy wavelet like operator [16], homotopy analysis method (HAM) [17], open fuzzy cubature rule [18], kernel iterative method [19], modified homotopy pertubation [20], block-pulse functions [21], optimal fuzzy quadrature formula [22], and finally, iterative method and fuzzy bivariate block-pulse functions [23]. Also, some researchers have solved one-dimensional fuzzy Fredholm integral equations by using fuzzy interpolation via iterative method such as: iterative interpolation method [9], Lagrange interpolation based on the extension principle [5], and spline interpolation [7].

As we know interpolation is one of the most substantial and the most applicable methods in numerical analysis. So, in this paper, we want to solve 2DLFFIEs by applying two-dimensional fuzzy Lagrange interpolation and iterative method. First of all an approximate solution of integral by applying Lagrange interpolation and iterative method is provided. Then, convergence analysis and numerical stability analysis of the proposed method in Theorems 11 and 12 are proved.

The paper is organized as follows: Some notations and theorems about the structure of fuzzy sets are reviewed in Section 2. In Section 3, firstly, we introduce two-dimensional fuzzy Lagrange interpolation. Also, we propose two-dimensional fuzzy Lagrange interpolation and iterative method for solving 2DLFFIEs. In Section 4, we verify convergence analysis for proposed method. Also, in Section 5, we prove numerical stability analysis for the method. Two numerical examples are presented in Section 6.

2. Preliminaries

At first, we review some basic definitions and necessary results about fuzzy set theory.

Definition 1 (see [24]). A fuzzy number is a function [??] : R [right arrow] [0,1] with the following properties:

(1) [there exists][x.sub.0] [member of] R such that [??]([x.sub.0]) = 1.

(2) [??]([eta]x + (1 - [eta])y) [greater than or equal to] min{[??](x), [??](y)}, [for all]x, y [member of] R, [for all][eta] [member of] [0,1].

(3) [for all][x.sub.0] [member of] R and [for all][epsilon] > 0, [there exists] neighborhood U([x.sub.0]) : [??](x) [less than or equal to] [??]([x.sub.0]) + [epsilon], [for all]x [member of] U([x.sub.0]).

(4) In R, the set [bar.supp([??])] is compact.

The set of all fuzzy numbers is denoted by [R.sub.F].

Definition 2 (see [24]). For [??] [member of] [R.sub.F] and 0 < [alpha] [less than or equal to] 1, define [[[??]].sup.0] := [bar.{x [member of] R : [??](x) > 0}] and

[[[??]].sup.[alpha]] := {x [member of] R : [??] (x) [greater than or equal to] [alpha]}. (1)

Then it is well known that, for each [alpha] [member of] [0, 1], [[[??]].sup.[alpha]] is a bounded and closed interval of R. We define uniquely the sum [??] [[direct sum]] [??] and the product [mu] [dot encircle] for [mathematical expression not reproducible] and [mu] [member of] R by

[mathematical expression not reproducible], (2)

where [mathematical expression not reproducible] means the usual addition of two intervals (as subsets of R) and [mu][[[??]].sup.[alpha]] means the usual product between a scalar and a subset of R. Notice 1 [dot encircle] [??] = [??] and it holds [mathematical expression not reproducible]. If 0 [less than or equal to] [[alpha].sub.1] [less than or equal to] [[alpha].sub.2] [less than or equal to] 1 then [mathematical expression not reproducible]. Actually [[[??]].sup.[alpha]] = [[[??]].sup.([alpha]).sub.-], [[[??]].sup.([alpha]).sub.+]], where [[[??].sup.([alpha]).sub.-] [less than or equal to] [[??].sup.([alpha]).sub.+], [[??].sup.([alpha]).sub.-], [[??].sup.([alpha]).sub.+] [member of] R, [for all][alpha] [member of] [0,1]. For [mu] > 0 one has [mu] [[??].sup.([alpha]).sub.[+ or -]] = [([mu] [dot encircle] f).sup.([alpha]).sub.[+ or -]], respectively.

Definition 3 (see [24]). Define D : [R.sub.F] x [R.sub.F] [right arrow] [R.sub.+] by

[mathematical expression not reproducible], (3)

where [mathematical expression not reproducible]. Clearly D is a metric on [R.sub.F]. Also ([R.sub.F], D) is a complete metric space, with the following properties [24]:

[mathematical expression not reproducible]. (4)

Definition 4 (see [24]). Suppose [mathematical expression not reproducible] be fuzzy number valued functions, then the distance between f and g is defined by

[mathematical expression not reproducible]. (5)

Lemma 5 (see [24]).

(1) If we denote [??] := [[chi].sub.{0}], then [mathematical expression not reproducible].

(2) With respect to [??], none of [mathematical expression not reproducible] has opposite in [R.sub.F].

(3) Let [alpha], [beta] [member of] R : [alpha].[beta] [greater than or equal to] 0, and any [??] [member of] [R.sub.F], we have ([alpha] + [beta]) [dot encircle] [??] = [alpha] [dot encircle] [??] [direct sum] [beta] [dot encircle] [??]. Notice that, for general [alpha], [beta] [member of] R, the above property is false.

(4) For any [gamma] [member of] R and any [mathematical expression not reproducible], we have [mathematical expression not reproducible].

(5) For any [gamma], [eta] [member of] R and any [??] [member of] [R.sub.F], we have [gamma] [dot encircle]([eta] [dot encircle] [??]) = ([gamma] [dot encircle] [eta]) [dot encircle] [??].

If we denote [mathematical expression not reproducible], then [[parallel]*[parallel].sub.F] has the properties of a usual norm on [R.sub.F], i.e.,

[mathematical expression not reproducible]. (6)

Notice that ([R.sub.F], [direct sum], [dot encircle]) is not a linear space over R, and consequently ([R.sub.F], [[parallel]*[parallel].sub.F]) is not a normed space. Here [[summation].sup.*] denotes the fuzzy summation.

Definition 6 (see [24]). A fuzzy valued function [??] : [a, b] [right arrow] [R.sub.F] is said to be continuous at [x.sub.0] [member of] [a, b], if for each [epsilon] > 0 there exists [delta] > 0 such that D([??](x), [??]([x.sub.0])) < [epsilon], whenever x [member of] [a, b] and [absolute value of x - [x.sub.0]] < [delta]. We say that [??] is fuzzy continuous on [a, b] if [??] is continuous at each [x.sub.0] [member of] [a, b] and denotes the space of all such functions by [C.sub.F][a, b].

Definition 7 (see [11]). Suppose that [??] : [a, b] x [c, d] [right arrow] [R.sub.f] is a bounded mapping. The function [[omega].sub.[a,b]x[c,d]]([??], *) : [R.sub.+] [union] {0} [right arrow] [R.sub.+] defined by

[[omega].sub.[a,b]x[c,d]] ([??], [delta]) = sup {D([??] (x, y), [??](s, t)); x, s

[member of] [a, b]; y, t [member of] [c, d]; [square root of [(x - s).sup.2] + [(y - t).sup.2]] [less than or equal to] [delta]}, (7)

is called modules of oscillation of [??] on [a, b] x [c, d]. Also, if [??] [member of] [C.sub.F]([a, b] x [c, d]), then [[omega].sub.[a,b]x[c,d]]([??], [delta]) is called uniform modules of continuity of [??].

Theorem 8 (see [11]). The following properties hold:

(1) D([??](x, y), [??](s,t)) [less than or equal to] [[omega].sub.[a,b]x[c,d]]([??], [square root of [(x - s).sup.2] + [(y - t).sup.2]), [for all]x, s [member of] [a, b], y, t [member of] [c, d];

(2) [[omega].sub.[a,b]x[c,d]]([??], [delta]) is a nondecreasing mapping in [delta];

(3) [[omega].sub.[a,b]x[c,d]]([??], 0) = 0;

(4) [[omega].sub.[a,b]x[c,d]]([??], [[delta].sub.1] + [[delta].sub.2]) [less than or equal to] [[omega].sub.[a,b]x[c,d]]([??], [[delta].sub.1]) + [[omega].sub.[a,b]x[c,d]]([??], [[delta].sub.2]), [for all][[delta].sub.1], [[delta].sub.2] [greater than or equal to] 0;

(5) [[omega].sub.[a,b]x[c,d]]([??], n[delta]) [less than or equal to] n[[omega].sub.[a,b]x[c,d]]([??], [delta]), [for all][delta] [greater than or equal to] 0, n [member of] N;

(6) [[omega].sub.[a,b]x[c,d]]([??], [mu][delta]) [less than or equal to] ([mu] + 1) [[omega].sub.[a,b]x[c,d]]([??], [delta]), [for all][mu], [delta] [greater than or equal to] 0.

Theorem 9 (see [11]). If [??] and [??] are Henstock integrable mapping on [a,b] x [c,d] and if [mathematical expression not reproducible] is Lebesgue integrable, then

[mathematical expression not reproducible]. (8)

3. The Main Result

In this section, first, we introduce two-dimensional fuzzy Lagrange interpolation. Then, we propose two-dimensional fuzzy Lagrange interpolation and iterative method for solving (9).

Consider 2DLFFIE as follows:

[mathematical expression not reproducible], (9)

where [mu] > 0, K(x, y, s, t) is an arbitrary positive function on [a, b] x [c, d] x [a, b] x [c, d] and [??] : [a, b] x [c, d] [right arrow] [R.sub.F]. We assume that K is continuous, and therefore it is uniformly continuous with respect to (s, t). So, there exists M > 0 such that M = [max.sub.s,x [member of][a,b]y,t [member of][c,d]] [absolute value of K(x, y, s, t)].

Two-dimensional Lagrange polynomials, [l.sub.ij](s, t), are defined as follows:

[l.sub.ij](s, t) = [l.sub.i](s) [cross product] [l.sub.j] (t), 0 [less than or equal to] i, j [less than or equal to] n, (10)

where [l.sub.i](s) is the Lagrange polynomial and is defined as follows:

[mathematical expression not reproducible], (11)

therefore,

[mathematical expression not reproducible]. (12)

So, the two-dimensional interpolation in the Lagrange form is (see [25])

p (x, y) = [n.summation over (j=0)] [n.summation over (i=0)] [??] ([s.sub.i], [t.sub.j]) [dot encircle] [l.sub.ij] (x, y), (13)

where the coefficients [??]([s.sub.i], [t.sub.j]) are the fuzzy numbers.

Here, we consider the two-dimensional fuzzy Lagrange interpolation in the given points a = [s.sub.0] < [s.sub.1] < ... < [s.sub.n] = b and c = [t.sub.0] < [t.sub.1] < ... < [t.sub.n] = d such that

K (x, y, s, t) [dot encircle] [??] (x, y)

[approximately equal to] [n.summation over (j=0)] [n.summation over (i=0)] [l.sub.ij] (x, y) [dot encircle] K ([s.sub.i], [t.sub.j], s, t) [dot encircle] [??] ([s.sub.i], [t.sub.j]). (14)

Now, we propose a numerical method to solve (9). To do this, we suppose the following iterative procedure to approximate the solution of (9) in point (s, t)

[mathematical expression not reproducible], (15)

where

[C.sub.ij] = [[integral].sup.d.sub.c] [[integral].sup.b.sub.a] [l.sub.ij] (x, y) dxdy. (16)

In Theorem 10, authors of [11] proved the existence and uniqueness solution of (9) by using the Banach fixed point theorem.

Theorem 10 (see [11]). Let the function K(x, y, s, t) be continuous and positive for x, s [member of] [a, b], and y, t [member of] [c, d], and let [??] : [a,b] x [c,d] [right arrow] [R.sub.F] be continuous on [a,b] x [c,d]. If B = [mu]M(d - c)(b - a) < 1 then the fuzzy integral equation (9) has a unique solution [[??].sup.*] [member of] X, where

X = {[??] : [a,b] x [c,d] [right arrow] [R.sub.F]; [??] is continous}, (17)

be the space of two-dimensional fuzzy continuous functions with the metric [D.sup.*] and it can be obtained by the following successive approximations method:

[mathematical expression not reproducible]. (18)

Moreover, the sequence of successive approximations [([[??].sub.k]).sub.k [greater than or equal to] 1] converges to the solution [[??].sup.*]. Furthermore, the following error bound holds:

[D.sup.*] ([[??].sup.*], [[??].sub.k]) [less than or equal to] [B.sup.k+1]/1 - B [M.sub.1], [for all]k [greater than or equal to] 1, (19)

where [M.sub.1] = [sup.sub.s[member of][a,b],t[member of][c,d]] [parallel][??](s, t)[parallel].

4. Convergence Analysis

In this section, we obtain an error estimate between the exact solution and the approximate solution of 2DLFFIE (9).

Theorem 11. Under the hypotheses of Theorem 10 and [mu] > 0, the iterative procedure (15) converges to the unique solution of (9), [[??].sup.*], and its error estimate is as follows:

[mathematical expression not reproducible]. (20)

Proof. Clearly, we have

[mathematical expression not reproducible]. (21)

From (18) and (15), we conclude that

[mathematical expression not reproducible]. (22)

By supposing [m.sub.k-1] = [sup.sub.(s,t)[member of][a,b]x[c,d]] [parallel][[??].sub.k-1](s, t)[parallel], we get

[mathematical expression not reproducible]. (23)

Also, we have

[mathematical expression not reproducible]. (24)

Therefore,

[mathematical expression not reproducible], (25)

where [??] = max[absolute value of [C.sub.ij]], i, j = 0, ..., n. Hence, we conclude that

[mathematical expression not reproducible]. (26)

So,

[mathematical expression not reproducible]. (27)

If [m.sub.l] = max{[m.sub.0], ..., [m.sub.k-1]}, then we obtain

[mathematical expression not reproducible], (28)

therefore

[mathematical expression not reproducible]. (29)

5. Numerical Stability Analysis

To show the numerical stability analysis of the proposed method in previous section, we consider another starting approximation [mathematical expression not reproducible] such that [there exists][epsilon] > 0 for which [mathematical expression not reproducible]. The obtained sequence of successive approximations is

[mathematical expression not reproducible], (30)

and using the same iterative method, the terms of produced sequence are

[mathematical expression not reproducible]. (31)

Theorem 12. The proposed method (15), under the assumptions of Theorem 11, is numerically stable with respect to the choice of the first iteration.

Proof. At first, we obtain that

[mathematical expression not reproducible]. (32)

However,

[mathematical expression not reproducible]. (33)

We conclude that

[mathematical expression not reproducible], (34)

and thus

[mathematical expression not reproducible]. (35)

So,

[mathematical expression not reproducible]. (36)

Therefore,

[mathematical expression not reproducible]. (37)

6. Numerical Examples

In this section, we use the proposed method in two-dimensional linear fuzzy Fredholm integral equations for solving two examples. By using the proposed method for n = 2, k = 5, 8, 16, and r [member of] {0.0,0.1,0.2,0.3,0.4,0.5,0.6, 0.7,0.8, 0.9,1.0} in (s, t) = (0.5,0.5), we present the absolute errors in Tables 1-4.

Example 13 (see [11]). Consider the linear integral equation

[mathematical expression not reproducible], (38)

with

K (x, y, s, t) = xyst,

[[??].bar] (s, t, r) = r (1/3 r + 8/3)(1 + s + t - 7/12 st),

[[bar.[??]] (s, t, r) = (2[r.sup.2] - 4r + 5)(1 + s + t - 7/12 st). (39)

The exact solution for 2DLFFIE (38) is

[[[??].sup.*.[bar]] (s,t,r) = r(1/3 r + 8/3) (s + t + 1),

[[bar.[??]].sup.*] (s, t, r) = (2[r.sup.2] - 4r + 5) (s + t +1). (40)

Example 14 (see [13]). Consider the following fuzzy Fredholm integral equation (38) with

[[??].bar] (s, t, r) = r (st + 1/676 ([s.sup.2] + [t.sup.2] - 2)),

[bar.[??]] (s, t, r) = (2 - r) (st + 1/676 ([s.sup.2] + [t.sup.2] - 2)), (41)

and kernel

K (x, y, s, t) = 1/169 ([x.sup.2] + [y.sup.2] - 2) ([s.sup.2] + [t.sup.2] - 2),

0 [less than or equal to] s, t, x, y [less than or equal to] 1. (42)

The exact solution is

[[[??].bar].sup.*] (s, t, r) = rst,

[[bar.[??]].sup.*] (s, t, r) = (2 - r) st. (43)

7. Conclusion

The 2DLFFIE is solved by utilizing two-dimensional fuzzy Lagrange interpolation and iterative method. As it was expected the method used to approximate the integral in this equation is a suitable one since convergence analysis and stability analysis have been proved and also absolute error in examples is nearly zero. As a result, considering the fact that the proposed method does not lead to solve fuzzy linear system, it can be utilized as an efficient method to solve this type of equations. As future researches, we can use finite and divided differences methods, fuzzy spline interpolation, and fuzzy quasi-interpolation for solving two-dimensional linear or nonlinear fuzzy Fredholm integral equations.

https://doi.org/10.1155/2018/5405124

Conflicts of Interest

The authors declare that there are no conflicts of interest related to this paper.

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[14] H. Nouriani and R. Ezzati, "Quadrature iterative method for numerical solution of two-dimensional linear fuzzy Fredholm integral equations," Mathematical Sciences, vol. 11, no. 1, pp. 63-72, 2017.

[15] R. Ezzati and S. Ziari, "Numerical solution of two-dimensional fuzzy Fredholm integral equations of the second kind using fuzzy bivariate Bernstein polynomials," International Journal of Fuzzy Systems, vol. 15, no. 1, pp. 84-89, 2013.

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H. Nouriani (iD) and R. Ezzati (iD)

Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran

Correspondence should be addressed to R. Ezzati; ezati@kiau.ac.ir

Received 26 October 2017; Accepted 19 May 2018; Published 2 December 2018

Academic Editor: Oscar Castillo
Table 1: The absolute errors on the level sets with n = 2, k = 5 for
Example 13 by using the proposed method (s, t) = (0.5,0.5).

r--level   [mathematical expression   [mathematical expression
              not reproducible]          not reproducible]

0.0                   0.                    1.23485e--5
0.1              6.66819e--7                 1.141 e--5
0.2               1.3501 e-6                1.05703e--5
0.3              2.04985e--6                9.82941e--6
0.4              2.76606e--6                9.18729e--6
0.5              3.49874e--6                8.64395e--6
0.6              4.24788e--6                8.19941e--6
0.7              5.01349e--6                7.85365e--6
0.8              5.79556e--6                7.60668e--6
0.9              6.5941 e--6                7.45849e--6
1.0              7.4091 e--6                7.4091 e--6

Table 2: The absolute errors on the level sets with n = 2, k = 16 for
Example 13 by using the proposed method in (s, t) = (0.5 ,0.5).

r--level   [mathematical expression   [mathematical expression
              not reproducible]          not reproducible]

0.0                   0.                    1.77636e--15
0.1              1.11022e--16               1.77636e--15
0.2                   0.                    1.77636e--15
0.3              2.22045e--16               8.88178e--16
0.4              4.44089e--16               8.88178e--16
0.5              4.44089e--16               8.88178e--16
0.6              4.44089e--16                    0.
0.7              8.88178e--16                    0.
0.8                   0.                         0.
0.9                   0.                         0.
1.0                   0.                         0.

Table 3: The absolute errors on the level sets with n = 2, k = 5 for
Example 14 by using the proposed method in (s, t) = (0.5, 0.5).

r--level   [mathematical expression   [mathematical expression
              not reproducible]          not reproducible]

0.0                   0.                    9.60565e--13
0.1              4.80276e--14               9.12548e--13
0.2              9.60551e--14               8.64531e--13
0.3              1.44079e--13               8.16458e--13
0.4              1.9211 e- 13               7.68441e--13
0.5              2.40141e--13               7.20479e--13
0.6              2.88158e--13               6.72407e--13
0.7              3.36203e--13               6.24334e--13
0.8              3.8422 e--13               5.76317e--13
0.9              4.32265e--13               5.283 e--13
1.0              4.80282e--13               4.80282e--13

Table 4: The absolute errors on the level sets with n = 2, k = 8 for
Example 14 by using the proposed method in (s, t) = (0.5, 0.5).

r--level   [mathematical expression   [mathematical expression
              not reproducible]          not reproducible]

0.0                   0.                         0.
0.1                   0.                         0.
0.2                   0.                    5.55112e--17
0.3                   0.                         0.
0.4                   0.                         0.
0.5                   0.                    5.55112e--17
0.6                   0.                         0.
0.7                   0.                         0.
0.8                   0.                         0.
0.9              2.77556e--17                    0.
1.0                   0.                         0.
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Title Annotation:Research Article
Author:Nouriani, H.; Ezzati, R.
Publication:Advances in Fuzzy Systems
Date:Jan 1, 2018
Words:3888
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