Abstract and Applied Analysis

Volume 2014 (2014), Article ID 413570, 18 pages

http://dx.doi.org/10.1155/2014/413570

Research Article

Quadrature Rules and Iterative Method for Numerical Solution of Two-Dimensional Fuzzy Integral Equations

S. M. Sadatrasoul and R. Ezzati

Department of Mathematics, College of Basic Sciences, Karaj Branch, Islamic Azad University, Alborz, Iran

Received 25 December 2013; Accepted 11 March 2014; Published 19 May 2014

Academic Editor: Soheil Salahshour

Copyright © 2014 S. M. Sadatrasoul and R. Ezzati. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We introduce some generalized quadrature rules to approximate two-dimensional, Henstock integral of fuzzy-number-valued functions. We also give error bounds for mappings of bounded variation in terms of uniform modulus of continuity. Moreover, we propose an iterative procedure based on quadrature formula to solve two-dimensional linear fuzzy Fredholm integral equations of the second kind (2DFFLIE2), and we present the error estimation of the proposed method. Finally, some numerical experiments confirm the theoretical results and illustrate the accuracy of the method.

1. Introduction

The concept of fuzzy numbers and arithmetic operations with these numbers were first introduced and investigated by Zadeh and others. The topic of fuzzy integrations is discussed in [1]. The Henstock and Riemann integral for fuzzy-number-valued functions was introduced and studied in [2, 3]. Their numerical computation was also proposed; see, for example, [3–6]. In [6], the authors obtained the upper estimates of error of some fuzzy quadrature rules for mappings of bounded variation and of Lipschitz type and gave some applications. In [7], the authors studied the Gaussian quadrature rules for fuzzy integrals. Also, in [8], Wu presented some optimal fuzzy quadrature formula for classes of fuzzy-number-valued functions of Lipschitz type. To study other works, see [9–12].

Since many real-valued problems in engineering and mechanics can be brought in the form of two-dimensional fuzzy integral equations, it is important that we develop quadrature rules and numerical methods for such integral equations. In this paper, we introduce two-dimensional fuzzy integrals and propose some generalized quadrature rules and their dependent theorems for mappings of bounded variation. Also, we present the conditions for existence of unique solution for 2DFFLIE2. Finally, we introduce an iterative method for solving 2DFFLIE2. The rest of the paper is organized as follows. In Section 2, we give basic information about the fuzzy set theory and develop them to two-dimensional space. Also, we define two-dimensional fuzzy integral equation and some other properties of it in this section. In Section 3, we derive the proposed method to obtain numerical solutions of 2DFFLIE2 based on an iterative procedure. The error estimation of the introduced method is presented in Section 4 in terms of uniform modulus of continuity to prove the convergence of the method. Some numerical experiments are presented in Section 5.

2. Preliminaries

In this section, we review some necessary basic definitions on fuzzy numbers, fuzzy-number-valued functions, and fuzzy integrals.

Definition 1 (see [13, 14]). A fuzzy number is a function having the following properties:

(i)is normal; that is, , such that ;(ii) is fuzzy convex set (i.e., , for all ;(iii) is upper semicontinuous on ;(iv)the support is a compact set, where denotes the closure of .

The set of all fuzzy numbers is denoted by . According to [2], any real number can be interpreted as a fuzzy number , and therefore . Also, the neutral element with respect to in is denoted by .

Definition 2 (see [2, 15]). For any , an arbitrary fuzzy number is represented in parametric form, by an ordered pair of functions , which satisfies the following properties:

(i) is bounded left continuous nondecreasing function over ;(ii) is bounded left continuous nonincreasing function over ;(iii).

Moreover, the addition and scalar multiplication of fuzzy numbers in are defined as follows:

(i)

(ii)

Also, according to [2, 16], the following algebraic properties for any hold:

(i); (ii); (iii)with respect to , none of ,    has opposite in ;(iv), for all with or ;(v), for all ;(vi), for all and .

Definition 3 (see [2, 17]). For arbitrary fuzzy numbers , , the quantity is the distance between and . Also, the following properties hold [6]:

(i) is a complete metric space;(ii) for all;(iii) for all ;(iv) for all ;(v) for all with and for all .

Throughout this paper, we denote that .

Theorem 4 (see [14]).    is a complete metric space.

The pair is a commutative semigroup with zero elements but cannot be a group for pure fuzzy numbers.

   has the properties of a usual norm on ; that is, if and only if , , and .

   and for any .

In [2], the authors introduced the concept of the Henstock integral for a fuzzy-number-valued function. We present a generalized definition of this concept for two-dimensional Henstock integrability for bivariate fuzzy-number-valued functions.

Definition 5. Suppose that is a bounded mapping, and then the function defined by

is called the modulus of oscillation of on .

Also, if   (i.e.,   is continuous on ), then is called uniform modulus of continuity of . The following properties will be very useful in what follows. The proofs of these properties in one-dimensional case are presented in [14] and those in two-dimensional case will be obtained in a similar way.

Theorem 6. The following properties hold:

(i) for any and ;(ii) is a nondecreasing mapping in ;(iii); (iv) for any ;(v) for any and ;(vi) for any .

Definition 7. Let , for and , be two partitions of the intervals and , respectively. Let one consider the intermediate points and , ; , and and . The divisions , , and , , denoted shortly by and are said to be -fine and -fine, respectively, if and .

The function is said to be two-dimensional Henstock integrable to if for every there are functions and such that for any -fine and -fine divisions we have , where denotes the fuzzy summation. Then, is called the two-dimensional Henstock integral of and is denoted by .

If the above and are constant functions, then one recaptures the concept of Riemann integral. In this case, will be called two-dimensional integral of on and will be denoted by .

Corollary 8. In [13], the authors proved that if , its definite integral exists, also , and . In a similar way, we can prove that if , its definite integral exists, and one has

Theorem 9. If and are Henstock integrable mappings on and if is Lebesgue integrable, then

Proof. In [2, 17], the authors demonstrated that for any integrable functions we have , and, clearly, we obtain

which completes the proof.

Theorem 10. If is an integrable bounded mapping, then for any fixed and the function , defined by , is Lebesgue integrable on .

Proof. Regarding [6], Lemma 1, part (ii), it is easy to see that if is two-dimensional Henstock integrable and bounded on , then and as real functions of are two-dimensional integrable and uniformly bounded with respect to ; that is, and are Lebesgue measurable (as functions of ) and uniformly bounded with respect to by

where , represent all the rational numbers in . By Lebesgue’s theorem of dominated convergence, it follows that is Lebesgue integrable on , and this ends the proof.

Definition 11. A function is said to be bounded if there exists such that for any .

Definition 12. A function is said to be of bounded variation if

where

is the variation of related to partitions , . The total variation of is defined to be, in this case, the number

It is known also that a function of bounded variation is Riemann integrable (see [18]), so it is Henstock integrable too.

Theorem 13. If , then for all .

If is of bounded variation, then for all .

Proof. (i) It is easy to see that

and, therefore, we obtain the required inequality.

(ii) Let and ; assume that , , , and . Taking supremum for any and with , we obtain the required inequality. It is obvious now that under this condition is bounded; therefore, we obtain

which completes the proof.

Definition 14. A function is said to be L-Lipschitz, if

for any and .

Definition 15. A function is said to be M-Condition, if

for any and .

Remark 16. We see that if is M-Condition function, then is of bounded variation and

Indeed, we have

and since

we obtain the required result.

3. Quadrature Rules for Two-Dimensional (2D) Henstock Integrals

In this section, we present some quadrature rules for 2D Henstock integral. The following theorem gives a unified approach to quadrature rules in 2D Henstock integrals.

Theorem 17. Let be Henstock integrable, bounded mappings. Then, for any divisions and and any points and , one has

Proof. It is known that the Henstock integrals are additive related to interval. This leads us to

and, by Definition 3 part (iv) and Theorem 9, we have

From part (i) of Theorem 6, we conclude that

which completes the proof.

From the above inequality, we infer some generalization of well-known trapezoidal-type, midpoint-type, and three-point-type inequalities with error estimations.

Corollary 18. Assume that is a Henstock integrable, bounded mapping. Then, with the notation

one has

(i)

for any ;

(ii)

for any , , , , , and ;

(iii)

for any , , , , , , , , , and with and .

Proof. (i) Taking in the previous theorem that ,  , and , we obtain the required inequality. Indeed,

(ii) Taking that , , , , , , and in Theorem 17, we obtain the required inequality. Indeed,

(iii) Considering and performing the similar way in part (ii), it is obvious that the inequality in previous theorem becomes the inequality stated above.

Corollary 19. Let be a two-dimensional Henstock integrable, bounded mapping. Then, the following inequalities hold:

Proof. (i) If we take and in the assertion (i) of Corollary 18, we obtain the required inequality. In other words, we have

(ii) Taking , , , , , and in the assertion (ii) of the previous corollary, we obtain

(iii) It is easy to see that the inequality follows from the corresponding assertion (iii) of the previous corollary by taking , , , , , , , , , , and . Indeed, we have

The next corollaries present simpler error estimation for the inequality stated in Theorem 17.

Corollary 20. Let be a two-dimensional Henstock integrable, bounded mapping. Then, for any divisions and , and , ; , one has

where is the norm of the divisions and .

Proof. Considering Theorem 17 and parts (i), (ii) of Theorem 13 and by regarding the definition of , we infer that

Corollary 21. Let be a two-dimensional Henstock integrable, bounded mapping. Then, for any divisions and , and , ; , one has

Proof. Since is the least upper bound of partitions and , we conclude that for any . Hence, the required inequality holds.

Remark 22. If is a two-dimensional Riemann integrable function, it is also Henstock integrable function. Therefore, the above quadrature rules hold for Riemann integrable function too.

Theorem 23. Let be a mapping of bounded variation. Then, for any divisions and , and , ; , one has

Proof. If we define such that for any , we see that is of bounded variation and we have

in other words,

Considering Theorem 13, Theorem 17, Corollary 21, and [18] and since any real valued function of bounded variation is Lebesgue integrable, we observe that

Theorem 24. If is L-Lipschitz mapping, then, for any divisions and , and , , one has

Proof. Analogous to the proof of Theorem 17 and by definition of L-Lipschitz mapping, we infer that

4. 2D Fuzzy Fredholm Integral Equations

Here, we consider the two-dimensional fuzzy Fredholm integral equations as follows:

where , is an arbitrary positive kernel on and . We assume that is continuous, and therefore it is uniformly continuous with respect to . This property implies that there exists such that

Now, we will prove the existence and uniqueness of the solution of (41) by the method of successive approximations. Let be the space of two-dimensional fuzzy continuous functions with the metric

that is called the uniform distance between two-dimensional fuzzy-number-valued functions. We define the operator by

Sufficient conditions for the existence of a unique solution of (41) are given in the following result.

Theorem 25. Let be continuous and positive for , , and , and let be continuous on . If , then the iterative procedure

converges to the unique solution of (41).

Moreover, the following error bound holds:

where

Proof. To prove this theorem, we investigate the conditions of the Banach fixed point principle. We first show that maps into (i.e., ). To the end, we show that the operator is uniformly continuous. Since is continuous on compact set of , we deduce that it is uniformly continuous, and hence for exists such that

As mentioned above, also is uniformly continuous; thus, for exists such that

Let , , and , with . According to Definition 3 and Theorem 9, we obtain

and by choosing and , we derive

This shows that is uniformly continuous for any and so continuous on , and hence .

Now, we prove that the operator is a contraction map. So, for , , and , we have

Therefore, we obtained

Since , the operator is a contraction on Banach space . Consequently, Banach’s fixed point principle implies that (41) has a unique solution in and we also have

therefore,

on the other hand,

so by (55) and (56), we obtained inequality (46), which completes the proof.

Now, we introduce a numerical method to solve (41). We consider (41) with continuous kernel having positive sign on and uniform partitions

with , , where , . Then, the following iterative procedure gives the approximate solution of (41) in point :

The above recursive relation can be written as follows:

4.1. Error Estimation

Here, we obtain an error estimate between the exact solution and the approximate solution for the given fuzzy Fredholm integral equation (41).

Theorem 26. Consider the 2DFFLIE2 (41) with continuous kernel having positive sign on and suppose that is continuous on . If , where , then the iterative procedure (59) converges to the unique solution of (41), , and the following error estimate holds true:

where

Proof. Considering iterative procedure (59), for all  , we have

Using part (ii) of Corollary 19, part (v) of Definition 3, and part (i) of Theorem 13, we obtain

By part (ii) of Theorem 6 and direct computation, it follows that

therefore, we obtain

Now, since , we infer that

therefore, we have

By induction for , using (45), (46), (59), and (62), we see that

taking supremum for from (70), we conclude that

and multiplying the above inequalities by , respectively, and summing them, we obtain

Since, for , with, , we have

we infer that

By this inequality and (72), we see that

By (62) and (63) and since , we obtain

therefore, we obtain

By inequalities (77) and (46), we deduce that

Remark 27. Since , it is easy to see that

which shows the convergence of the method.

5. Numerical Experiments

The proposed iterative method of successive approximations was tested on three numerical examples to provide the accuracy and the convergence of the method and illustrate the correctness of the theoretical results. In these examples, we assumed that , , and we performed the algorithm in point .

Example 1. Assume that

where

the exact solution is given by

To obtain numerical solution, we apply the proposed method. To compare numerical and exact solutions, see Table 1.

Table 1: Numerical results on the level sets for Example 1 in .

-level
,
,
,
,

0.00
0.000000
0.000657
0.000000
0.000661
0.000000
0.000008
0.000000
0.000003

0.25
0.000067
0.000427
0.000051
0.000617
0.000011
0.000053
0.000001
0.000005

0.50
0.000086
0.000586
0.000024
0.000258
0.000066
0.000815
0.000043
0.000001

0.75
0.000150
0.000423
0.000022
0.000367
0.000069
0.000499
0.000008
0.000006

1.00
0.000229
0.000329
0.000131
0.000221
0.000154
0.000154
0.000005
0.000005

Example 2. Consider (80) with

and exact solution

We perform the proposed method and obtain numerical solution. Comparison of these two results is presented in Table 2.

Table 2: Numerical results on the level sets for Example 2 in .

r-level
,
,

0.0
0.000000
0.000000
0.000000
0.000000

0.2
0.000003
0.000008
0.000000
0.000002

0.4
0.000007
0.000011
0.000000
0.000008

0.6
0.000004
0.000006
0.000001
0.000000

0.8
0.000000
0.000005
0.000000
0.000000

1.0
0.000000
0.000000
0.000000
0.000000

Example 3. The integral equation (80) with

has the exact solution

For this linear example, we apply our proposed iterative method and obtain numerical results that can be viewed in Table 3.

Table 3: Numerical results on the level sets for Example 3 in .

-level
,
,

0.0
0.000047
0.000028
0.000008
0.000003

0.2
0.000039
0.000086
0.000006
0.000009

0.4
0.000009
0.000008
0.000003
0.000000

0.6
0.000004
0.000005
0.000001
0.000000

0.8
0.000012
0.000033
0.000007
0.000002

1.0
0.000006
0.000004
0.000000
0.000000

6. Conclusions

In this paper, we introduced 2D fuzzy mappings and defined 2D fuzzy integrals. Quadrature rules to approximate the solution of 2D fuzzy integrals are given. We established the theorem of existence of unique solution of 2DFFLIE2, and we have proved it by using Banach’s fixed point principle. Moreover, to approximate the solution of 2DFFLIE2, we have proposed an iterative algorithm based on method of successive approximations. The convergence to the unique solution in our iterative method is investigated. The presented numerical experiments show that the method applies well for 2DFFLIE2.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the editor and anonymous referees for various suggestions which have led to an improvement in both the quality and clarity of the paper.

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