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

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 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 Correspondence should be addressed to R. Ezzati; 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. 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 𝑢 : 𝑅 → [0, 1] having the following properties: (i) 𝑢 is normal; that is, ∃𝑥0 ∈ 𝑅, such that 𝑢(𝑥0 ) = 1; (ii) 𝑢 is fuzzy convex set (i.e., 𝑢(𝜆𝑥 + (1 − 𝜆)𝑦) ≥ min{𝑢(𝑥), 𝑢(𝑦)}, for all 𝑥, 𝑦 ∈ 𝑅, 𝜆 ∈ [0, 1]); (iii) 𝑢 is upper semicontinuous on 𝑅; 2 Abstract and Applied Analysis (iv) the support {𝑥 ∈ 𝑅 : 𝑢(𝑥) > 0} is a compact set, where 𝐴 denotes the closure of 𝐴. The set of all fuzzy numbers is denoted by 𝑅ϝ . According to [2], any real number 𝛼 ∈ 𝑅 can be interpreted as a fuzzy number 𝛼 = 𝜒{𝛼} , and therefore 𝑅 ⊂ 𝑅ϝ . Also, the neutral element with respect to ⊕ in 𝑅ϝ is denoted by 0̃ = 𝜒{0} . Definition 2 (see [2, 15]). For any 0 < 𝑟 ≤ 1, an arbitrary fuzzy number is represented in parametric form, by an ordered pair of functions (𝑢(𝑟), 𝑢(𝑟)), which satisfies the following properties: (i) 𝑢(𝑟) is bounded left continuous nondecreasing function over [0, 1]; (ii) 𝑢(𝑟) is bounded left continuous nonincreasing function over [0, 1]; (iii) 𝑢(𝑟) ≤ 𝑢(𝑟). Moreover, the addition and scalar multiplication of fuzzy numbers in 𝑅ϝ are defined as follows: Theorem 4 (see [14]). (i) (𝑅ϝ , 𝐷) is a complete metric space. (ii) The pair (𝑅ϝ , 𝐷) is a commutative semigroup with 0̃ = 𝜒0 zero elements but cannot be a group for pure fuzzy numbers. (iii) ‖ ⋅ ‖ϝ has the properties of a usual norm on 𝑅ϝ ; that is, ‖ ⋅ ‖ϝ = 0 if and only if 𝑢 = 0, ‖𝜆 ⊙ 𝑢‖ϝ = |𝜆|‖𝑢‖ϝ , and ‖𝑢 ⊕ V‖ϝ ≤ ‖𝑢‖ϝ + ‖V‖ϝ . (iv) |‖𝑢‖ϝ − ‖V‖ϝ | ≤ 𝐷(𝑢, V) and 𝐷(𝑢, V) ≤ ‖𝑢‖ϝ + ‖V‖ϝ for any 𝑢, V ∈ 𝑅ϝ . 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 𝑓 : [𝑎, 𝑏] × [𝑐, 𝑑] → 𝑅ϝ is a bounded mapping, and then the function 𝜔[𝑎,𝑏]×[𝑐,𝑑] (𝑓, ⋅) : 𝑅+ ∪ 0 → 𝑅+ defined by 𝜔[𝑎,𝑏]×[𝑐,𝑑] (𝑓, 𝛿) = sup {𝐷 (𝑓 (𝑥, 𝑦) , 𝑓 (𝑠, 𝑡)) ; 𝑥, 𝑠 ∈ [𝑎, 𝑏] ; 𝑦, 𝑡 ∈ [𝑐, 𝑑] ; (i) (𝑢 ⊕ V) (𝑟) = (𝑢 (𝑟) + V (𝑟) , 𝑢 (𝑟) + V (𝑟)) , (𝜆𝑢 (𝑟) , 𝜆𝑢 (𝑟)) (𝜆 ⊙ V) (𝑟) = { (𝜆𝑢 (𝑟) , 𝜆𝑢 (𝑟)) 𝜆 ≥ 0, 𝜆 < 0. (2) Also, according to [2, 16], the following algebraic properties for any 𝑢, V, 𝑤 ∈ 𝑅ϝ hold: (i) 𝑢 ⊕ (V ⊕ 𝑤) = (𝑢 ⊕ V) ⊕ 𝑤; (ii) 𝑢 ⊕ 0̃ = 0̃ ⊕ 𝑢 = 𝑢; ̃ none of 𝑢 ∈ (𝑅ϝ − 𝑅), 𝑢 ≠ 0̃ has (iii) with respect to 0, opposite in (𝑅ϝ , +); (iv) (𝑎 ⊕ 𝑏) ⊙ 𝑢 = 𝑎 ⊙ 𝑢 ⊕ 𝑏 ⊙ 𝑢, for all 𝑎, 𝑏 ∈ 𝑅 with 𝑎𝑏 ≥ 0 or 𝑎𝑏 ≤ 0; (v) 𝑎 ⊙ (𝑢 ⊕ V) = 𝑎 ⊙ 𝑢 ⊕ 𝑎 ⊙ V, for all 𝑎 ∈ 𝑅; (vi) 𝑎 ⊙ (𝑏 ⊙ 𝑢) = (𝑎𝑏) ⊙ 𝑢, for all 𝑎 ∈ 𝑅 and 1 ⊙ 𝑢 = 𝑢. Definition 3 (see [2, 17]). For arbitrary fuzzy numbers 𝑢 = (𝑢(𝑟), 𝑢(𝑟)), V = (V(𝑟), V(𝑟)), the quantity 𝐷(𝑢, V) = sup𝑟∈[0,1] max{|𝑢(𝑟) − V(𝑟)|, |𝑢(𝑟) − V(𝑟)|} is the distance between 𝑢 and V. Also, the following properties hold [6]: (i) (𝑅ϝ , 𝐷) is a complete metric space; (ii) 𝐷(𝑢 ⊕ 𝑤, V ⊕ 𝑤) = 𝐷(𝑢, V) for all 𝑢, V, 𝑤 ∈ 𝑅ϝ ; (iii) 𝐷(𝑘⊙𝑢, 𝑘⊙V) = |𝑘|𝐷(𝑢, V) for all 𝑢, V ∈ 𝑅ϝ for all 𝑘 ∈ 𝑅; (iv) 𝐷(𝑢⊕V, 𝑤⊕𝑒) ≤ 𝐷(𝑢, 𝑤)+𝐷(V, 𝑒) for all 𝑢, V, 𝑤, 𝑒 ∈ 𝑅ϝ ; (v) 𝐷(𝑘1 ⊙ 𝑢, 𝑘2 ⊙ 𝑢) = |𝑘1 − 𝑘2 |𝐷(𝑢, 0̃) for all 𝑘1 , 𝑘2 ∈ 𝑅 with 𝑘1 𝑘2 ≥ 0 and for all 𝑢 ∈ 𝑅ϝ . Throughout this paper, we denote that ‖ ⋅ ‖ϝ = 𝐷(⋅, 0). √(𝑥 − 𝑠)2 + (𝑦 − 𝑡)2 ≤ 𝛿} (1) (ii) (3) is called the modulus of oscillation of 𝑓 on [𝑎, 𝑏] × [𝑐, 𝑑]. Also, if 𝑓 ∈ 𝐶ϝ ([𝑎, 𝑏] × [𝑐, 𝑑]) (i.e., 𝑓 : [𝑎, 𝑏] × [𝑐, 𝑑] → 𝑅ϝ is continuous on [𝑎, 𝑏] × [𝑐, 𝑑]), then 𝜔[𝑎,𝑏]×[𝑐,𝑑] (𝑓, 𝛿) is called uniform modulus of continuity of 𝑓. The following properties will be very useful in what follows. The proofs of these properties in on (...truncated)