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Approximation of Fuzzy Integrals Using Fuzzy Bernstein Polynomials

Approximation of Fuzzy Integrals Using Fuzzy Bernstein Polynomials Fuzzy Inf. Eng. (2012) 4: 415-423 DOI 10.1007/s12543-012-0124-y ORIGINAL ARTICLE Approximation of Fuzzy Integrals Using Fuzzy Bernstein Polynomials Reza Ezzati · Shokrollah Ziari Received: 6 July 2011/ Revised: 2 September 2012/ Accepted: 20 October 2012/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract In this paper, we approximate the integration of continuous fuzzy real number valued function of one and two variables. To do this, we use Bernstein-type fuzzy polynomials. Moreover, we obtain the error estimates for these approximations in terms of the modulus of continuity. Keywords Fuzzy integral· Bernstein polynomial· Modulus of continuity 1. Introduction The concept of fuzzy measures and fuzzy integrals for single-valued mappings which are useful in several applied fields like mathematical economics, optimal control the- ory and engineering was initiated by Sugeno [18]. Particularly, they have been studied by Ralescu and Adams [23], Wang [24], and others. In [20-22], the authors studied several kinds of integrals based on the classical Lebesgue integral for set-valued map- pings. Using the Lebesque type concept of integration, Kaleva [4] defined the integral of fuzzy function. A survey on the subject of fuzzy integration was done by many authors [2, 3, 5, 7, 9-11]. Some numerical methods for solving fuzzy integrals were presented in [6, 13-16]. In [16], the authors presented quadrature rules to approximate fuzzy multiple inte- grals. The fuzzy-Riemann integral and its numerical integration was investigated by Wu in [9]. In [12], the authors introduced some quadrature rules for the integral of fuzzy number valued functions. The Newton Cot’s methods for the integration of fuzzy functions were presented in [13]. The Romberg integration of fuzzy functions were applied in [14]. Guassian quadratures to approximate fuzzy integrals were con- sidered in [15]. Reza Ezzati () · Shokrollah Ziari Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran email: [email protected] 416 Reza Ezzati · Shokrollah Ziari (2012) The main aim of this paper is to present approximation of the integration of con- tinuous fuzzy real number valued functions of one and two variables using Bernstein- type fuzzy polynomials. Moreover, we obtain the error estimates for these approxi- mations in terms of the modulus of continuity. This paper is organized as follows. Some theoretical background that are needed in the rest of the paper are briefly reviewed in Section 2. In Section 3, first, the approximate formulas for evaluating fuzzy-Riemann integrals using Bernstein-type fuzzy polynomials are introduced. Then, error bounds for the formulas are obtained by using the modulus of continuity. Illustrative examples for the efficiency of the proposed method presented in Section 3 are given in Section 4. Finally, this paper is concluded in Section 5. 2. Preliminaries Definition 2.1 A fuzzy number is a mapping u : R → I = [0, 1] with the following properties (see [2]): 1) u is normal, i.e., ∃x ∈ R such that u(x ) = 1; 0 0 2) u is fuzzy convex set (i.e., u(λx + (1 − λ)y) ≥ min{u(x), u(y)}∀x, y ∈ R,λ ∈ [0, 1]); 3) u is upper semi-continuous on R; 4) {x ∈ R : u(x) > 0} is compact, where A denotes the closure of A. The set of fuzzy numbers on R is denoted by R . Definition 2.2 [6] Suppose that u ∈ R . The α-level set of u is denoted by [u] and α 0 defined by [u] = {x ∈ R; u(x) ≥ α}, where 0<α ≤ 1. Also, [u] is called the support of u and it is given as [u] = {x ∈ R; u(x) > 0}. It follows that the level sets of u are closed and bounded intervals in R. It is well-known that the addition and multiplication operations of real numbers can be extended to R . In other words, for u, v ∈ R and λ ∈ R, we define uniquely F F the sum u⊕ v and the product λ u by α α α α α [u⊕ v] = [u] + [v] , [λ u] = λ [u] , ∀α ∈ [0, 1], α α α where [u] +[v] means the usual addition of two intervals (as subsets of R) andλ[u] means the usual product between a scalar and a subset of R. We use the same symbol both for the sum of real numbers and for the sum ⊕ (when the terms are fuzzy numbers). Definition 2.3 [6] An arbitrary fuzzy number is represented, in parametric form, by an ordered pair of functions (u (r), u (r)), 0 ≤ r ≤ 1, which satisfy the following − + requirements: 1) u (r) is a bounded left continuous nondecreasing function over [0,1], 2) u (r) is a bounded left continuous nonincreasing function over [0,1], + Fuzzy Inf. Eng. (2012) 4: 415-423 417 3) u (r) ≤ u (r), 0 ≤ r ≤ 1. − + Definition 2.4 [11] For arbitrary fuzzy numbers u, v ∈ R , the quantity α α α α D(u, v) = sup max{|u − v |,|u − v |} − − + + α∈[0,1] α α α α α α is the distance between u and v, where [u] = [u , u ], [v] = [v , v ]. It is proved − + − + that (R , D) is a complete metric space with the properties 1) D(u⊕ w, v⊕ w) = D(u, v) ∀ u, v, w ∈ R , 2) D(ku, kv) = |k| D(u, v) ∀ u, v ∈ R ∀k ∈ R, 3) D(u⊕ v, w⊕ e) ≤ D(u, w)+ D(v, e) ∀ u, v, w, e ∈ R . Let f, g :[a, b] −→ R be fuzzy real number valued functions. The uniform distance between f, g is defined by D ( f, g) = sup{D( f (x), g(x); x ∈ [a, b]}. Definition 2.5 [10] Let f :[a, b] −→ R be a bounded function. Then the function ω ( f,·): R ∪{0}−→ R [a,b] + + ω ( f,δ) = sup{D( f (x), f (y)); x, y ∈ [a, b], |x− y|≤ δ}, [a,b] where R is the set of positive real numbers, is called the modulus of continuity of f on [a, b]. Some properties of the modulus of continuity are presented below: Theorem 2.1 [10] The following properties holds: 1) D( f (x), f (y)) ≤ ω ( f,|x− y|) f or any x, y ∈ [a, b], [a,b] 2) ω ( f,δ) is increasing f unction o f δ, [a,b] 3) ω ( f, 0) = 0, [a,b] 4) ω ( f,δ +δ ) ≤ ω ( f,δ )+ω ( f,δ ) f or any δ ,δ ≥ 0, [a,b] 1 2 [a,b] 1 [a,b] 2 1 2 ( ) ( ) 5) ω f, nδ ≤ nω f,δ f or any δ ≥ 0,n ∈ N, [a,b] [a,b] 6) ω ( f,λδ) ≤ (λ+ 1)ω ( f,δ) f or any δ,λ ≥ 0, [a,b] [a,b] 7) If [c, d] ⊆ [a, b], then ω ( f,δ) ≤ ω ( f,δ). [c,d] [a,b] A fuzzy real number valued function f :[a, b] −→ R is said to be continuous in x ∈ [a, b], if for eachε> 0, there isδ> 0 such that D( f (x), f (x ))<ε, whenever x ∈ 0 0 [a, b] and |x− x |<δ. We say that f is fuzzy continuous on [a, b]if f is continuous at each x ∈ [a, b], and denote the space of all such functions by C ([a, b]) [11]. 0 F Definition 2.6 [17] Let f :[a, b] −→ R . f is fuzzy-Riemann integrable to I ∈ R F F if for anyε> 0, there existsδ> 0 such that for any division P = [u, v];ξ of [a, b] { } with the norms Δ(p)<δ, we have ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ∗ ⎟ ⎜ ⎟ D ⎜ (v− u) f (ξ), I⎟ <ε, ⎝ ⎠ P 418 Reza Ezzati · Shokrollah Ziari (2012) where denotes the fuzzy summation. In this case, it is denoted by I = (FR) f (x) dx. Lemma 2.1 [17] If f, g :[a, b] ⊆ R → R are fuzzy continuous functions, then the function F :[a, b] → R by F(x) = D( f (x), g(x)) is continuous on [a, b] and b b b D (FR) f (x)dx, (FR) g(x)dx ≤ D( f (x), g(x))dx. a a a Definition 2.7 [8] For f :[a, b] → R , modulus of continuity ω 2( f,δ) is defined F [a,b] as follows: ω ( f,δ) [a,b] 2 2 2 = sup{D( f (u, v), f (x, y)) (u, v), (x, y) ∈ [a, b] , (u− x) + (v− y) ≤ δ}. For f ∈ C ([0, 1]), let us consider the Bernstein-type fuzzy polynomials (F) ∗ B ( f )(x) = f p (x) , n ∈ N , x ∈ [0, 1], n,k k=0 ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ k n−x ∗ ⎜ ⎟ where P (x) = ⎜ ⎟ x (1− x) and means addition with respect to⊕ in R [17]. n,k F ⎝ ⎠ It is obvious that P (x) ≥ 0, ∀x ∈ [0, 1], and P (x), P (x),··· , P (x) are n,k n,0 n,1 n,n linearly independent algebraic polynomials of degree ≤ n [17]. For f ∈ C [0, 1] , we recall the fuzzy two-dimensional Bernstein operators as follows: m n (F) k ∗ ∗ 2 2 B ( f )(x, y) = f ( , ) P (x)P (y), ∀(x, y) ∈ [0, 1] , ∀(m, n) ∈ N , m,n m, j n,k m n j=0 k=0 ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ m⎟ ⎜ m⎟ ⎜ ⎟ j m− j ⎜ ⎟ k n−k ∗ ⎜ ⎟ ⎜ ⎟ where P (x) = ⎜ ⎟ x (1− x) and P (y) = ⎜ ⎟ y (1− y) and denotes the m, j m,k ⎝ ⎠ ⎝ ⎠ j k fuzzy means addition with respect to ⊕ in R , for more details see [17]. 3. Main Results Now, we obtain approximation value of (FR) f (x)dx by using the Bernstein-type fuzzy polynomials. We observe that 1 1 (F) (FR) f (x)dx  (FR) B ( f )(x)dx 0 0 = (FR) f p (x)dx n,k k=0 = f P (x)dx n,k k=0 ⎛ ⎞ ⎜ ⎟ k ⎜ n⎟ ∗ ⎜ ⎟ k n−k ⎜ ⎟ = f x (1− x) dx. ⎜ ⎟ ⎝ ⎠ k=0 We know that m!n! n m x (1− x) dx = . (1) (m+ n+ 1)! 0 Fuzzy Inf. Eng. (2012) 4: 415-423 419 Using (1) in the above equation, we have ⎛ ⎞ n n ⎜ ⎟ k ⎜ n⎟ 1 k!(n− k)! k 1 ∗ ⎜ ⎟ ∗ ⎜ ⎟ (FR) f (x)dx  f ⎜ ⎟ = f · ⎝ ⎠ n n+ 1 n! n n+ 1 k=0 k=0 Therefore, we conclude that 1 k (FR) f (x)dx  f · (2) n+ 1 n k=0 Theorem 3.1 For any f ∈ C ([0, 1]), we have ⎛ ⎞ ⎜ ⎟ 1 k 1 ⎜ ⎟ ⎜ ∗ ⎟ ⎜ ⎟ D ⎜ (FR) f (x)dx, f ⎟ ≤ ω f, . [0,1] ⎝ ⎠ n+ 1 n n+ 1 k=0 Proo f We observe that ⎛ ⎞ ⎜ ⎟ ⎜ 1 k ⎟ ⎜ ∗ ⎟ ⎜ ⎟ D (FR) f (x)dx, f ⎜ ⎟ ⎝ ⎠ n+ 1 n k=0 ⎛ ⎞ k+1 n n ⎜ ⎟ n+1 1 k ⎜ ⎟ ⎜ ∗ ∗ ⎟ ⎜ ⎟ = D ⎜ (FR) f (x)dx, f ⎟ ⎝ ⎠ n+ 1 n k=0 n+1 k=0 ⎛ ⎞ k+1 k+1 ⎜ n+1 n+1 ⎟ ⎜ k ⎟ ⎜ ⎟ ⎜ ⎟ ≤ D (FR) f (x)dx, (FR) 1 f dx ⎜ ⎟ ⎝ ⎠ k k n+1 n+1 k=0 k+1 n+1 ≤ D f (x), f dx k=0 n+1 k+1 n+1 k k ≤ sup{D f (x), f ; |x− |≤ δ}dx. n n n+1 k=0 Since k k+ 1 −k k k+ 1 k k+ 1 k ≤ x ≤ ⇒ ≤ x− ≤ − ≤ − . n+ 1 n+ 1 n(n+ 1) n n+ 1 n n+ 1 n+ 1 Hence k 1 |x− |≤ . n n+ 1 Thus k+1 n+1 1 1 1 n k D (FR) f (x)dx, f ≤ ω k k+1 f, dx n+1 k=0 n [ , ] n+1 n+1 n+ 1 k=0 n+1 1 1 = ω k k+1 f, [ , ] n+1 n+1 n+ 1 n+ 1 k=0 ≤ ω f, . [0,1] n+ 1 Now, we obtain approximation value of 1 1 (FR) f (x, y)dxdy 0 0 420 Reza Ezzati · Shokrollah Ziari (2012) by using bivariate Bernstein-type fuzzy polynomials. We have 1 1 1 1 (F) (FR) f (x, y)dxdy  (FR) B ( f )(x, y)dxdy m,n 0 0 0 0 ⎛ ⎞ ⎛ ⎞ m n 1 1 ⎜ ⎟ ⎜ ⎟ j k m n ⎜ ⎟ ⎜ ⎟ ∗ ∗ ⎜ ⎟ ⎜ ⎟ j m− j k n−k ⎜ ⎟ ⎜ ⎟ = (FR) f , ⎜ ⎟ ⎜ ⎟ x (1− x) y (1− y) dxdy ⎝ ⎠ ⎝ ⎠ m n j k 0 0 j=0 k=0 ⎛ ⎞ ⎛ ⎞ m n 1 1 ⎜ ⎟ ⎜ ⎟ j k ⎜ m⎟ ⎜ n⎟ ⎜ ⎟ ⎜ ∗ ∗ ⎟ j m− j k n−k ⎜ ⎟ ⎜ ⎟ = f , ⎜ ⎟ ⎜ ⎟ x (1− x) y (1− y) dxdy ⎝ ⎠ ⎝ ⎠ m n j k 0 0 j=0 k=0 m n 1 j k ∗ ∗ = f , . (m+ 1)(n+ 1) m n j=0 k=0 So, we have m n 1 1 1 j k ∗ ∗ (FR) f (x, y)dxdy  f , . (3) (m+ 1)(n+ 1) m n 0 0 j=0 k=0 Theorem 3.2 Let f ∈ C ([0, 1] ). Then ⎛ ⎞ m n ⎜ 1 1   ⎟ ⎜ ⎟ 1 j k ⎜ ⎟ ∗ ∗ ⎜ ⎟ ⎜ ⎟ D (FR) f (x, y)dxdy, f , ⎜ ⎟ ⎝ ⎠ (m+ 1)(n+ 1) m n 0 0 j=0 k=0 m+ n+ 2 ≤ ω 2( f, ). [0,1] (m+ 1)(n+ 1) Proo f Clearly, we have ⎛ ⎞ m n 1 1 ⎜   ⎟ ⎜ 1 j k ⎟ ⎜ ⎟ ∗ ∗ ⎜ ⎟ ⎜ ⎟ D (FR) f (x, y)dxdy, f , ⎜ ⎟ ⎝ ⎠ (m+ 1)(n+ 1) m n 0 0 j=0 k=0 ⎛ ⎞ j+1 k+1 m n m n ⎜     ⎟ m+1 n+1 ⎜ ⎟ 1 j k ⎜ ⎟ ∗ ∗ ∗ ∗ ⎜ ⎟ = D ⎜ (FR) f (x, y)dxdy, f , ⎟ ⎜ ⎟ ⎝ ⎠ j k (m+ 1)(n+ 1) m n m+1 n+1 j=0 k=0 j=0 k=0 ⎛ ⎞ j+1 j+1 k+1 k+1 m n ⎜ ⎟ m+1 n+1 m+1 n+1 j k ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ≤ D ⎜ (FR) f (x, y)dxdy, (FR) 1 f , dxdy⎟ ⎝ ⎠ j j k k m n j=0 k=0 m+1 n+1 m+1 n+1 j+1 k+1 m n m+1 n+1 j k ≤ D f (x, y), f , dxdy j k m n j=0 k=0 m+1 n+1 j+1 k+1 m n m+1 n+1 j k j k ≤ sup{D f (x, y), f , ; x− + y− ≤ δ}dxdy. j k m n m n j=0 k=0 m+1 n+1 j j+ 1 k k+ 1 According to this fact that ≤ x ≤ and ≤ y ≤ , we conclude m+ 1 m+ 1 n+ 1 n+ 1 that j 1 k 1 |x− |≤ , |y− |≤ . m m+ 1 n n+ 1 Hence j k 1 1 x− + y− ≤ + . m n m+ 1 n+ 1 Fuzzy Inf. Eng. (2012) 4: 415-423 421 Thus ⎛ ⎞ m n ⎜ 1 1   ⎟ ⎜ ⎟ 1 j k ⎜ ⎟ ∗ ∗ ⎜ ⎟ ⎜ ⎟ D f (x, y)dxdy, f , ⎜ ⎟ ⎝ ⎠ (m+ 1)(n+ 1) m n 0 0 j=0 k=0 m n 1 1 1 ≤ ω 2( f, + ) [0,1] (m+ 1)(n+ 1) m+ 1 n+ 1 j=0 k=0 m+ n+ 2 = ω ( f, ). [0,1] (m+ 1)(n+ 1) 4. Numerical Examples In this section, we apply the proposed methods in Section 3 in some examples. We compare results with exact solutions. Example 4.1 Consider the following fuzzy integral (FR) k( )dx, k = (r, 2− r), r ∈ [0, 1]. 2+ sin(10πx) The exact solution in this case is given by (1.1547)(r, 2 − r). By using proposed method in (2), we present approximate solution to this example for different values of n in Table 1. Table 1: Numerical results for Examples 4.1-4.2. n Approximate solution for Example 4.1 Approximate solution for Example 4.2 10 1.0000 (r , 2-r) 0.7818 (r-1 , 1-r) 20 1.1587 (r , 2-r) 0.7836 (r-1 , 1-r) 30 1.1547 (r , 2-r) 0.7842 (r-1 , 1-r) Example 4.2 Consider the following fuzzy integral (FR) k dx, k = (r− 1, 1− r), r ∈ [0, 1]. x + 1 The exact solution of this example is (r− 1, 1− r). To compare approximate solu- tion for different values of n and exact solution, see Table 1. Example 4.3 Consider the following fuzzy integral 1 1 (FR) kye dxdy, k = (r− 1, 1− r), r ∈ [0, 1]. 0 0 (e− 1) The exact solution of this example is (r− 1, 1− r). By using proposed method in (3), we present approximate solution to this example for different values of n in Table 2. 422 Reza Ezzati · Shokrollah Ziari (2012) Table 2: Approximate solutions for Examples 4.3-4.4. nm Example 4.3 Example 4.4 10 10 0.8662 (r-1 , 1-r) 0.2870 (r-1 , 1-r) 20 20 0.8627 (r-1 , 1-r) 0.3025 (r-1 , 1-r) 30 30 0.8615 (r-1 , 1-r) 0.3078 (r-1 , 1-r) Example 4.4 Consider the following fuzzy integral 1 1 (FR) ky sin(πx)dxdy, k = (r− 1, 1− r), r ∈ [0, 1]. 0 0 The exact solution is (r− 1, 1− r). For comparing approximate and exact solutions, see Table 2. 5. Conclusion In this paper, by using monotone and bivariate Bernstein-type fuzzy polynomials, we present numerical methods for approximate fuzzy-Riemann integrals. Then we prove the error estimates of the methods in terms of modulus of continuity. Acknowledgments The authors are grateful to anonymous referees for their constructive comments and suggestions. References 1. Dubois D, Prade H (1982) Towards fuzzy differential calculus. Fuzzy Sets and Systems 8: 225-233 2. Goetschel R, Voxman W (1986) Elementary fuzzy calculus. Fuzzy Sets and Systems 18: 31-43 3. Puri P, Ralescu D (1986) Fuzzy random variables. J. Math. Anal. Appl. 114: 409-422 4. Kaleva O (1987) Fuzzy differential equations. Fuzzy Sets and Systems 24: 301-317 5. Nanda S (1989) On integration of fuzzy mappings. Fuzzy Sets and Systems 32: 95-101 6. Friedman M, Ma M, Kandel A (1996) Numerical methods for calculating the fuzzy integral. Fuzzy Sets and Systems 83: 57-62S 7. Zimmerman H J (1996) Fuzzy set theory and its applications. New York: Kluwer Academic 8. Gal S G (2000) Approximation theory in fuzzy setting. Chapter 13 in Handbook of Analytic- Computational Methods in Applied Mathematics, Edited by G. Anastassiou, New York: Chapman & CRC 9. Wu H C (2000) The fuzzy Riemann integral and its numerical integration. Fuzzy Sets and Systems 110: 1-25 10. Anastassiou G A, Gal S G (2001) On a fuzzy trigonometric approximation theorem of Weierstrass- type. Journal of Fuzzy Mathematics 9(3): 701-708 11. Wu C X, Gong Z T (2001) On Henstock integral of fuzzy-number-valued functions (I). Fuzzy Sets and Systems 120: 523-532 12. Bede B, Gal S G (2004) Quadrature rules for integrals of fuzzy-number-valued functions. Fuzzy Sets and Systems 145: 359-380 13. Allahviranloo T (2005) Newton Cot’s methods for integration of fuzzy functions. Applied Mathe- matics and Computation 166: 339-348 14. Allahviranloo T (2005) Romberg integration for fuzzy functions. Applied Mathematics and Compu- tation 168: 866-876 Fuzzy Inf. Eng. (2012) 4: 415-423 423 15. Allahviranloo T, Otadi M (2005) Gaussian quadratures for approximate of fuzzy integrals. Applied Mathematics and Computation 170: 874-885 16. Allahviranloo T, Otadi M (2006) Gaussian quadratures for approximate of fuzzy multiple integrals. Applied Mathematics and Computation 172: 175-187 17. Anastassiou G A (2010) Fuzzy mathematics: Approximation theory. Springer-Verlag Berlin Heidel- berg 18. Sugeno M (1974) Theory of fuzzy integrals and its applications. Ph.D. Dissertation, Tokyo Inst. of Tech. 19. Aumann j (1965) Integrals of set-valued functions. J. Math. Anal. Appl. 12: 1-12 20. Dubois D, Prade H (1980) Fuzzy sets and systems: theory and applications. New York: Academic Press 21. Hiai F, Umegaki H (1977) Integrals, conditional expectations, and martingales of multivalued func- tions. J. Multivariate Anal. 7: 149-182 22. Papagergiou N S (1985) On the theory of Banach space valued multifunctions. 1. Integration and conditional expectation. J. Multivariate Anal. 17: 185-206 23. Ralescu D, Adams G (1980) The fuzzy integrals, J. Math. Anal.Appl. 75: 562-570 24. Wang Z (1984) The autocontinuity of set-function and the fuzzy integral. J. Math. Anal. Appl. 99: 195-218 25. Salahshour S, Allahviranloo T (2012) Application of fuzzy differential transformation method for solving fuzzy integral equations. Applied Mathematical Modelling. In Press http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Fuzzy Information and Engineering Taylor & Francis

Approximation of Fuzzy Integrals Using Fuzzy Bernstein Polynomials

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Fuzzy Inf. Eng. (2012) 4: 415-423 DOI 10.1007/s12543-012-0124-y ORIGINAL ARTICLE Approximation of Fuzzy Integrals Using Fuzzy Bernstein Polynomials Reza Ezzati · Shokrollah Ziari Received: 6 July 2011/ Revised: 2 September 2012/ Accepted: 20 October 2012/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract In this paper, we approximate the integration of continuous fuzzy real number valued function of one and two variables. To do this, we use Bernstein-type fuzzy polynomials. Moreover, we obtain the error estimates for these approximations in terms of the modulus of continuity. Keywords Fuzzy integral· Bernstein polynomial· Modulus of continuity 1. Introduction The concept of fuzzy measures and fuzzy integrals for single-valued mappings which are useful in several applied fields like mathematical economics, optimal control the- ory and engineering was initiated by Sugeno [18]. Particularly, they have been studied by Ralescu and Adams [23], Wang [24], and others. In [20-22], the authors studied several kinds of integrals based on the classical Lebesgue integral for set-valued map- pings. Using the Lebesque type concept of integration, Kaleva [4] defined the integral of fuzzy function. A survey on the subject of fuzzy integration was done by many authors [2, 3, 5, 7, 9-11]. Some numerical methods for solving fuzzy integrals were presented in [6, 13-16]. In [16], the authors presented quadrature rules to approximate fuzzy multiple inte- grals. The fuzzy-Riemann integral and its numerical integration was investigated by Wu in [9]. In [12], the authors introduced some quadrature rules for the integral of fuzzy number valued functions. The Newton Cot’s methods for the integration of fuzzy functions were presented in [13]. The Romberg integration of fuzzy functions were applied in [14]. Guassian quadratures to approximate fuzzy integrals were con- sidered in [15]. Reza Ezzati () · Shokrollah Ziari Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran email: [email protected] 416 Reza Ezzati · Shokrollah Ziari (2012) The main aim of this paper is to present approximation of the integration of con- tinuous fuzzy real number valued functions of one and two variables using Bernstein- type fuzzy polynomials. Moreover, we obtain the error estimates for these approxi- mations in terms of the modulus of continuity. This paper is organized as follows. Some theoretical background that are needed in the rest of the paper are briefly reviewed in Section 2. In Section 3, first, the approximate formulas for evaluating fuzzy-Riemann integrals using Bernstein-type fuzzy polynomials are introduced. Then, error bounds for the formulas are obtained by using the modulus of continuity. Illustrative examples for the efficiency of the proposed method presented in Section 3 are given in Section 4. Finally, this paper is concluded in Section 5. 2. Preliminaries Definition 2.1 A fuzzy number is a mapping u : R → I = [0, 1] with the following properties (see [2]): 1) u is normal, i.e., ∃x ∈ R such that u(x ) = 1; 0 0 2) u is fuzzy convex set (i.e., u(λx + (1 − λ)y) ≥ min{u(x), u(y)}∀x, y ∈ R,λ ∈ [0, 1]); 3) u is upper semi-continuous on R; 4) {x ∈ R : u(x) > 0} is compact, where A denotes the closure of A. The set of fuzzy numbers on R is denoted by R . Definition 2.2 [6] Suppose that u ∈ R . The α-level set of u is denoted by [u] and α 0 defined by [u] = {x ∈ R; u(x) ≥ α}, where 0<α ≤ 1. Also, [u] is called the support of u and it is given as [u] = {x ∈ R; u(x) > 0}. It follows that the level sets of u are closed and bounded intervals in R. It is well-known that the addition and multiplication operations of real numbers can be extended to R . In other words, for u, v ∈ R and λ ∈ R, we define uniquely F F the sum u⊕ v and the product λ u by α α α α α [u⊕ v] = [u] + [v] , [λ u] = λ [u] , ∀α ∈ [0, 1], α α α where [u] +[v] means the usual addition of two intervals (as subsets of R) andλ[u] means the usual product between a scalar and a subset of R. We use the same symbol both for the sum of real numbers and for the sum ⊕ (when the terms are fuzzy numbers). Definition 2.3 [6] An arbitrary fuzzy number is represented, in parametric form, by an ordered pair of functions (u (r), u (r)), 0 ≤ r ≤ 1, which satisfy the following − + requirements: 1) u (r) is a bounded left continuous nondecreasing function over [0,1], 2) u (r) is a bounded left continuous nonincreasing function over [0,1], + Fuzzy Inf. Eng. (2012) 4: 415-423 417 3) u (r) ≤ u (r), 0 ≤ r ≤ 1. − + Definition 2.4 [11] For arbitrary fuzzy numbers u, v ∈ R , the quantity α α α α D(u, v) = sup max{|u − v |,|u − v |} − − + + α∈[0,1] α α α α α α is the distance between u and v, where [u] = [u , u ], [v] = [v , v ]. It is proved − + − + that (R , D) is a complete metric space with the properties 1) D(u⊕ w, v⊕ w) = D(u, v) ∀ u, v, w ∈ R , 2) D(ku, kv) = |k| D(u, v) ∀ u, v ∈ R ∀k ∈ R, 3) D(u⊕ v, w⊕ e) ≤ D(u, w)+ D(v, e) ∀ u, v, w, e ∈ R . Let f, g :[a, b] −→ R be fuzzy real number valued functions. The uniform distance between f, g is defined by D ( f, g) = sup{D( f (x), g(x); x ∈ [a, b]}. Definition 2.5 [10] Let f :[a, b] −→ R be a bounded function. Then the function ω ( f,·): R ∪{0}−→ R [a,b] + + ω ( f,δ) = sup{D( f (x), f (y)); x, y ∈ [a, b], |x− y|≤ δ}, [a,b] where R is the set of positive real numbers, is called the modulus of continuity of f on [a, b]. Some properties of the modulus of continuity are presented below: Theorem 2.1 [10] The following properties holds: 1) D( f (x), f (y)) ≤ ω ( f,|x− y|) f or any x, y ∈ [a, b], [a,b] 2) ω ( f,δ) is increasing f unction o f δ, [a,b] 3) ω ( f, 0) = 0, [a,b] 4) ω ( f,δ +δ ) ≤ ω ( f,δ )+ω ( f,δ ) f or any δ ,δ ≥ 0, [a,b] 1 2 [a,b] 1 [a,b] 2 1 2 ( ) ( ) 5) ω f, nδ ≤ nω f,δ f or any δ ≥ 0,n ∈ N, [a,b] [a,b] 6) ω ( f,λδ) ≤ (λ+ 1)ω ( f,δ) f or any δ,λ ≥ 0, [a,b] [a,b] 7) If [c, d] ⊆ [a, b], then ω ( f,δ) ≤ ω ( f,δ). [c,d] [a,b] A fuzzy real number valued function f :[a, b] −→ R is said to be continuous in x ∈ [a, b], if for eachε> 0, there isδ> 0 such that D( f (x), f (x ))<ε, whenever x ∈ 0 0 [a, b] and |x− x |<δ. We say that f is fuzzy continuous on [a, b]if f is continuous at each x ∈ [a, b], and denote the space of all such functions by C ([a, b]) [11]. 0 F Definition 2.6 [17] Let f :[a, b] −→ R . f is fuzzy-Riemann integrable to I ∈ R F F if for anyε> 0, there existsδ> 0 such that for any division P = [u, v];ξ of [a, b] { } with the norms Δ(p)<δ, we have ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ∗ ⎟ ⎜ ⎟ D ⎜ (v− u) f (ξ), I⎟ <ε, ⎝ ⎠ P 418 Reza Ezzati · Shokrollah Ziari (2012) where denotes the fuzzy summation. In this case, it is denoted by I = (FR) f (x) dx. Lemma 2.1 [17] If f, g :[a, b] ⊆ R → R are fuzzy continuous functions, then the function F :[a, b] → R by F(x) = D( f (x), g(x)) is continuous on [a, b] and b b b D (FR) f (x)dx, (FR) g(x)dx ≤ D( f (x), g(x))dx. a a a Definition 2.7 [8] For f :[a, b] → R , modulus of continuity ω 2( f,δ) is defined F [a,b] as follows: ω ( f,δ) [a,b] 2 2 2 = sup{D( f (u, v), f (x, y)) (u, v), (x, y) ∈ [a, b] , (u− x) + (v− y) ≤ δ}. For f ∈ C ([0, 1]), let us consider the Bernstein-type fuzzy polynomials (F) ∗ B ( f )(x) = f p (x) , n ∈ N , x ∈ [0, 1], n,k k=0 ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ k n−x ∗ ⎜ ⎟ where P (x) = ⎜ ⎟ x (1− x) and means addition with respect to⊕ in R [17]. n,k F ⎝ ⎠ It is obvious that P (x) ≥ 0, ∀x ∈ [0, 1], and P (x), P (x),··· , P (x) are n,k n,0 n,1 n,n linearly independent algebraic polynomials of degree ≤ n [17]. For f ∈ C [0, 1] , we recall the fuzzy two-dimensional Bernstein operators as follows: m n (F) k ∗ ∗ 2 2 B ( f )(x, y) = f ( , ) P (x)P (y), ∀(x, y) ∈ [0, 1] , ∀(m, n) ∈ N , m,n m, j n,k m n j=0 k=0 ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ m⎟ ⎜ m⎟ ⎜ ⎟ j m− j ⎜ ⎟ k n−k ∗ ⎜ ⎟ ⎜ ⎟ where P (x) = ⎜ ⎟ x (1− x) and P (y) = ⎜ ⎟ y (1− y) and denotes the m, j m,k ⎝ ⎠ ⎝ ⎠ j k fuzzy means addition with respect to ⊕ in R , for more details see [17]. 3. Main Results Now, we obtain approximation value of (FR) f (x)dx by using the Bernstein-type fuzzy polynomials. We observe that 1 1 (F) (FR) f (x)dx  (FR) B ( f )(x)dx 0 0 = (FR) f p (x)dx n,k k=0 = f P (x)dx n,k k=0 ⎛ ⎞ ⎜ ⎟ k ⎜ n⎟ ∗ ⎜ ⎟ k n−k ⎜ ⎟ = f x (1− x) dx. ⎜ ⎟ ⎝ ⎠ k=0 We know that m!n! n m x (1− x) dx = . (1) (m+ n+ 1)! 0 Fuzzy Inf. Eng. (2012) 4: 415-423 419 Using (1) in the above equation, we have ⎛ ⎞ n n ⎜ ⎟ k ⎜ n⎟ 1 k!(n− k)! k 1 ∗ ⎜ ⎟ ∗ ⎜ ⎟ (FR) f (x)dx  f ⎜ ⎟ = f · ⎝ ⎠ n n+ 1 n! n n+ 1 k=0 k=0 Therefore, we conclude that 1 k (FR) f (x)dx  f · (2) n+ 1 n k=0 Theorem 3.1 For any f ∈ C ([0, 1]), we have ⎛ ⎞ ⎜ ⎟ 1 k 1 ⎜ ⎟ ⎜ ∗ ⎟ ⎜ ⎟ D ⎜ (FR) f (x)dx, f ⎟ ≤ ω f, . [0,1] ⎝ ⎠ n+ 1 n n+ 1 k=0 Proo f We observe that ⎛ ⎞ ⎜ ⎟ ⎜ 1 k ⎟ ⎜ ∗ ⎟ ⎜ ⎟ D (FR) f (x)dx, f ⎜ ⎟ ⎝ ⎠ n+ 1 n k=0 ⎛ ⎞ k+1 n n ⎜ ⎟ n+1 1 k ⎜ ⎟ ⎜ ∗ ∗ ⎟ ⎜ ⎟ = D ⎜ (FR) f (x)dx, f ⎟ ⎝ ⎠ n+ 1 n k=0 n+1 k=0 ⎛ ⎞ k+1 k+1 ⎜ n+1 n+1 ⎟ ⎜ k ⎟ ⎜ ⎟ ⎜ ⎟ ≤ D (FR) f (x)dx, (FR) 1 f dx ⎜ ⎟ ⎝ ⎠ k k n+1 n+1 k=0 k+1 n+1 ≤ D f (x), f dx k=0 n+1 k+1 n+1 k k ≤ sup{D f (x), f ; |x− |≤ δ}dx. n n n+1 k=0 Since k k+ 1 −k k k+ 1 k k+ 1 k ≤ x ≤ ⇒ ≤ x− ≤ − ≤ − . n+ 1 n+ 1 n(n+ 1) n n+ 1 n n+ 1 n+ 1 Hence k 1 |x− |≤ . n n+ 1 Thus k+1 n+1 1 1 1 n k D (FR) f (x)dx, f ≤ ω k k+1 f, dx n+1 k=0 n [ , ] n+1 n+1 n+ 1 k=0 n+1 1 1 = ω k k+1 f, [ , ] n+1 n+1 n+ 1 n+ 1 k=0 ≤ ω f, . [0,1] n+ 1 Now, we obtain approximation value of 1 1 (FR) f (x, y)dxdy 0 0 420 Reza Ezzati · Shokrollah Ziari (2012) by using bivariate Bernstein-type fuzzy polynomials. We have 1 1 1 1 (F) (FR) f (x, y)dxdy  (FR) B ( f )(x, y)dxdy m,n 0 0 0 0 ⎛ ⎞ ⎛ ⎞ m n 1 1 ⎜ ⎟ ⎜ ⎟ j k m n ⎜ ⎟ ⎜ ⎟ ∗ ∗ ⎜ ⎟ ⎜ ⎟ j m− j k n−k ⎜ ⎟ ⎜ ⎟ = (FR) f , ⎜ ⎟ ⎜ ⎟ x (1− x) y (1− y) dxdy ⎝ ⎠ ⎝ ⎠ m n j k 0 0 j=0 k=0 ⎛ ⎞ ⎛ ⎞ m n 1 1 ⎜ ⎟ ⎜ ⎟ j k ⎜ m⎟ ⎜ n⎟ ⎜ ⎟ ⎜ ∗ ∗ ⎟ j m− j k n−k ⎜ ⎟ ⎜ ⎟ = f , ⎜ ⎟ ⎜ ⎟ x (1− x) y (1− y) dxdy ⎝ ⎠ ⎝ ⎠ m n j k 0 0 j=0 k=0 m n 1 j k ∗ ∗ = f , . (m+ 1)(n+ 1) m n j=0 k=0 So, we have m n 1 1 1 j k ∗ ∗ (FR) f (x, y)dxdy  f , . (3) (m+ 1)(n+ 1) m n 0 0 j=0 k=0 Theorem 3.2 Let f ∈ C ([0, 1] ). Then ⎛ ⎞ m n ⎜ 1 1   ⎟ ⎜ ⎟ 1 j k ⎜ ⎟ ∗ ∗ ⎜ ⎟ ⎜ ⎟ D (FR) f (x, y)dxdy, f , ⎜ ⎟ ⎝ ⎠ (m+ 1)(n+ 1) m n 0 0 j=0 k=0 m+ n+ 2 ≤ ω 2( f, ). [0,1] (m+ 1)(n+ 1) Proo f Clearly, we have ⎛ ⎞ m n 1 1 ⎜   ⎟ ⎜ 1 j k ⎟ ⎜ ⎟ ∗ ∗ ⎜ ⎟ ⎜ ⎟ D (FR) f (x, y)dxdy, f , ⎜ ⎟ ⎝ ⎠ (m+ 1)(n+ 1) m n 0 0 j=0 k=0 ⎛ ⎞ j+1 k+1 m n m n ⎜     ⎟ m+1 n+1 ⎜ ⎟ 1 j k ⎜ ⎟ ∗ ∗ ∗ ∗ ⎜ ⎟ = D ⎜ (FR) f (x, y)dxdy, f , ⎟ ⎜ ⎟ ⎝ ⎠ j k (m+ 1)(n+ 1) m n m+1 n+1 j=0 k=0 j=0 k=0 ⎛ ⎞ j+1 j+1 k+1 k+1 m n ⎜ ⎟ m+1 n+1 m+1 n+1 j k ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ≤ D ⎜ (FR) f (x, y)dxdy, (FR) 1 f , dxdy⎟ ⎝ ⎠ j j k k m n j=0 k=0 m+1 n+1 m+1 n+1 j+1 k+1 m n m+1 n+1 j k ≤ D f (x, y), f , dxdy j k m n j=0 k=0 m+1 n+1 j+1 k+1 m n m+1 n+1 j k j k ≤ sup{D f (x, y), f , ; x− + y− ≤ δ}dxdy. j k m n m n j=0 k=0 m+1 n+1 j j+ 1 k k+ 1 According to this fact that ≤ x ≤ and ≤ y ≤ , we conclude m+ 1 m+ 1 n+ 1 n+ 1 that j 1 k 1 |x− |≤ , |y− |≤ . m m+ 1 n n+ 1 Hence j k 1 1 x− + y− ≤ + . m n m+ 1 n+ 1 Fuzzy Inf. Eng. (2012) 4: 415-423 421 Thus ⎛ ⎞ m n ⎜ 1 1   ⎟ ⎜ ⎟ 1 j k ⎜ ⎟ ∗ ∗ ⎜ ⎟ ⎜ ⎟ D f (x, y)dxdy, f , ⎜ ⎟ ⎝ ⎠ (m+ 1)(n+ 1) m n 0 0 j=0 k=0 m n 1 1 1 ≤ ω 2( f, + ) [0,1] (m+ 1)(n+ 1) m+ 1 n+ 1 j=0 k=0 m+ n+ 2 = ω ( f, ). [0,1] (m+ 1)(n+ 1) 4. Numerical Examples In this section, we apply the proposed methods in Section 3 in some examples. We compare results with exact solutions. Example 4.1 Consider the following fuzzy integral (FR) k( )dx, k = (r, 2− r), r ∈ [0, 1]. 2+ sin(10πx) The exact solution in this case is given by (1.1547)(r, 2 − r). By using proposed method in (2), we present approximate solution to this example for different values of n in Table 1. Table 1: Numerical results for Examples 4.1-4.2. n Approximate solution for Example 4.1 Approximate solution for Example 4.2 10 1.0000 (r , 2-r) 0.7818 (r-1 , 1-r) 20 1.1587 (r , 2-r) 0.7836 (r-1 , 1-r) 30 1.1547 (r , 2-r) 0.7842 (r-1 , 1-r) Example 4.2 Consider the following fuzzy integral (FR) k dx, k = (r− 1, 1− r), r ∈ [0, 1]. x + 1 The exact solution of this example is (r− 1, 1− r). To compare approximate solu- tion for different values of n and exact solution, see Table 1. Example 4.3 Consider the following fuzzy integral 1 1 (FR) kye dxdy, k = (r− 1, 1− r), r ∈ [0, 1]. 0 0 (e− 1) The exact solution of this example is (r− 1, 1− r). By using proposed method in (3), we present approximate solution to this example for different values of n in Table 2. 422 Reza Ezzati · Shokrollah Ziari (2012) Table 2: Approximate solutions for Examples 4.3-4.4. nm Example 4.3 Example 4.4 10 10 0.8662 (r-1 , 1-r) 0.2870 (r-1 , 1-r) 20 20 0.8627 (r-1 , 1-r) 0.3025 (r-1 , 1-r) 30 30 0.8615 (r-1 , 1-r) 0.3078 (r-1 , 1-r) Example 4.4 Consider the following fuzzy integral 1 1 (FR) ky sin(πx)dxdy, k = (r− 1, 1− r), r ∈ [0, 1]. 0 0 The exact solution is (r− 1, 1− r). For comparing approximate and exact solutions, see Table 2. 5. Conclusion In this paper, by using monotone and bivariate Bernstein-type fuzzy polynomials, we present numerical methods for approximate fuzzy-Riemann integrals. Then we prove the error estimates of the methods in terms of modulus of continuity. Acknowledgments The authors are grateful to anonymous referees for their constructive comments and suggestions. References 1. 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Salahshour S, Allahviranloo T (2012) Application of fuzzy differential transformation method for solving fuzzy integral equations. Applied Mathematical Modelling. In Press

Journal

Fuzzy Information and EngineeringTaylor & Francis

Published: Dec 1, 2012

Keywords: Fuzzy integral; Bernstein polynomial; Modulus of continuity

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