Romberg integration for fuzzy functions

Applied Mathematics and Computation 168 (2005) 866–876 www.elsevier.com/locate/amc Romberg integration for fuzzy functions Tofigh Allahviranloo Department of Mathematics, Science and Research Branch, Islamic Azad University, Post code 14778, Tehran, Hesarak, Poonak, Iran Abstract In this paper the integration formulas of Newton CotÕs methods with positive coefficient for fuzzy integrations in [T. Allahviranloo, Newton CotÕs methods for integration of fuzzy functions, in press] are considered and then are followed by Romberg integration to obtain improvements of the approximations of fuzzy integrations. The proposed algorithm is illustrated by solving some numerical examples. Ó 2004 Elsevier Inc. All rights reserved. Keywords: Romberg integration; Newton CotÕs methods; Fuzzy integral 1. Introduction With consideration of approximation theory, the integration problem plays major role in various areas such as mathematics, physics, statistics, engineering and social sciences. Since in many applications at least some of the systemÕs parameters and measurements are represented by fuzzy rather than crisp numbers, it is important to develop fuzzy integration and solve them. The concept of fuzzy numbers and arithmetic operations with these numbers were first introduced and investigated by Zadeh [3] and others. The topic of E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2004.09.036 T. Allahviranloo / Appl. Math. Comput. 168 (2005) 866–876 867 the Richardson extrapolation method for improving of the approximations of fuzzy differential equations is first introduced by Abbasbandy and Allahviranloo [2]. The topic of fuzzy integration is discussed by [9] and then is followed by numerical methods (Newton CotÕs) in [1]. The structure of this paper is organized as follows: In Section 2, we bring some basic definitions and results on fuzzy numbers and fuzzy integrations and the integration formulas of Newton CotÕs methods for fuzzy integration with PeanoÕs error representation theorem and convergence theorem. In Section 3, we introduce the Romberg method for improving of the solutions of fuzzy integrations. The proposed algorithm is illustrated by solving some examples in Section 4 and conclusions are drawn in Section 5. 2. Preliminaries We represent an arbitrary fuzzy number by an ordered pair of functions ðuðrÞ; uðrÞÞ; 0 6 r 6 1; which satisfy the following requirements: 1. u(r) is a bounded left continuous non decreasing function over [0,1]. 2. uðrÞ is a bounded left continuous non increasing function over [0,1]. 3. uðrÞ 6 uðrÞ; 0 6 r 6 1. A crisp number a is simply represented by uðrÞ ¼ uðrÞ ¼ a; 0 6 r 6 1: The set of all the fuzzy numbers is denoted by E1. Lemma 2.1. Let v, w 2 E1 and s be real number. Then for 0 6 r 6 1 u = v if and only if u(r) = v(r) and uðrÞ ¼ vðrÞ, v þ w ¼ ðvðrÞ þ wðrÞ; vðrÞ þ wðrÞÞ, v  w ¼ ðvðrÞ  wðrÞ; vðrÞ  wðrÞÞ, v:w ¼ ðminfvðrÞ  wðrÞ; vðrÞ  wðrÞ; vðrÞ  wðrÞ; vðrÞ  wðrÞg, maxfvðrÞ  wðrÞ; vðrÞ  wðrÞ; vðrÞ  wðrÞ; vðrÞ  wðrÞgÞ, sv ¼ sðvðrÞ; vðrÞÞ: See [8]. E1 with addition and multiplication as defined by Lemma 2.1 is a convex cone which is then embedded isomorphically and isometrically into a Banach space. Definition 2.1. For arbitrary fuzzy numbers u ¼ ðu; uÞ and u ¼ ðv; vÞ the quantity 868 T. Allahviranloo / Appl. Math. Comput. 168 (2005) 866–876 Dðu; vÞ ¼ sup fmax½juðrÞ  vðrÞj; juðrÞ  vðrÞj g 06r61 is the distance between u and v. It is shown [6] that E1, D is a complete metric space. Definition 2.2. A function f : R1 ! E1 is called a fuzzy function. If for arbitrary fixed t0 2 R1 and  > 0, a d > 0 such that jt  t0 j < d ) D½f ðtÞ; f ðt0 Þ <  exists, f is said to be continuous. Throughout this work we also consider fuzzy functions which are defined only over a finite interval [a, b]. We now follow Goetschel and Voxman [5] and define the integral of a fuzzy function using the Rieman integral concept. Definition 2.3. Let f : [a, b] ! E1. For each partition P = {t0, t1, . . . , tn} of [a, b] and for arbitrary ni : ti1 6 ni 6 ti, 1 6 i 6 n, let n X f ðni Þðti  ti1 Þ: Rp ¼ i¼1 The definite integral of f(t) over [a, b] is Z b f ðtÞ dt ¼ lim Rp ; max jti  ti1 j ! 0 16i6n a provided that this limit exists in the metric D. If the fuzzy function f(t) is continuous in the metric D, its definite integral exists [5]. Furthermore, ! Z Z b b f ðt; rÞ dt; f ðt; rÞ dt ¼ a Z b f ðt; rÞ dt a ! a ¼ Z ð2:1Þ b f ðt; rÞ dt: a It should be noted that the fuzzy integral can be also defined using the Lebesgue-type approach [4,7]. More details about properties of the fuzzy integral are given in [5,4]. From [1], let f be a fuzzy function. For any natural number n, the Newton CotÕs formulas Z b n X ba ; ð2:2Þ fi ai þ E; f i ¼ f ða þ ihÞ; h ¼ f ðxÞ dx ¼ h n a i¼1 T. Allahviranloo / Appl. Math. Comput. 168 (2005) 866–876 provide approximate values for follows: Z b f ðx; rÞ dx ¼ h a Z n X Rb a f ðxÞ dx. The parametric form of (2.2) is as ai f ðxi ; rÞ þ Eðf ; rÞ; i¼1 b f ðx; rÞ dx ¼ h a n X 869 ð2:3Þ ai f ðxi ; rÞ þ Eðf ; rÞ; 0 6 r 6 1: i¼1 P The weights ai, i = 1, . . . , n, are rational numbers with the property ni¼1 ai ¼ n: This follows (2.3) when applied to f ðx; rÞ ¼ f ðx; rÞ ¼ 1: It can be shown that the approximation error may be expressed as follows: Eðf ; rÞ ¼ hpþ1  K  f ðpÞ ðn; rÞ; Eðf ; rÞ ¼ hpþ1  K  f ðpÞ n 2 ða; bÞ; ð2:4Þ ðn; rÞ; n 2 ða; bÞ; 0 6 r 6 1: Here (a, b) denotes the open interval from a to b. The values of p and K depend only on n but not on the integrand f. For large n, some of the values ai become negative and the corresponding formulas are unsuitable for numerical purposes, as cancellations tend to occur in computing the sum (2.4). Let Qðf ; rÞ ¼ h n X ai f ðxi ; rÞ; i¼1 Qðf ; rÞ ¼ h n X ð2:5Þ ai f ðxi ; rÞ; 0 6 r 6 1: i¼1 Thus from (2.3) we have Iðf ; rÞ ¼ Z b f ðx; rÞdx ¼ Qðf ; rÞ þ Eðf ; rÞ; a Iðf ; rÞ ¼ Z ð2:6Þ b f ðx; rÞdx ¼ Qðf ; rÞ þ Eðf ; rÞ; 0 6 r 6 1: a The following theorem is proving that Qðf ; rÞ; Qðf ; rÞ converge to Iðf ; rÞ; Iðf ; rÞ respectively. Theorem 2.1. If f(t) is continuous (in the metric D). The convergence of Qðf ; rÞ; Qðf ; rÞ to Iðf ; rÞ; Iðf ; rÞ, respectively is uniform in r. Proof. See [1]. h 870 T. Allahviranloo / Appl. Math. Comput. 168 (2005) 866–876 2.1. PeanoÕs error representation From (2.3) we have Eðf ; rÞ ¼ Iðf ; rÞ  Qðf ; rÞ; Eðf ; rÞ ¼ Iðf ; rÞ  Qðf ; rÞ; ð2:7Þ 0 6 r 6 1: The integration error E(f) is a linear operator in parametric form Eðaf ðrÞ þ gðrÞÞ ¼ aEðf ; rÞ þ Eðg; rÞ; Eðaf ðrÞ þ gðrÞÞ ¼ aEðf ; rÞ þ Eðg; rÞ; for f ; g 2 Ve ; a 2 R; 0 6 r 6 1 on some suitable linear fuzzy function space Ve . The following elegant fuzzy integral representation of the error E(f) is a classical result due to Peano. Theorem 2.2. Suppose E(p) = 0 holds for all p 2 Pn, that is, every polynomial whose degree does not exceed n is integrated exactly. Them for all fuzzy functions in parametric form f ðrÞ; f ðrÞ 2 C nþ1 ½a; b ; 0 6 r 6 1, Z b Eðf ; rÞ ¼ f nþ1 ðt; rÞKðtÞ dt; a ð2:8Þ Z b nþ1 f ðt; rÞKðtÞ dt; 0 6 r 6 1; Eðf ; rÞ ¼ a where 1 n KðtÞ :¼ Ex ½ðx  tÞþ ; n! ðx  n tÞþ :¼  n ðx  tÞ ; x P t; 0; x < t; n and Ex ½ðx  tÞþ when the latter is considered as a function in x. Proof. See [1]. h 3. Romberg integration Romberg integration uses the Composite Trapezoidal rule to give preliminary approximations and then applies the Richardson extrapolation process to obtain improvements of the approximations. Recall from [2] that Richardson extrapolation can be performed on any approximation procedure of the form M  N ðhÞ ¼ K 1 h þ K 2 h2 þ    þ K n hn ; where the K1, K2, . . . , Kn are constants and N(h) is an approximation to the unknown value M. The truncation error in this formula is dominated by K1h 871 T. Allahviranloo / Appl. Math. Comput. 168 (2005) 866–876 when h is small, so this formula gives O(h) approximations. RichardsonÕs extrapolation uses an averaging technique to produce formulas with higherorder truncation error. In this section we will use extrapolation to approximate definite fuzzy integral. To begin the presentation of the Romberg integration scheme, recall that the Composite Trapezoidal rule for approximate definite integral of a fuzzy function f on interval [x0, xm] = [a, b] using m subintervals is " # m1 X h Iðf ; rÞ ¼ f ðx0 ; rÞ þ 2 f ðxi ; rÞ þ f ðxm ; rÞ þ Eðf ; rÞ; 2 i¼1 " # m1 X h f ðx0 ; rÞ þ 2 f ðxi ; rÞ þ f ðxm ; rÞ þ Eðf ; rÞ; 0 6 r 6 1: Iðf ; rÞ ¼ 2 i¼1 ð3:1Þ where h2 ðb  aÞf ð2Þ ðn; rÞ; 12 h2 ð2Þ Eðf ; rÞ ¼  ðb  aÞf ðn; rÞ; 12 Eðf ; rÞ ¼  ð3:2Þ n; n 2 ½x0 ; xn ; 0 6 r 6 1 [1]: We first obtain Composite Trapezoidal rule approximations with m1 = 1, m2 = 2, m3 = 4, . . . , and mn = 2n1, where n is positive integer. The values of the step size hk corresponding to mk are hk = (ba)/mk = (ba)/2k1. With this notion the Trapezoidal rule rule becomes " # k1 2X 1 hk Iðf ; rÞ ¼ f ðx0 ; rÞ þ 2 f ðxi ; rÞ þ f ðxm ; rÞ þ Eðf ; rÞ; 2 i¼1 " # k1 2X 1 hk f ðx0 ; rÞ þ 2 f ðxi ; rÞ þ f ðxm ; rÞ þ Eðf ; rÞ; 0 6 r 6 1; Iðf ; rÞ ¼ 2 i¼1 ð3:3Þ where h2k ðb  aÞf ð2Þ ðn; rÞ; 12 h2 ð2Þ Eðf ; rÞ ¼  k ðb  aÞf ðn; rÞ; 12 Eðf ; rÞ ¼  ð3:4Þ n; n 2 ½x0 ; xn ; 0 6 r 6 1: If the notions Qk,1(f;r) and Qk,1(f;r) are introduced to denote the portion of Eqs. (3.3) and (3.4) used for the Trapezoidal approximation, then: Q1;1 ðf ; rÞ ¼ h1 ½f ðx0 ; rÞ þ f ðxn ; rÞ ; 2 Q1;1 ðf ; rÞ ¼ h1 ½f ðx0 ; rÞ þ f ðxn ; rÞ ; 2 872 T. Allahviranloo / Appl. Math. Comput. 168 (2005) 866–876 h2 ½f ðx0 ; rÞ þ 2f ðx0 þ h2 Þ þ f ðxn ; rÞ 2 i xn  x0 h xn  x0 ¼ ;r f ðx0 ; rÞ þ f ðxn ; rÞ þ 2f x0 þ 4 2 1 ¼ ½Q1;1 ðf ; rÞ þ h1 f ðx0 þ h2 Þ ; 2 Q2;1 ðf ; rÞ ¼  h2  f ðx0 ; rÞ þ 2f ðx0 þ h2 Þ þ f ðxn ; rÞ 2 i xn  x0 h xn  x0 ;r f ðx0 ; rÞ þ f ðxn ; rÞ þ 2f x0 þ ¼ 4 2  1 ¼ Q1;1 ðf ; rÞ þ h1 f ðx0 þ h2 Þ ; 2 Q2;1 ðf ; rÞ ¼ 1 Q3;1 ðf ; rÞ ¼ fQ2;1 ðf ; rÞ þ h2 ½f ðx0 þ h3 Þ þ f ðx0 þ 3h3 Þ g; 2   1 Q3;1 ðf ; rÞ ¼ fQ2;1 ðf ; rÞ þ h2 f ðx0 þ h3 Þ þ f ðx0 þ 3h3 Þ g; 2 and, in general ( ) 2k2 X 1 Qk1;1 ðf ; rÞ þ hk1 Qk;1 ðf ; rÞ ¼ f ðx0 þ ð2j  1Þhk Þ ; 2 j¼1 ( ) 2k2 X 1 Qk1;1 ðf ; rÞ þ hk1 Qk;1 ðf ; rÞ ¼ f ðx0 þ ð2j  1Þhk Þ ; k ¼ 2; 3; . . . ; n: 2 j¼1 Richardson extrapolation will be used to speed the convergence. If f 2 C1[a, b], the Composite Trapezoidal rule can be written with an alternative error term in the form Iðf ; rÞ  Qk;1 ðf ; rÞ ¼ 1 X K i hk2i ¼ K 1 h2k þ Iðf ; rÞ  Qk;1 ðf ; rÞ ¼ K i h2i k ; i¼2 i¼1 1 X 1 X K 0i hk2i ¼ K 01 h2k þ 1 X ð3:5Þ K 0i h2i k ; i¼2 i¼1 where K i ; K 0i for each i are independent of hk and depends only on ð2i1Þ ð2i1Þ f ð2i1Þ ðaÞ; f ðaÞ and f ð2i1Þ ðbÞ; f ðbÞ. With the Composite Trapezoidal rule in this form, we can eliminate the term involving h2k by combining this equation with its counterpart with hk replaced by hkþ1 ¼ h2k 873 T. Allahviranloo / Appl. Math. Comput. 168 (2005) 866–876 Iðf ; rÞ  Qkþ1;1 ðf ; rÞ ¼ 1 X K i h2i k ; 4i 2i K 0i hkþ1 K 0i h2i k : 4i i¼1 Iðf ; rÞ  Qkþ1;1 ðf ; rÞ ¼ 1 X 1 1 X K i h2i K 1 h2k X k þ ¼ 4 22i i¼2 i¼1 1 1 0 2i 0 2 X X K i hk K h ¼ ¼ 1 kþ 2i 4 2 i¼1 i¼2 2i K i hkþ1 ¼ i¼1 ð3:6Þ Subtracting Eq. (3.5) from 4 times (3.6) and simplifying gives the Oðh4k Þ formula " #   1 X Qkþ1;1 ðf ; rÞ  Qk;1 ðf ; rÞ K i h2i 2i k  hk Iðf ; rÞ  Qkþ1;1 ðf ; rÞ þ ¼ 3 4i1 3 i¼2   1 X K i 1  4i1 2i ¼ hk ; 3 4i1 i¼2 " #   1 X Qkþ1;1 ðf ; rÞ  Qk;1 ðf ; rÞ K 0i h2i 2i k Iðf ; rÞ  Qkþ1;1 ðf ; rÞ þ  hk ¼ 3 3 4i1 i¼2   1 X K 0i 1  4i1 2i ¼ hk : 3 4i1 i¼2 Extrapolation can now be applied to this formula to obtain an Oðh6k Þ result, and so on. To simplify the notion we define Qk;1 ðf ; rÞ  Qk1;1 ðf ; rÞ ; 3 Qk;1 ðf ; rÞ  Qk1;1 ðf ; rÞ Qk;2 ðf ; rÞ ¼ Qk;1 ðf ; rÞ þ 3 Qk;2 ðf ; rÞ ¼ Qk;1 ðf ; rÞ þ for each k = 2, 3, . . . , n, and apply the Richardson extrapolation procedure to these values. Continuing this notion, we have, for each k = 2, 3, . . . , n and j = 2, . . . , k, an Oðh2j k Þ approximation formula defined by Qk;j ðf ; rÞ ¼ Qk;j1 ðf ; rÞ þ Qk;j ðf ; rÞ ¼ Qk;j1 ðf ; rÞ þ Qk;j1 ðf ; rÞ  Qk1;j1 ðf ; rÞ 4j1  1 Qk;j1 ðf ; rÞ  Qk1;j1 ðf ; rÞ 4j1  1 ; : We generate approximations until not only jQn1;n1 ðf ; rÞ  Qn;n ðf ; rÞj; jQn1;n1 ðf ; rÞ  Qn;n ðf ; rÞj are within the tolerance, but also jQn2;n2 ðf ; rÞ  Qn1;n1 ðf ; rÞj; jQn2;n2 ðf ; rÞ  Qn1;n1 ðf ; rÞj: 874 T. Allahviranloo / Appl. Math. Comput. 168 (2005) 866–876 Although not a universal safeguard, this will ensure that two differently generated sets of approximations agree within the specified tolerance before Qn;n ðf ; rÞ; Qn;n ðf ; rÞ, are accepted as sufficiently accurate. 4. Numerical examples In this section we illustrate and compared the methods in Sections 2 and 3 by solving some numerical examples. Example 4.1 [1]. Consider the following fuzzy integral: Z 1 0 e kx2 dx; e k ¼ ðr  1; 1  rÞ; the exact solution is 13 ðr  1; 1  rÞ. h ¼ 1; 1 h¼ ; 2 1 Q1;1 ðf ; rÞ ¼ ðr  1Þ; 2 3 Q2;1 ðf ; rÞ ¼ ðr  1Þ; 8 Q1;2 ðf ; rÞ ¼ 4Q2;1 ðf ; rÞ  Q1;1 ðf ; rÞ 1 ¼ ðr  1Þ; 3 3 Fig. 1. ð4:1Þ 875 T. Allahviranloo / Appl. Math. Comput. 168 (2005) 866–876 and 1 Q1;1 ðf ; rÞ ¼ ðr  1Þ; 2 1 3 h ¼ ; Q2;1 ðf ; rÞ ¼ ðr  1Þ; 2 8 4Q2;1 ðf ; rÞ  Q1;1 ðf ; rÞ 1 Q1;2 ðf ; rÞ ¼ ¼ ðr  1Þ: 3 3 h ¼ 1; It is clear that Q1,2 is the exact solution. The exact solution and Trapezoidal solutions with h = 1 and Romberg method are plotted and compared in Fig. 1. Example 4.2 [1]. Consider the following fuzzy integral: Z 1 0 e kx4 dx; e k ¼ ðr; 2  rÞ; ð4:2Þ the exact solution is 15 ðr; 2  rÞ. h ¼ 1; Q1;1 ðf ; rÞ ¼ 0:5r; 1 h ¼ ; Q2;1 ðf ; rÞ ¼ 0:28125r; Q1;2 ðf ; rÞ ¼ 0:208333r; 2 1 h ¼ ; Q3;1 ðf ; rÞ ¼ 0:220703r; Q2;2 ðf ; rÞ ¼ 0:200521r; 4 Q1;3 ðf ; rÞ ¼ 0:2r; and h ¼ 1; Q1;1 ðf ; rÞ ¼ 0:5ð2  rÞ; 1 h ¼ ; Q2;1 ðf ; rÞ ¼ 0:28125ð2  rÞ; Q1;2 ðf ; rÞ ¼ 0:208333ð2  rÞ; 2 1 h ¼ ; Q3;1 ðf ; rÞ ¼ 0:220703ð2  rÞ; Q2;2 ðf ; rÞ ¼ 0:200521ð2  rÞ; 4 Q1;3 ðf ; rÞ ¼ 0:2ð2  rÞ: Q1,3 is the exact solution. 5. Conclusion In this work first we applied the Newton CotÕs method with positive coefficients to solve fuzzy integral over a finite interval [a, b] and then improved solutions by using Romberg method with arbitrary tolerance. Since coefficients of Newton CotÕs are positive so we have two crisp equations and we can use Romberg method for two crisp equations. 876 T. Allahviranloo / Appl. Math. Comput. 168 (2005) 866–876 References [1] T. 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