Newton Cot’s methods for integration of fuzzy functions

Applied Mathematics and Computation 166 (2005) 339–348 www.elsevier.com/locate/amc Newton CotÕs methods for integration of fuzzy functions Tofigh Allahviranloo Department of Mathematics, Science and Research Branch, Islamic Azad University, Post code 14778, Hesarak, Poonak, Tehran, Iran Abstract In this paper the integration formulas of Newton CotÕs methods with positive coefficient for fuzzy integrations are discussed and then is followed by PeanoÕs error representation theorem and convergence theorem. The proposed algorithms are illustrated by solving some numerical examples. Ó 2004 Elsevier Inc. All rights reserved. Keywords: 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 E-mail addresses: [email protected], [email protected] 0096-3003/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2004.04.110 340 T. Allahviranloo / Appl. Math. Comput. 166 (2005) 339–348 introduced and investigated by Zadeh [1] and . . . The topic of fuzzy integration is discussed by [8]. 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. In Section 3, we introduce the integration formulas of Newton CotÕs methods for fuzzy integration with PeanoÕs error representation theorem and convergence theorem. The proposed algorithms are 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]. ðrÞ, 0 6 r 6 1. 3. uðrÞ 6 u 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 uðrÞ ¼ vðrÞ and  uðrÞ ¼ vðrÞ,  ðrÞÞ, v þ w ¼ ðvðrÞ þ wðrÞ; vðrÞ þ w  ðrÞ; vðrÞ
wðrÞÞ, v
w ¼ ðvðrÞ
w  ðrÞ; vðrÞ  wðrÞ; vðrÞ  w  ðrÞg; v  w ¼ ðminfvðrÞ  wðrÞ; vðrÞ  w  ðrÞ; vðrÞ  wðrÞ; vðrÞ  w  ðrÞgÞ, maxfvðrÞ  wðrÞ; vðrÞ  w sv ¼ sðvðrÞ; vðrÞÞ: See [6]. E1 with addition and multiplication as defined by Lemma 2.1 is a convex cone which is then embedded isomorphically and isometrically in to a Banach space. Definition 2.1. For arbitrary fuzzy numbers u ¼ ðu; uÞ and u ¼ ðv; vÞ the quantity uðrÞ
vðrÞjg Dðu; vÞ ¼ sup fmax½juðrÞ
vðrÞj; j 06r61 is the distance between u and v. It is shown [4] that E1, D is a complete metric space. T. Allahviranloo / Appl. Math. Comput. 166 (2005) 339–348 341 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 [3] 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 : ti
1 6 ni 6 ti, 1 6 i 6 n, let Rp ¼ n X f ðni Þðti
ti
1 Þ: i¼1 The definite integral of f(t) over [a, b] is Z b f ðtÞ dt ¼ lim Rp ; max jti
ti
1 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 [3]. Further more, ! Z Z b b f ðt; rÞ dt a Z b f ðt; rÞ dt a ¼ ! f ðt; rÞ dt; a ¼ Z b ð2:1Þ f ðt; rÞ dt: a It should be noted that the fuzzy integral can be also defined using the Lebesgue-type approach [2,5]. More details about properties of the fuzzy integral are given in [3,2]. 3. Newton cot’s methods Let f be a fuzzy function. For any natural number n, the Newton CotÕs formulas Z b n X b
a ; ð3:2Þ fi ai þ E; fi ¼ f ða þ ihÞ; h ¼ f ðxÞ dx ¼ h n a i¼1 342 T. Allahviranloo / Appl. Math. Comput. 166 (2005) 339–348 Rb provide approximate values for a f ðxÞ dx. The parametric form of (3.2) is as follows: Z b n X ai f ðxi ; rÞ þ Eðf ; rÞ; f ðx; rÞ dx ¼ h a Z i¼1 b f ðx; rÞ dx ¼ h a n X ð3:3Þ ai f ðxi ; rÞ þ Eðf ; rÞ; 0 6 r 6 1: i¼1 PnThe weights ai, i = 1, . . ., n, are rational numbers  with the property i¼1 ai ¼ n. This follows (3.3) 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; n 2 ða; bÞ; ðpÞ Eðf ; rÞ ¼ hpþ1  K  f ð n; rÞ;  n 2 ða; bÞ; 0 6 r 6 1: ð3:4Þ 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 (3.4). Let n X ai f ðxi ; rÞ; Qðf ; rÞ ¼ h i¼1 Qðf ; rÞ ¼ h n X ð3:5Þ ai f ðxi ; rÞ; 0 6 r 6 1: i¼1 Thus from (3.3) we have Z b f ðx; rÞ dx ¼ Qðf ; rÞ þ Eðf ; rÞ; Iðf ; rÞ ¼ a Z b  f ðx; rÞ dx ¼ Qðf ; rÞ þ Eðf ; rÞ; Iðf ; rÞ ¼ ð3:6Þ 0 6 r 6 1: a The following theorem is proving that Qðf ; rÞ, Qðf ; rÞ converge to Iðf ; rÞ,  Iðf ; rÞ respectively. Theorem 3.1. 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. The continuity of f(t) guarantees the existence of the definite of f(t), [3]. Thus Rp in Definition 2.3 converges to this integral in the metric D if   lim maxðti
ti
1 Þ ¼ 0: ð3:7Þ n!1 16i6n T. Allahviranloo / Appl. Math. Comput. 166 (2005) 339–348 343 It is easily seen that the formulas (3.5) can be represented in the form Rp in (2.3). For arbitrary Rp ¼ ðRp ; Rp Þ and Iðf Þ ¼ ðIðf ; rÞ; Iðf ; rÞÞ we have n h io DðRp ; f Þ ¼ sup max jRp ðrÞ
Iðf ; rÞj; jRp ðrÞ
Iðf ; rÞj 06r61 and since lim DðRp ; Iðf ÞÞ ¼ 0; n!1 maxðti
ti
1 Þ ! 0; 16i6n we obtain that Rp ðrÞ; Rp ðrÞ converge uniformly to Iðf ; rÞ; Iðf ; rÞ, respectively. Consequently, Qðf ; rÞ and Qðf ; rÞ (which a particular case of Rp ; Rp , since (ti
ti
1) :¼ hai, i = 1, . . ., n) converge uniformly to Iðf ; rÞ; Iðf ; rÞ as well. This concludes the proof of Theorem 3.1. h 3.1. Peano’s error representation From (3.3) we have Eðf ; rÞ ¼ Iðf ; rÞ
Qðf ; rÞ; Eðf ; rÞ ¼ Iðf ; rÞ
Qðf ; rÞ; ð3:8Þ 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 Eð g; 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 3.2. Suppose E(p) = 0 holds for all p 2 Õn, that is, every polynomial whose degree does not exceed n is integrated exactly. Then 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 Eðf ; rÞ ¼ Z b ð3:9Þ nþ1 f ðt; rÞKðtÞ dt; 0 6 r 6 1; a where 1 n KðtÞ :¼ Ex ½ðx
tÞþ ; n! n ðx
n tÞþ :¼  n ðx
tÞ ; x P t; 0; x<t and Ex ½ðx
tÞþ  when the latter is considered as a function in x. 344 T. Allahviranloo / Appl. Math. Comput. 166 (2005) 339–348 Proof. Consider the Taylor expansion of f(x), x 2 [a, b] at a, in parametric form: f ðnÞ ða; rÞ ðx
aÞn þ M n ðx; rÞ; n! ðnÞ f ða; rÞ 0 f ðx; rÞ ¼ f ða; rÞ þ f ða; rÞðx
aÞ þ    þ ðx
aÞn þ M n ðx; rÞ; n! f ðx; rÞ ¼ f ða; rÞ þ f 0 ða; rÞðx
aÞ þ    þ 0 6 r 6 1: ð3:10Þ Its reminder term can be expressed in the form Z x Z 1 1 b nþ1 n n f nþ1 ðt; rÞðx
tÞ dt ¼ f ðt; rÞðx
tÞþ dt; n! a n! a Z x Z 1 1 b  nþ1 nþ1 n  M n ðx; rÞ ¼ f ðt; rÞðx
tÞ dt ¼ f ðt; rÞðx
tÞnþ dt; n! a n! a M n ðx; rÞ ¼ Applying the linear operator E to (3.10) gives Z b  1 Eðf ; rÞ ¼ EðM n ; rÞ ¼ Ex f nþ1 ðt; rÞðx
tÞnþ dt ; n! a Z b  1 nþ1 Eðf ; rÞ ¼ EðM n ; rÞ ¼ Ex f ðt; rÞðx
tÞnþ dt ; n! a 0 6 r 6 1: ð3:11Þ 0 6 r 6 1: ð3:12Þ Since E(p) = 0 for p 2 Õn. For any 0 6 r 6 1, from [7], the entire operator Ex commutes with integration and we obtain the desired result (3.9). h 3.2. Trapozedial integration rule From (3.3) we have: " # n
1 X 1 1 Iðf ; rÞ ¼ h f ðx0 ; rÞ þ f ðxi ; rÞ þ f ðxn ; rÞ þ Eðf ; rÞ; 2 2 i¼1 " # n
1 X 1 1   Iðf ; rÞ ¼ h f ðx0 ; rÞ þ f ðxi ; rÞ þ f ðxn ; rÞ þ Eðf ; rÞ; 2 2 i¼1 0 6 r 6 1; ð3:13Þ where h2 ðb
aÞf ð2Þ ðn; rÞ; 12 h2 ð2Þ  rÞ; Eðf ; rÞ ¼
ðb
aÞf ðn; 12 Eðf ; rÞ ¼
ð3:14Þ n;  n 2 ½x0 ; xn ; 0 6 r 6 1: T. Allahviranloo / Appl. Math. Comput. 166 (2005) 339–348 345 The following variant of the trapozedial sum is already a method of order 3: " # n
2 X 5 13 13 5 Iðf ; rÞ ¼ h f ðx0 ; rÞ þ f ðx1 ; rÞ f ðxi ; rÞ þ f ðxn
1 ; rÞ þ f ðxn ; rÞ; 12 12 12 12 i¼2 " # n
2 X 5 13 13  5     f ðxi ; rÞ þ f ðxn
1 ; rÞ þ f ðxn ; rÞ; f ðx0 ; rÞ þ f ðx1 ; rÞ Iðf ; rÞ ¼ h 12 12 12 12 i¼2 0 6 r 6 1: ð3:15Þ 3.3. Simpson integration rule From (3.3) we have: " # n
1 n 1 4X 2X 1 f ðx0 ; rÞ þ f ðx2iþ1 ; rÞ þ f ðx2i ; rÞ f ðxn ; rÞ þ Eðf ; rÞ; 3 3 i¼0 3 i¼1 3 " # n
1 n X X 1 4 2 1 f ðx2iþ1 ; rÞ þ f ðx2i ; rÞ f ðxn ; rÞ þ Eðf ; rÞ; Iðf ; rÞ ¼ h f ðx0 ; rÞ þ 3 3 i¼0 3 i¼1 3 Iðf ; rÞ ¼ h 0 6 r 6 1; ð3:16Þ where h4 ðb
aÞf ð4Þ ðn; rÞ; 180 h4 ð4Þ ðb
aÞf ð n; rÞ; Eðf ; rÞ ¼ 180 Eðf ; rÞ ¼ ð3:17Þ n;  n 2 ½x0 ; xn ; 0 6 r 6 1: 4. Numerical examples In this section we illustrate the methods in Section 3 by solving some numerical examples. Example 4.1. Consider the following fuzzy integral: Z 1 e k ¼ ðr
1; 1
rÞ; kx2 dx; e 0 the exact solution is 13 ðr
1; 1
rÞ. From trapozedial rule with h = 1: 1 Qðf ; rÞ ¼ ðr
1Þ; 2 f 00 ¼ 2ðr
1Þ; 1 Qðf ; rÞ ¼ ð1
rÞ; 2 00 f ¼ 2ð1
rÞ ð4:18Þ 346 T. Allahviranloo / Appl. Math. Comput. 166 (2005) 339–348 and 1 1 Eðf ; rÞ ¼
ðr
1Þ; Eðf ; rÞ ¼
ð1
rÞ; 6 6 it is clear that Eq. (3.8) holds. Now with h ¼ 12: 3 Iðf ; rÞ ¼ ðr
1Þ; 8 3 Iðf ; rÞ ¼ ð1
rÞ; 8 1 1 ðr
1Þ; Eðf ; rÞ ¼
ð1
rÞ: 24 24 Eq. (3.8) holds too. The exact solution and trapozedial solutions with h = 1 and h ¼ 12 are plotted and compared in Fig. 1. Eðf ; rÞ ¼
Example 4.2. Consider the following fuzzy integral: Z 1 e k ¼ ðr; 2
rÞ; kx4 dx; e 0 the exact solution is 15 ðr; 2
rÞ. From Simpson rule with h ¼ 12: Qðf ; rÞ ¼ 5 r; 24 f 00 ¼ 24r; Qðf ; rÞ ¼ 5 ð2
rÞ; 24 00 f ¼ 24ð2
rÞ and Eðf ; rÞ ¼
1 r; 120 Eðf ; rÞ ¼
1 ð2
rÞ; 120 Fig. 1. ð4:19Þ T. Allahviranloo / Appl. Math. Comput. 166 (2005) 339–348 347 Fig. 2. it is clear that Eq. (3.8) is hold. Now with h ¼ 14: Iðf ; rÞ ¼ 154 r; 768 Eðf ; rÞ ¼
3 r; 5760 Iðf ; rÞ ¼ 154 ð2
rÞ; 768 Eðf ; rÞ ¼
3 ð2
rÞ: 5760 Eq. (3.8) holds too. The exact solution and trapozedial solutions with h ¼ 12 and h ¼ 14 are plotted and compared in Fig. 2. Example 4.3. Consider the following fuzzy integral: Z
2 e kx dx; e k ¼ ðr; 2
rÞ; ð4:20Þ
1 the exact solution is 32 ðr
2;
rÞ. From trapozedial rule with h = 1: 3 Qðf ; rÞ ¼ ðr
2Þ; 2 3 Qðf ; rÞ ¼ ð
rÞ; 2 Eðf ; rÞ ¼ Eðf ; rÞ ¼ 0: 5. Conclusion In this work we applied the Newton CotÕs method with positive coefficients to solve fuzzy integral over a finite interval [a, b]. Since this integration yields fuzzy number in parametric form, we used the parametric form of methods. The integration of triangular fuzzy number is a triangular fuzzy number. 348 T. Allahviranloo / Appl. Math. Comput. 166 (2005) 339–348 References [1] L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning, Inform. Sci. 8 (1975) 199–249. [2] O. Kaleva, Fuzzy differential equations, FSS 24 (1987) 301–317. [3] R. Goetschel, W. Voxman, Elementary calculus, FSS 18 (1986) 31–43. [4] M.L. Puri, D. Ralescu, Fuzzy random variables, J. Math. Anal. Appl. 114 (1986) 409–422. [5] M. Matloka, On fuzzy integrals, Proc. 2nd Polish Symp. on Interval and Fuzzy Mathematics, Polite chnika poznansk, (1987) 167–170. [6] S. Seikkala, On the fuzzy initial value problem, Fuzzy Sets and Systems 24 (1987) 319–330. [7] J. Stoer, R. Bulirsch, Introduction to Numerical Analysis, Springer-Verlag, 1980. [8] H.J. Zimmerman, Fuzzy Set Theory and its Applications, Kluwer Academic, New York, 1996.