Hindwi Pulishing Corportion Mthemticl Prolems in Engineering, Article ID 873498, 7 pges http://dx.doi.org/0.55/04/873498 Reserch Article Composite Guss-Legendre Formuls for Solving Fuzzy Integrtion Xioin Guo, Dequn Shng, nd Xioqun Lu 3 College of Mthemtics nd Sttistics, Northwest Norml University, Lnzhou 730070, Chin Deprtment of Pulic Courses, Gnsu College of Trditionl Chinese Medicine, Lnzhou 730000, Chin 3 College of Chemistry nd Chemicl Engineering, Northwest Norml University, Lnzhou 730070, Chin Correspondence should e ddressed to Xioqun Lu; luxq@nwnu.edu.cn Received 9 Decemer 03; Revised 6 My 04; Accepted 6 My 04; Pulished 9 My 04 Acdemic Editor: Vlentin Emili Bls Copyright 04 Xioin Guo et l. This is n open ccess rticle distriuted under the Cretive Commons Attriution License, which permits unrestricted use, distriution, nd reproduction in ny medium, provided the originl work is properly cited. Two numericl integrtion rules sed on composition of Guss-Legendre formuls for solving integrtion of fuzzy numers-vlued functions re investigted in this pper. The methods constructions re presented nd the corresponding convergence theorems re shown in detil. Two numericl exmples re given to illustrte the proposed lgorithms finlly.. Introduction Numericl integrtion is one of the sic contents in numericl mthemtics, nd it lwys plys vitl role in engineering nd science clcultion. Numericl integrtion methods re introduced in detil []. Numericl integrtion is lwys crried out y mechnicl qudrture nd its sic scheme []issfollows: f (x) dx n A k f(x k ), () where A k 0, k = 0,,...,n,ndx k [, ], k = 0,,...,n, re clled coefficients nd nodes for mechnicl qudrture, respectively. Once the coefficients nd nodes re set down, the scheme () cn e determined. Over yers, some works hve emerged out the symptotic properties of numericl integrtion methods. However, their results re concise, ut resoning processes re very complicted [3 6]. The topic of fuzzy integrtion ws first discussed in [7]. In 005, Allhvirnloo [8] mde good ttempt to use Newton Cot s methods with positive coefficients for integrtion of fuzzy functions. For instnce, he designed Trpezoidl integrtion rule nd Simpson integrtion rule for fuzzy integrl. Lter, they pplied the Gussin qudrture method nd Romerg method for pproximtion of fuzzy integrl nd fuzzy multiple integrl, uilt series of formuls for intricte fuzzy integrl [8 ], nd otined some good results. But their methods did not hve high convergence order. In this pper, we set up clss of high lgeric ccurcy numericl integrtion methods which re proposed y compositing the two-point nd three-point Guss-Legendre formuls. We design these formuls to clculte integrtion of fuzzy functions. We lso present the methods reminder terms nd give corresponding convergence theorems. Compred with some pproches for pproximting fuzzy integrtions efore, our methods re superior to those formuls on oth mount of clcultion nd qudrture error. The structure of this pper is s follows. In Section, we recll some sic definitions nd results on integrtion of fuzzy functions. In Section 3,weintroduce the two-point nd three-point Guss-Legendre formuls nd their composite method. Then we design them to solve fuzzy integrtion. We lso put up methods reminder term representtions nd convergence theorems. The proposed lgorithms re illustrted y solving two exmples in Section 4 nd the conclusion is drwn in Section 5.. Preliminries.. Integrtion of Fuzzy Function. Let E e the set of ll rel fuzzy numers which re norml, upper semicontinuous, convex, nd compctly supported fuzzy sets.
Mthemticl Prolems in Engineering Definition (see []). A fuzzy numer u in prmetric form is pir (u, u) of functions u(r), u(r), 0 r, which stisfies the following requirements: () u(r) is ounded monotonic incresing left continuous function, () u(r) is ounded monotonic decresing left continuous function, (3) u(r) u(r), 0 r. Bsed on Zdeh s principle of extension, Goetschel et l. [3] presented fuzzy numers ddiction nd multipliction y k which re s follows. Let x = (x(r), x(r)), y = (y(r), y(r)) E, 0 r,ndrelnumerk, s follows: () xy=(x(r) y(r), x(r) y(r)), () x y=(x(r) y(r), x(r) y(r)), (kx(r),kx(r)), k 0, (3) kx = { (kx(r),kx(r)), k<0. For ritrry fuzzy numers x = (x(r), x(r)), y = (y(r), y(r)) E,thequntity D(x,y)= sup {mx [ 0 r x (r) y(r), x (r) y (r) ]} () is the distnce etween x nd y. Afunctionf:R E is clled fuzzy function. If for ritrry fixed t 0 R nd ε>0,δ>0such tht t t 0 <δ D[f(t),f(t 0)]<ε, (3) exists, f is sid to e continuous. Definition (see [4]). Assume f:[,] E.Forech prtition P = {t 0,t,...,t n } of [, ] nd for ritrry ξ i : t i ξ i t i, i n,let R p = n i= f(ξ i )(t i t i ). (4) The definite integrl of f(t) over [, ] is f (t) dt = lim R p, mx i n t i t i 0, (5) provided tht this limit exists in the metric D. If the fuzzy function f(t) is continuous in the metric D, its definite integrl exists. Furthermore, ( ( f (t; r) dt) = f (t; r) dt, f (t; r) dt) = f (t; r) dt. It should e noted tht the fuzzy integrl lso cn e defined using the Leesgue-type pproch [5, 6]. More detils out properties of fuzzy integrl re given in [7]. (6).. Guss-Legendre Formuls. Guss qudrture formul is the highest lgeric ccurcy of interpoltion qudrture formul. By resonly selecting qudrture nodes nd qudrture coefficients of the form of f (x) dx n A k f (x k ), (7) we cn otin the interpoltion qudrture formul with the highest lgeric ccurcy; tht is, n. Using root nodes of norder Legendre orthogonl polynomil on specil intervl [, ], we cn propose Guss-Legendre qudrture formul (7). Lemm 3 (see [6]). The reminder term of Guss-Legendre qudrture formul (7) is E(f)= f (x) dx n A k f(x k ) f (n) (ξ) = (n )! ω (x) dx, ξ [, ], where ω(x) = n (x x i) nd x i re Guss nodes. In prticulr, when n=, (7) is two-point Guss-Legendre qudrture formul: f (x) dx (f (x ( 3 3 )) f (x ( 3 3 ))), (9) where x(t) = ( )/ (( )/)t nd its reminder term is E(f)= (8) f (4) (ξ) (x x( 3 4! 3 )) (x x( 3 3 )) dx, ξ [, ]. (0) When n=, (7) is three-point Guss-Legendre qudrture formul: f (x) dx ( 5 9 f(x( 5 5 )) nd its reminder term is E(f)= 8 9 f( )5 9 f(x( 5 5 ))), () f (6) (ξ) (x x( 5 6! 5 )) (x ) (x x( 5 5 )) dx, ξ [, ]. ()
Mthemticl Prolems in Engineering 3.3. Convergence Order of Composite Method Definition 4 (see [8]). Suppose I= f(x)dx, ndi n is composite numericl integrtion method. If h 0, it stisfies I I lim n h 0 h p =C, C=0, (3) nd we cll I n p order convergent method. For instnce, composite Trpezoid method f (x) dx nd composite Simpson f (x) dx h 6 (f (x i )f(x i )) (4) (f (x i )4f(x i/ )f(x i )) (5) hve two-order nd four-order convergence property, respectively. 3. Guss-Legendre Formuls for Solving Fuzzy Integrl 3.. Composite Formuls nd Their Error Reminders Theorem 5. Let f(x) C (6) [, ], x k =x 0 kh, k = 0,,,...,n, h = ( )/n; then the reminder term of composite three-point formuls Guss-Legendre is f (x) dx h ( 5 9 f(x i/ 5h 0 )8 9 f(x i/) 0 )) (6) ( ) h6 E (f (x)) = 6 3500 f(6) (ξ), ξ [, ], (7) nd it hs six-order convergence property. Proof. We first consider the reminder term of three-point Guss-Legendre. By Lemm 3, E(g)= g (t) dt ( 5 9 g( 5 5 )8 9 g (0) 5 9 g( 5 5 )) Let g(t) = t 6,nd =cg (6) (ξ), ξ [, ]. g (6) (t) 70, t 6 dt = 7, I(t 6 )= 5 6 9 ( 5 5 ) 5 6 9 ( 5 5 ) = 6 5. (8) (9) We hve E(t 6 )= t 6 dt I (t 6 )= 7 6 = 70c, (0) 5 nd c = /5750. Thus, we otin the reminder term of threepoint formuls Guss-Legendre: E(g)= 5750 g(6) (ξ), ξ [, ]. () Now we study the integrl reminder term of composite three-point Guss-Legendre: f (x) dx = x i f (x) dx, () x i where P={x 0,x,...,x n } is out equidistnt prtition on [, ].Letx(t) = (x i x i )/ (h/)t, x i/ =(x i x i )/; then x i f (x) dx = h x i f(x i/ h t) dt. (3) By mens of three-point Guss-Legendre formul nd its reminder term (), we hve f (x) dx = h = h = h = h h7 7 f(x i/ h t) dt ( 5 9 f(x i/ 3h 5 )8 9 f(x i/) 5 9 f(x i/ 3h 5 ) h 6 6 5750 f(6) (ξ i )), ξ i [x i,x i ], ( 5 9 f(x i/ 3h 5 )8 9 f(x i/) 5 9 f(x i/ 3h 5 )) 5750 f(6) (ξ i ), ( 5 9 f(x i/ 3h 5 )8 9 f(x i/) 5 9 f(x i/ 3h 5 )) ( ) h6 6 3500 f(6) (ξ), ξ [, ]. (4)
4 Mthemticl Prolems in Engineering So we get the reminder term of composite three-point Guss- Legendre s follows: ( ) h6 E (f (x)) = 6 3500 f(6) (ξ), ξ [, ]. (5) Since I I lim n ( ) h 0 h 6 = 6 3500 f(6) (ξ), (6) we know it hs six-order convergence property. In similr wy, we otin the reminder term of composite two-point formuls Guss-Legendre s follows. f (x; r) dx h f (x; r) dx h [f (x i/ 3h 6 ;r) f (x i/ 3h 6 ;r)], [f(x i/ 3h 6 ;r) f(x i/ 3h 6 ;r)], 0 r. (30) Theorem 6. Let f(x) C (4) [, ], x k = x 0 kh, k = 0,,,..., n, h = ( )/n; then the reminder term of composite two-point formuls Guss-Legendre is f (x) dx h (f (x i/ 3h 6 )f(x i/ 3h 6 )), (7) ( ) h4 E(f(x)) = 4 70 f(4) (ξ), ξ [, ], (8) nd it hs four-order convergence property. 3.. Guss-Legendre Formuls to Fuzzy Integrtion. In this susection, we pply composite Guss-Legendre formuls to solve fuzzy integrtion nd give their reminder terms nd convergence theorems. Applying formuls (6)nd(7) to numericl integrtion for fuzzy function (3), we hve f (x; r) dx h f (x; r) dx h [ 5 9 f (x i/ 5h 8 9 f (x i/;r) 5 9 f (x i/ 5h ], [ 5 9 f(x i/ 5h 8 9 f(x i/;r) ], 0 r (9) Theorem 7. Suppose f(x; r) C 6 [, ] out x nd f (6) (x; r) 0, f (6) (x; r) 0, 0 r ; then the reminder terms of composite three-point Guss-Legendre formuls (9) for fuzzy integrtion re E(f(x; r)) = E(f (x :r)) = where ξ, ξ [,], 0 r. ( ) h6 6 3500 f(6) (ξ;r), ( ) h6 6 3500 f(6) (ξ:r), Proof. From formuls (6)nd(7), we get f (x) dx = h (3) [ 5 9 f(x i/ 5h 0 )8 9 f(x i/) 0 )] ( ) h6 6 3500 f(6) (ξ), ξ [, ]. (3) Using the ove formul for fuzzy integrtion in prmetric form, we hve f (x; r) dx = h f (x; r) dx = h [ 5 9 f (x i/ 5h 8 9 f (x i/;r) 5 9 f (x i/ 5h ] ( ) h6 6 3500 f(6) (ξ;r), [ 5 9 f(x i/ 5h 8 9 f(x i/;r)
Mthemticl Prolems in Engineering 5 ] ( ) h6 6 3500 f(6) (ξ:r), (33) where ξ, ξ [,], 0 r. So the reminder terms of composite three-point Guss- Legendre formuls (9) for fuzzy integrtion re (3). Theorem 8. Let f(x; r) C 6 [, ] out x, 0 r ;then lim [h h 0 ( 5 9 f (x i/ 5h 8 9 f (x i/;r) = f (x; r) dx, lim [h h 0 5 9 f (x i/ 5h )] ( 5 9 f(x i/ 5h 8 9 f(x i/;r) = f (x; r) dx. )] Proof. From Theorem 7,we hve E(f(x; r)) = f (x; r) dx = h ( 5 9 f (x i/ 5h 8 9 f (x i/;r) 5 9 f (x i/ 5h ) ( ) h6 6 3500 f(6) (ξ;r), E(f (x :r)) = f (x; r) dx h ( 5 9 f(x i/ 5h 8 9 f(x i/;r) (34) = ) ( ) h6 6 3500 f(6) (ξ:r), (35) where ξ, ξ [,], 0 r. Since f (6) (x; r), f (6) (x; r) re ounded over [, ], we esily get the following fct: if h 0. E(f;r) 0, E(f; r) 0, (36) Similrly, we hve the following convergence theorem. Theorem 9. Suppose f(x; r) C 4 [, ] out x nd f (4) (x; r) 0, f (4) (x; r) 0, 0 r ; then the reminder termsofcompositetwo-pointguss-legendreformuls(30) for fuzzy integrtion re E(f(x; r)) = E(f (x :r)) = where ξ, ξ [,], 0 r.and h lim h 0 h lim h 0 ( ) h4 4 70 f(4) (ξ;r), ( ) h4 4 70 f(4) (ξ:r), [f (x i/ 3h 6 ;r)f(x i/ 3h 6 ;r)] = f (x; r) dx, [f(x i/ 3h 6 ;r)f(x i/ 3h 6 ;r)] = f (x; r) dx. 4. Numericl Exmples Exmple. Consider the following fuzzy integrl: 0 (37) (38) kx 4 dx, k = (r, r). (39) The exct solution is (3/5)(r, r) = 6.400000(r, r). From the two-point Guss-Legendre formul: Q(f; r) = 6.r, Q (f; r) = 6. ( r), E(f; r) = 0.77778r, E (f; r) = 0.77778 ( r), (40) it is cler tht formul (37)holds.
6 Mthemticl Prolems in Engineering Tle : Numericl solutions nd errors etween different methods (h = ). Methods Numericl solution Error Trpezoidl formul 0.6839397( r, r) 0.05897 Simpson formul 0.6333368( r, r) 0.000303 Two-point Guss-L 0.6397875( r, r) 0.000480 Three-point Guss-L 0.6305( r, r) 0.00000040 By the two-point Guss-Legendre formul with h= Q(f; r) = 6.388888r, Q (f; r) = 6.388888 ( r), E(f; r) = 0.0r, E (f; r) = 0.0 ( r), (4) it is cler tht formul (37)holds.Nowwithh=/, Q(f; r) = 6.399305r, Q (f; r) = 6.399305 ( r), E(f; r) = 0.000695r, E (f; r) = 0.000695 ( r). (4) Eqution (38)holdstoo. Exmple. Consider the following fuzzy integrl: 0 ke x dx, k =( r, r). (43) The exct solution is ( /e)( r, r) = 0.63055 ( r, r). We clculte numericlly the ove integrl using Trpezoidl formul, Simpson formul, composite two-point Guss-Legendre, nd three-point Guss-Legendre methods with h=, h=/,ndh=/4. Some comprisons out the numericl solutions nd the errors etween the different methods re shown in Tles,,nd3. All dt re denoted with eight-it significnt digits nd errors re clculted y the distnce etween exct solution nd numericl solution. From the ove tles figures, we cn clerly see tht our methods hve etter pproximtion thn the Trpezoidl formul nd Simpson formul on the sme fuzzy integrtion, in which the composite three-point Guss-Legendre is relly the cse. 5. Conclusion In this work, we pplied composite Guss-Legendre formuls to solve fuzzy integrl over finite intervl [, ]. Since this integrtion yields fuzzy numer in prmetric form, we use the prmetric form of the methods. The integrtion of tringulr fuzzy numer is tringulr fuzzy numer. Numericl exmples showed tht our methods re prcticl nd efficient while computing fuzzy integrl on lrger intervl [, ]. Tle : Numericl solutions nd errors etween different methods (h = 0.5). Methods Numericl solution Error Trpezoidl formul 0.645359( r, r) 0.03464 Simpson formul 0.63348( r, r) 0.0000363 Two-point Guss-L 0.6348( r, r) 0.00000877 Three-point Guss-L 0.63055( r, r) 0.00000000 Tle 3: Numericl solutions nd errors etween different methods (h = 0.5). Methods Numericl solution Error Trpezoidl formul 0.63540943( r, r) 0.0038888 Simpson formul 0.634( r, r) 0.00000086 Two-point Guss-L 0.63998( r, r) 0.00000057 Three-point Guss-L 0.63055( r, r) 0.00000000 Conflict of Interests The uthors declre tht there is no conflict of interests regrding the puliction of this pper. Acknowledgments The work is supported y the Nturl Scientific Funds of Chin (nos. 660 nd 7508) nd the Youth Reserch Aility Project of Northwest Norml University (NWNU- LKQN-0). References [] P. J. Dvis nd P. Rinowitz, Methods of Numericl Integrtion, Acdemic Press, Orlndo, Fl, USA, nd edition, 984. [] R. L. Burden nd J. Dougls, Numericl Anlyisis, Thomson Lerning, Boston, Mss, USA, 00. [3] B. Q. Liu, Asymptotic nlysis for some numericl integrl formuls, Communiction on Applied Mthemtics nd Computtion,vol.4,no.,pp.83 87,000. [4] B. Q. Liu, Limit properties out clss of Gussin integrtion formuls, Journl of Engineering Mthemtics, vol. 0, no. 4, pp. 37 39, 003. [5] S.F.QiundZ.W.Wng, Asymptoticpropertiesoftheintermedite point to numericl integrtion nd its Applictions, Mthemtics in Prctice nd Theory,vol.36,no.5,pp.8 3, 006. [6] Q. H. Zho, Correction formuls for numericl integrl, Mthemtics in Prctice nd Theory,vol.37,pp.07 08,007. [7] H. J. Zimmermn, Fuzzy sets theory nd its pplictions, Fuzzy Sets nd Systems,vol.4,pp.39 330,987. [8] T. Allhvirnloo, Newton Cot s methods for integrtion of fuzzy functions, Applied Mthemtics nd Computtion, vol. 66, no., pp. 339 348, 005. [9] T. Allhvirnloo, Romerg integrtion for fuzzy function, Applied Mthemtics nd Computtion, vol. 68, pp. 886 876, 005. [0] T. Allhvirnloo nd M. Otdi, Gussin qudrtures for pproximte of fuzzy integrls, Applied Mthemtics nd Computtion,vol.70,no.,pp.874 885,005.
Mthemticl Prolems in Engineering 7 [] T. Allhvirnloo nd M. Otdi, Gussin qudrtures for pproximte of fuzzy multiple integrls, Applied Mthemtics nd Computtion,vol.7,no.,pp.75 87,006. [] C. X. Wu nd M. M, On emdedding prolem of fuzzy numer spce prt I, Fuzzy Sets nd Systems,vol.44,no.,pp.33 38, 99. [3] R.Goetschel,Jr.ndW.Voxmn, Elementryclculus, Fuzzy Sets nd Systems,vol.8,no.,pp.3 43,986. [4] M. L. Puri nd D. A. Rlescu, Fuzzy rndom vriles, Journl of Mthemticl Anlysis nd Applictions, vol.4,no.,pp. 409 4, 986. [5] O. Klev, Fuzzy differentil equtions, Fuzzy Sets nd Systems,vol.4,no.3,pp.30 37,987. [6] M. Mtłok, On fuzzy integrls, in Proceedings of the nd Polish Symposium on Intervl nd Fuzzy Mthemtics, vol.8, pp. 67 70, Plite Chnick Poznnsk, 987. [7] J. Stoer nd R. Bulirsch, Introduction to Numericl Anlysis, Springer,NewYork,NY,USA,980. [8] Q.Y.Li,N.C.Wng,ndD.Y.Yi,Numericl Anlyisis, Tsuing University Press, Springer, 003.
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