Numerical Solution of Fuzzy Fredholm Integral Equations of the Second Kind using Bernstein Polynomials

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1 Jourl of Al-Nhri Uiversity Vol.5 (), Mrch,, pp.-9 Sciece Numericl Solutio of Fuzzy Fredholm Itegrl Equtios of the Secod Kid usig Berstei Polyomils Srmd A. Altie Deprtmet of Computer Egieerig d Iformtio Techology, Uiversity of Techology, Bghdd-Irq. Astrct I this pper, umericl method is give for solvig fuzzy Fredholm itegrl equtios of the secod kid, y usig Berstei piecewise polyomil, whose coefficiets determied through solvig dul fuzzy lier system. Numericl exmples re preseted to illustrte the proposed method, whose clcultios were implemeted y usig the Computer softwre MthCdV.. Keywords: Fuzzy Itegrl Equtio, Dul Fuzzy Lier System, Berstei Polyomils. Itroductio I recet yers, the iterest i fuzzy itegrl equtio shve ee rpidly growig d drwig ttetio y scietists, due to its importce i pplictios, such s, fuzzy cotrol, pproximte resoig, fuzzy ficil d ecoomic systems, etc. Cosequetly, the topic of umericl methods for solvig fuzzy itegrl equtios hve ee cosidered thoroughly, ecuse of the difficulty of fidig the lyticl solutio i my cses, to these equtios. Some umericl methods for fuzzy itegrl equtios illustrted y[] usig itertive method to the fuzzy fuctio, lso [5] used differetil trsformtio method to solve fuzzy itegrl, d [] gve lgorithm to solve the fuzzy itegrl equtios y usig the trpezoidl rule to compute the Riem itegrls tht covert it to lier system its ukows re to e determied, lso [8] used itertive iterpoltio, d [7] with fiite differeces d divided differeces methods. The first prt of this pper, is dedicted to give some ecessry theoreticl ckgroud mterils tht led to the uderstdig of the proposed method, while the secod prt dels with illustrtig the proposed pproch for solvig fuzzy Fredholm itegrl equtios of the secod kid, followed y umericl exmples d illustrtive figures, whose clcultios were implemeted y usig the Computer softwre MthCd Versio. - Berstei Polyomils []: The geerl form of the Berstei polyomils of the th degree over the itervl, is defied y: B i, t = i t i t i i =,,,,.Note tht ech of these + polyomils hvig degree stisfies the followig properties: i) B i, t =, if i < or i >. ii) i= B i, t =. iii) B i, = B i, =, i. Remrk () []:. Ay Berstei polyomil of degree my e writte i terms of the power sis, s follows, B i, t = j i j j t i=j i j. The st derivtive of th degree Berstei polyomil c e expressed s follows: d dt B i, t = B i, t B i, t i =,,,. - Dul Fuzzy Lier System: This sectio will grdully rech its purpose of fidig the solutio of dul fuzzy lier system through givig some ecessry required defiitios, d s follows: Defiitio.[]: A ritrry fuzzy umer is ordered pir of fuctios v = v α, v α, α, which stisfy the followig requiremets:. v α is ouded left cotiuous o decresig fuctio over,.

2 Srmd A. Altie. v α is ouded left cotiuous o icresig fuctio over,.. v α v α, α. For d ritrry fuzzy umers u = u α, u α, v = v α, v α d k R, we defie the dditio d sclr multiplictio y k s: u + v = u α + v α, u α + v α ku = ku α, ku α, k d ku = ku α, ku α, k < The set of ll these fuzzy umers is deoted y E. Defiitio. [9]: The fuzzy system Ax = Bx + y the coefficiets mtrix A = ij, d B = ij, i m, j re crisp m mtrices, d m, x = x, x,, x T d y = y, y,, y m T re fuzzy umer vectors. The ove system is clled the dul fuzzy lier system. Remrk () [9]: Usully, there is o iverse elemet for ritrry fuzzy umer E, i.e. there exists o elemet E such tht + =. Actully, for ll o-crisp fuzzy umer E we hve +. So the ove system cot e equivletly replced y the fuzzy system, A B x = y. Defiitio.[6]: A fuzzy umer vector x = x, x,, x T is give y: x i = x i α, x i α, i, α is clled solutio of the system i Defiitio. if ij x j = ij x j = y i ij x j = ij x j = ij x j = ij x j = ij x j = y i = ij x j for prticulr i, ij > d ij >, j, we simply get ij x j ij x j = y i + ij x j = y i + ij x j Remrk (): Follow the work i [], [6] d cosider the dul fuzzy lier system, Ax = Bx + y d trsform its coefficiet mtrices A, d B ito crisp lier system: SX = TX + Y S T X = Y X = S T Y the coefficiets mtrix S = s ij, d T = t ij, i, j. The elemets s ij d t ij, re determied s follows: If ij s ij = ij d s i+ j + = ij. If ij < s i j + = ij d s i+ j = ij. If ij t ij = ij d t i+ j + = ij. If ij < t i j + = i,j d t i+ j = ij. d y elemet s ij d t ij which hs o ssiged vlue from the coefficiet mtrices A, d B is set s zero.also, the vriles vectors re: X = x x x x x x T d Y = y y y y y y T - Fuzzy Fredholm Itegrl Equtios [7]: I this sectio, the defiitio of Fuzzy Fredholm itegrl equtio of the secod kid, will e studied, d s follows: y(x; α) = f(x; α) + λ k x, t y(t; α)dt () λ >, d k x, t is ritrry fuctio clled the kerel over the rectgle x, t, f(x; α) = f(x; α), f(x; α) fuzzy fuctios o the itervl,, α, d y x; α = y x; α, y(x; α) is hece equtio () c e replced y two equtios, y(x; α) = f(x; α) + U(t; α)dt, d y(x; α) = f(x; α) + U(t; α)dt

3 Jourl of Al-Nhri Uiversity Vol.5 (), Mrch,, pp.-9 Sciece U t; α = d U t; α = k s, t y(t; α) k s, t k s, t y(t; α) k s, t < k s, t y(t; α) k s, t k s, t y(t; α) k s, t < Sufficiet coditios for the existece of uique solutio to the fuzzy Fredholm itegrl equtios of the secod kid, hve ee give i []. - Mi Results: I this sectio, the solutio of equtio () y sustitutig the Berstei polyomils i y(x; α) to give y(x; α) = i= i B i, (x), d hece: i= i B i, (x) = f(x; α) + λ k x, t i B i, (t) dt or equivletly, i B i, (x) i= = i= f(x; α) + λ i= i k x, t B i, (t)dt () Now, i order to fied i, choose x i,, i =,,,, d sustitute them ito equtio () to oti the dul fuzzy lier system of the form: A = f + B A = i,j, B = i,j, i, j, i,j = B j, (x i ), i,j = λ k x i, t B j, (t)dt d f = f(x ) f(x ) T is ritrry fuzzy umer vector. Now, trsform the coefficiet mtrices A d B ito crisp lier mtrices S d T respectively s metioed i sectio () to oti the vlues of, s follows: S = T + F d hece S T = F = S T F Where F = f f f f f f T re fuctios whose vlues must e determied t x i,, i =,,,, d the fuzzy pproximte solutio of equtio () will e give y: y x; α = y x; α, y(x; α) the lower solutio y x; α = i= i B i, (x) d the upper solutio y x; α = i B i, (x) i=. 5- Numericl Exmples: Now, two exmples will e preseted d solved,logside illustrtig grphs to compre with the exct solutio to illustrte the proposed method. Exmple (): Cosider the fuzzy Fredholm itegrl equtio of the secod kid, with λ =, =, d =, such tht: y(x; α) = f(x; α) + k x, t y(t; α)dt () k x, t = x + t, x, t f x; α = α x, f x; α = ( α) x y x; α = αx is the exct lower solutio, d y x; α = ( α)x is the exct upper solutio []. i= Now, y sustitutig y x; α = i B i, (x), ito equtio () we get: i= i B i, (x) or equivletly: i B i, (x) = f(x; α) + k x, t i B i, (t) dt i= i= = i= i k x, t B i, (t)dt () f(x; α) + d i order to fied i, i =,,,, let us tke x =, x =, x =, d x = the sustitute them ito equtio () to oti the dul fuzzy lier system: A = f + B A = = d , f = f(x ) f(x ) f(x ) f(x ), 5

4 Srmd A. Altie B = Now, trsform the ove coefficiet mtrices A d B ito 8 8 crisp lier Mtrices S, T respectively s metioed i sectio (), to get: S = d T = to oti the vlues of, s follows: = S T F =.α.667α.999α.α α. α Hece =,, =.α,.667.α, =.667α,..667α, =.999α, α, therefore the fuzzy pproximte solutio y x; α = y x; α, y(x; α), will e give s: y x; α = x + x x + x x + x, d y x; α = x + x x + x x + x Figs. (.), (.), (.c), d (.d) gives compriso etwee the exct d pproximte solutios for α =.5,.5,.75, d respectively (.) α = (.) α = (.c) α =.75 6

5 Jourl of Al-Nhri Uiversity Vol.5 (), Mrch,, pp.-9 Sciece (.d) α = Fig. () The exct solutio logside the umericl solutio of exmple () for differet vlues of α. Exmple (): Cosider the fuzzy Fredholm itegrl equtio of the secod kidwith λ =, =, d = π, such tht: π y x; α = f x; α + k x, t y t; α dt k x, t =. si t si( x), x, t π f x; α = 5 α + α f x; α = 5 α + α + 5 α α si( x) (5) + 5 α α si( x) Where the exct lower solutio is y x; α = α + α si( x), d the exct upper solutio is y x; α = α α si( x)[]. Now, y sustitutig y x; α = i B i, (x), ito equtio (5) we get: i= i= i B i, (x) π = f(x; α) + k x, t i B i, (t) dt or equivletly, i B i, (x) i= i= = π i= (6) f(x; α) + i k x, t B i, (t)dt Now, i order to fied i, i =,,,, let us tke x = π, x = π, x = 5π, dx = 7π, the sustitute them ito equtio (6) to oti the dul fuzzy lier system: A = f + B A =, =, f(x ) f(x f = ) f(x ) f(x ) d B = Now, trsform the ove coefficiet mtrices A d B ito 8 8 crisp lier Mtrices S, T respectively s metioed i sectio (), to get: S = d T = to oti the vlues of, s follows: = S T F =..75α.8α +.8α.8 +.α +.α.α.8 +.α +.α.α..75α.8α +.8α..75α +.8α.8α.98 +.α.α +.α.98 +.α.α +.α..75α +.8α.8α Hece: =..75α.8α +.8α,. +.75α.8α +.8α, =.8 +.α +.α.α,.98.α +.α.α, =.8 +.α +.α.α,.98.α +.α.α, =..75α.8α +.8α,. +.75α.8α +.8α, Therefore the fuzzy pproximte solutio is give y y x; α = y x; α, y(x; α),, 7

6 Srmd A. Altie y x; α = π x + π x π x + π π x π x + π x, d y x; α = π x + π x π x + π π x π x + π x, Figs. (.), (.), (.c), d (.d) gives compriso etwee the exct d pproximte solutios for α =.5,.5,.75, d respectively (.c) α = (.) α = (.) α = (.d) α = Fig.() The exct solutio logside the umericl solutio of exmple () for differet vlues of α. Coclusios I this pper very simple d stright forwrd method for pproximtig the solutio of the give fuzzy itegrl equtio usig Berstei polyomil sis d depedig o solvig dul fuzzy lier system of equtios is preseted, d the results show i exmple () d () re very good if compred with the exct solutio. Refereces [] E. Boli d M. Pripour, Numericl Solvig of Geerl Fuzzy Lier Systems, TritMollem Uiversity, th Semir o Alger, - Ordiehesht, 88 (9), pp.-. [] M. Brkhordry, N.A. Kii, A. R. Bozorgmesh,"A Method for Solvig Fuzzy Fredholm Itegrl Equtios of The Secod Kid" Itertiol Ceter For 8

7 Jourl of Al-Nhri Uiversity Vol.5 (), Mrch,, pp.-9 Sciece Scietific Reserch d Studies, Vol., No., Septemer, 8. []Mehem Friedm, Mig M, Arhm Kdel, Numericl solutios of fuzzy differetil d itegrl equtios, Fuzzy Sets d Systems 6 (999), pp.5-8. [] Keeth I. Joy, Berstei Polyomils, Visuliztio d Grphics Reserch Group, Deprtmet of Computer Sciece, Uiversity of Clifori, Dvis,. [5] Y. Nejtkhsh, T. Allhvirloo, N. A. Kii," Solvig Fuzzy Itegrl equtios y Differetil Trsformtio Method" First Joit Cogress o Fuzzy d Itelliget Systems, Ferdowsi Uiversity of Mshhd, Ir, Aug. 7. [6] M. Otdi, S. Asdy, d M. Mosleh," System of lier fuzzy differetil equtios" First Joit Cogress o Fuzzy d Itelliget Systems, Ferdowsi Uiversity of Mshhd, Ir, Aug. 7. [7] N. Prdi, M. A. FriorziArghi, The umericl solutio of lier fuzzy Fredholm itegrl equtios of the secod kid y usig fiite d divided differeces methods, Spriger- Verlg(), pp [8] N. Prdi, M. A. FriorziArghi, The Approximte Solutio of Lier Fuzzy Fredholm Itegrl Equtios of the Secod Kid y Usig Itertive Iterpoltio, World Acdemy of Sciece, Egieerig d Techology Vol.9, (9), pp [9] Rez Ezzti, "A Method for Solvig Dul Fuzzy Geerl Lier Systems" Appl. Comput.Mth.7 (8), No., pp.5-. []Shiri, A. d Islm, M. S., Numericl Solutios of Fredholm Itegrl Equtios Usig Berstei Polyomils, J. Sci. Res. (),(), pp.6-7. MthCd V. 9