NUMERICAL METHODS FOR SOLVING FUZZY FREDHOLM INTEGRAL EQUATION OF THE SECOND KIND. Muna Amawi 1, Naji Qatanani 2

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1 Interntionl Journl of Applied Mthemtics Volume 28 No , ISSN: (printed version); ISSN: (on-line version) doi: NUMERICAL METHODS FOR SOLVING FUZZY FREDHOLM INTEGRAL EQUATION OF THE SECOND KIND Mun Amwi 1, Nji Qtnni 2 1,2 Deprtment of Mthemtics An-Njh Ntionl University Nblus, PALESTINE Abstrct: In this rticle some numericl methods, nmely: the Tylor expnsion method nd the Trpezoidl method, hve been investigted nd implemented to solve fuzzy Fredholm integrl eqution of the second kind Consequently, we convert liner fuzzy Fredholm integrl eqution of the second kind into liner system of integrl equtions of the second kind in crisp cse To demonstrte the credibility of these numericl schemes we consider numericl test exmple The numericl results show to be in close greement with the exct solution AMS Subject Clssifiction: 45B05, 45A05, 65R20, 41A58 Key Words: fuzzy function, fuzzy integrl, Tylor series, trpezoidl rule 1 Introduction The fuzzy integrl equtions hve ttrcted the ttention of mny scientists nd reserchers in recent yers, due to their wide rnge of pplictions, such s Fuzzy control, Fuzzy finnce, pproximte resoning nd economic systems, etc Received: December 24, 2014 Correspondence uthor c 2015 Acdemic Publictions

2 178 M Amwi, N Qtnni The concept of integrtion of fuzzy functions ws first introduced by Dubois nd Prde [5] Then lterntive pproches were lter suggested by Goetschel nd Voxmn [8], Klev [13], Mtlok [17], Nnd [19], nd others While Goetschel nd Voxmn [8], nd lter Mtlok [17], preferred Riemnn integrl type pproch, Klev [13], choose to define the integrl of fuzzy function using the Lebesgue type concept for integrtion Prk et l [22] hve considered the existence of solution of fuzzy integrl eqution in Bnch spce Wu nd M [4] investigted the Fuzzy Fredholm integrl eqution of the second kind, which is one of the first pplictions of fuzzy integrtion Due to the complexity of solving Fuzzy Fredholm integrl equtions nlyticlly, numericl methods hve been proposed For instnce, Mleknejd [16] solved the first kind Fredholm integrl eqution by using the sinc function Prndin nd Arghi [21] estblished method to pproximte the solution using finite nd divided differences methods Jfrzdeh [12], solved liner fuzzy Fredholm integrl eqution with Upper bound on error by Splinders Interpoltion Altie [1] used the Bernstein piecewise polynomil Prndin nd Arghi [11] proposed the pproximte solution by using n itertive interpoltion Lotfi nd Mhdini [15] used Fuzzy Glerkin method with error nlysis Attri nd Yzdni [3] studied the ppliction of Homotopy perturbtion method Mirzee, Pripour nd Yri [18] presented direct method using Tringulr Functions Gohry nd Gohry [10], found n pproximte solution for system of liner fuzzy Fredholm integrl eqution of the second kind with two vribles which exploit hybrid Legendre nd block-pulse functions, nd Legendre wvelets Ziri, Ezzti nd Abbsbndy [23], used Fuzzy Hr Wvelet Ghnbri, Toushmlni nd Kmrni [9], presented numericl method bsed on block-pulse functions (BPFs) Amwi [2], hs investigted some nlyticl nd numericl solutions for the fuzzy Fredholm integrl eqution of the second kind In this rticle some numericl methods, nmely: the Tylor expnsion method nd the Trpezoidl method, hve been investigted nd implemented to solve fuzzy Fredholm integrl eqution of the second kind Using the prmetric form of fuzzy numbers, the fuzzy liner Fredholm integrl eqution of the second kind cn be converted to liner system of Fredholm integrl equtions of the second kind in the crisp cse The pper is orgnized s follows: In Section 2, fuzzy Fredholm integrl eqution of the second kind is introduced The Tylor expnsion method used to pproximte solution of fuzzy Fredholm integrl eqution of the second kind is ddressed in Section 3 In Section 4 we present the Trpezoidl method s uniformly convergent itertive procedure The proposed numericl methods

3 NUMERICAL METHODS FOR SOLVING FUZZY 179 re implemented using numericl exmple by pplying the MAPLE softwre in Section 5, conclusions re given in Section 6 2 Fuzzy Fredholm Integrl Eqution A stndrd form of the Fredholm integrl eqution of the second kind is given by [11] g(t) = f(t)+λ k(s, t)g(s)ds, (21) where λ is positive prmeter, k[s,t] is function clled the kernel of the integrl eqution defined over the squre G : [,b] [,b] nd f(t) is given function of t [,b] Now, if f(t) is crisp function then (21) possess crisp solution nd the solution is fuzzy if f(t) is fuzzy function We introduce prmetric form of fuzzy Fredholm integrl eqution of the second kind Let (f(t,r),f(t,r)) nd (g(t,r),g(t,r)),0 r 1 nd t [,b] re prmetric forms of f(t) nd g(t) respectively, then the prmetric form of fuzzy Fredholm integrl eqution of the second kind is s follows: where nd g(t,r) = f(t,r)+λ U(s,r)ds, g(t,r) = f(t,r)+λ U(s,r)ds, (22) U(t,r) = { k(s,t)g(s,r), k(s,t) 0 k(s,t)g(s,r), k(s,t) < 0 (23) U(t,r) = { k(s,t)g(s,r), k(s,t) 0 k(s,t)g(s,r), k(s,t) < 0 (24) for ech 0 r 1 nd s,t b We cn see tht (22) is crisp system of liner Fredholm integrl equtions for ech 0 r 1 nd t b Definition 21 ([14]) The fuzzy Fredholm integrl equtions system of the second kind is of the form: m g i (t) = f i (t)+ (λ ij k ij (s,t)g i (s)ds),i = 1,,m, (25)

4 180 M Amwi, N Qtnni where s,t,λ re rel constnts nd s,t [,b],λ ij 0 for i,j = 1,,m In the system (25), g(t) = [g 1 (t),,g m (t)] T is unknown function Moreover, the fuzzy function f i (t) nd kernel k ij (s,t) re known nd ssumed to be sufficiently differentible functions with respect to ll their rguments on the intervl [, b] Now, let the prmetric forms of f i (t) nd g i (t) re (f i (t,r),f i (t,r)) nd (g i (t,r),g i (t,r)), 0 r 1, t [,b], respectively We write the prmetric form of the given fuzzy Fredholm integrl equtions system s follows: g i (t,r) = f i (t,r)+ ( m ) b λ ij U i,j(s,r)ds, g i (t,r) = f i (t,r)+ ( m ) b λ ij U i,j(s,r)ds, i = 1,,m, (26) where nd U i,j (s,r) = U i,j (s,r) = { ki,j (s,t)g j (s,r), k i,j (s,t) 0 k i,j (s,t)g j (s,r), k i,j (s,t) < 0 { ki,j (s,t)g j (s,r), k i,j (s,t) 0 k i,j (s,t)g j (s,r), k i,j (s,t) < 0 (27) (28) 3 Tylor Expnsion Method This method is bsed on differentiting p-times both sides of the liner fuzzy Fredholm integrl eqution of the second kind nd then substitute the Tylor series expnsion for the unknown function into the integrl eqution As result, we obtin liner system for which the solution of this system yields the unknown Tylor coefficients of the solution functions If we ssume tht [11] { λij k i,j (s,t) 0, s c i,j, λ ij k i,j (s,t) < 0, c i,j s b,

5 NUMERICAL METHODS FOR SOLVING FUZZY 181 then system (26) cn be trnsformed into g i (t,r) = f i (t,r) m ( c + λ ij k i,j (s,t)g j (s,r)ds+ g i (t,r) = f i (t,r) m ( c + λ ij k i,j (s,t)g j (s,r)ds+ c c ) k i,j (s,t)g j (s,r)ds, ) k i,j (s,t)g j (s,r)ds, i = 1,,m (31) Differentiting both sides of ech eqution of the system (31) N-times with respect to t, we get (p) g i (t,r) t p = (p) f i (t,r) ( t m p c (p) k i,j (s,t) + λ i,j t p g j (s,r)ds+ (p) g i (t,r) t p = (p) f i (t,r) ( t m p c (p) k i,j (s,t) + λ i,j t p g j (s,r)ds+ c c ) (p) k i,j (s,t) t p g j (s,r)ds, ) (p) k i,j (s,t) t p g j (s,r)ds, i,j = 1,,m, p = 0,,N (32) Let us introduce the following nottions for shortness: g (p) i = (p) g i (t,r) t p t=z, g (p) i = (p) g i (t,r) t p t=z, f (p) i = (p) f i (t,r) (p) t p t=z, f i = (p) f i (t,r) t p t=z, k (p) i,j = (p) k i,j (t,r) t p t=z, i,j = 1,,m (33) Now, we expnd the unknown functions g i (s,r) nd g i (s,r) in Tylor series for multivrite vrible t rbitrry point z, neglecting the trunction error, nd get g j,n = N 1 q! g (p) (s,r)(s z) p, i g j,n = N 1 q! g i (p) (s,r)(s z) p (34), z b,j = 1,,m

6 182 M Amwi, N Qtnni Substituting equtions (34) nd (33) into eqution (32) yields: m N g (p) i = f (p) i + λ c ij k (p) i,j (s,z)g q! j (s z) p ds + g (p) (p) m N i = f i + By using the nottions: + N N λ ij q! λ ij q! λ ij q! c c c k (p) i,j (s,z)g j (s z) p ds, k i,j (p) (s,z)g j (s z) p ds k (p) i,j (s,z)g j (s z) p ds (35) w (i,j) p,q = λ ij q! w (i,j) p,q = λ ij q! ci,j eqution (35) cn be written in the form k (p) i,j (s,z)(s z) p ds, i,j = 1,,m, c i,j k (p) i,j (s,z)(s z) p ds, p,q = 0,,N, g (p) i =f (p) i m N + w (i,j) p,q g j + g (p) (p) i =f i m N + w (i,j) p,q g j + We cn rerrnge (37) s follows: f (p) i = g (p) i m N + w (i,j) p,q g j + (p) f i (p) = gi m N + w (i,j) p,q g j + N w (i,j) p,q g j, N w (i,j) p,q g j N w (i,j) p,q g j, N w (i,j) p,q g j (36) (37) (38)

7 NUMERICAL METHODS FOR SOLVING FUZZY 183 When p = q, (38) becomes: f i (p) = m N ( wp,q (i,j) 1 ) g j, f i (p) = m N ( wp,q (i,j) 1 ) g j (39) Consequently, eqution (37) cn be written in the following mtrix form: where G = g 1 g i (N) g 1 g i (N) g m g m (N) g m g m (N) WG = F, (310) f 1 f (N) i f 1 (N) f i W (1,1) W (1,m),F =,W = f m W (m,1) W (m,m) f (N) m f m (N) f m The prochil mtrices W (i,j) re defined by the following elements (see [11]) [ ] W (i,j) (i,j) (i,j) W1,1 W = 1,2 (i,j) (i,j), W 2,1 W 2,2 where W (i,j) (i,j) 1,1 = W 2,2 W (i,j) (i,j) 0,0 1 W 0,1 (i,j) W 0,N 1 (i,j) W 0,N (i,j) W 1,0 W (i,j) (i,j) (i,j) 1,1 1 W 1,N 1 W 1,N =, (i,j) (i,j) W N 1,0 W N 1,1 W (i,j) (i,j) N 1,N 1 1 W N 1,N W (i,j) N,0 1 (i,j) W N,1 (i,j) W N,N 1 W (i,j) N,N 1

8 184 M Amwi, N Qtnni W (i,j) 1,2 = W (i,j) 2,1 W (i,j) 0,0 W (i,j) 0,1 W (i,j) 0,N 1 W (i,j) 0,N W (i,j) 1,0 W (i,j) 1,1 W (i,j) 1,N 1 W (i,j) 1,N = W (i,j) N 1,0 W (i,j) N 1,1 W (i,j) N 1,N 1 W (i,j) N 1,N W (i,j) N,0 W (i,j) N,1 W (i,j) N,N 1 W (i,j) N,N The solution of system (310) is given s: g(t,r) = N g(t,r) = N p=0 1 p! p=0 1 (p) g(t,r) p! t p t=z (t z) p, (p) g(t,r) t p t=z (t z) p, z b (311) 31 Convergence Anlysis In order to show the efficiency of the Tylor expnsion method, one cn show tht the pproximte solution converges to the exct solution of system (26) (see [11] for more detils) Theorem 31 ([11]) If g j,n (t,r) nd g j,n (t,r) re Tylor polynomils of degree N nd their coefficients hve been found by solving the liner system (310), then they converge to the exct solution of system (26), when N 4 Trpezoidl Method We compute the integrl of fuzzy function g(t,r) using the Riemnn integrl of g(t,r) nd g(t,r) by pplying the trpezoidl rule We consider g(t,r) nd g(t,r)overtheintervl[,b], thensubdividetheintervl[,b]intonsubintervls of equl width h = (b ) n using eqully spced nodes: We define (see [6]) = t 0 < t 1 < < t n 1 < t n = b, t i = +ih,t i t i 1 = h,1 i n [ f(,r)+f(b,r) S n (r) = h 2 + ] n 1 i=1 f(t i,r), [ S n (r) = h f(,r)+f(b,r) 2 + ] n 1 i=1 f(t i,r) (41) (42)

9 NUMERICAL METHODS FOR SOLVING FUZZY 185 Then, for rbitrry fixed r, we hve lim S n n(r) = g(r) = f(t,r)dt, lim S n(r) = g(r) = n f(t,r)dt (43) Theorem 41 ([7]) If f(t) is continuous in the metric D, then S n (r) nd S n (r) converge uniformly in r to g(t,r) nd g(t,r) respectively Definition 41 ([6]) A fuzzy number u(r) = (u(r),u(r)) belongs to CE is defined s CE = {(u(r),u(r)) : u(r),u(r) C[0,1]}, where CE is subclss of E Theorem 42 ([7]) Let g(t,r) = (g(t,r),g(t,r)) be fuzzy continuous function in t for fixed r nd belong to CE, then its pproximte solutions S n (r) nd S n (r) converge uniformly The exct itertive process for finding the exct solution for eqution (21) is given by g 0 (t) = f(t), g m (t) = f(t)+λ k(s,t)g (44) m 1(s)ds, m 1 However, the numericl process provide us with pproximte fuzzy function for g m (t) If we denote it S (m) n t the m-th itertion using n integrtion nodes, then we hve S n (m) (t,r) = f(t,r)+λ k(s,t)s n 1 (m 1) ds+δ n (t,r), (45) where δ n (t,r) = (δ n (1) (t,r),δ n (2) (t,r)) re uniformly convergent to 0 s n,m Now, let δ n (t,r) = (δ n (t,r),δ n (t,r)) nd neglect δ n (t,r) in eqution (45), we obtin nd S n (0) (t,r) = f(t,r), S n (m) (t,r) = f(t,r)+λ k(s,t)s n 1 (m 1) ds (46) (0) S n (t,r) = f(t,r), (m) (m 1) (47) S n (t,r) = f(t,r)+λ k(s,t)s n 1 ds

10 186 M Amwi, N Qtnni Theorem 43 ([7]) Let S n (m) (t) be n pproximtion to g m (t) using the trpezoidl rule with m eqully spced integrtion nodes, then S n (m) (t) converges uniformly to the unique solution g(t) when n,m 5 Numericl Exmples nd Results In order to test the efficiency of our numericl schemes, we hve crried out some numericl experiments We compre numericl results with exct solutions using the metric of Definition 51(see Tbles 1 nd 2) Further comprison between the pproximte solutions nd the exct solutions cn be seen in Figures 1 nd 2 for fixed t = 1 Definition 51 ([23]) For rbitrry fuzzy numbers u,v E the quntity D(u,v) = sup {mx u(r) v(r),mx u(r) v(r) } 0 r 1 defines the distnce between u nd v Exmple 1 (Tylor expnsion method) The following fuzzy Fredholm integrl equtions: g(t,r) = (r +1)(exp( t)+t sint) sintg(s,s)ds, g(t,r) = (3 r)(exp( t)+t sint) sintg(s,s)ds (51) hve the exct solutions g(t,r) = (r +1)(exp( t)+t), g(t,r) = (3 r)(exp( t)+t) (52) Here we expnd the unknown functions g(t,r) nd g(t,r) in Tylor series t z = 1 2 Algorithm (51): 1 input,b,λ i,j,z,m,k i,j (s,t),f i (t,r),f i (t,r) 2 input the Tylor expnsion degree N 3 clculte (p) k i,j (s,t) t p 4 clculte w (i,j) p,q = λ ij q!, (p) f i (t,r) t p ci,j, (p) f i (t,r) t,p,q = 0,,N p k (p) i,j (s,z)(s z) p ds,i,j = 1,,m

11 NUMERICAL METHODS FOR SOLVING FUZZY clculte w (i,j) p,q = λ ij q! 6 put W (i,j) (i,j) 1,1 = W 2,2 c i,j k i,j (p) (s,z)(s z) p ds,p,q = 0,,N W (i,j) (i,j) (i,j) (i,j) 0,0 1 W 0,1 W 0,N 1 W 0,N (i,j) W 1,0 W (i,j) (i,j) (i,j) 1,1 1 W 1,N 1 W 1,N = (i,j) (i,j) W N 1,0 W N 1,1 W (i,j) (i,j) N 1,N 1 1 W N 1,N W (i,j) (i,j) (i,j) N,0 1 W N,1 W N,N 1 W (i,j) N,N 1 7 put W (i,j) 1,2 = W (i,j) 2,1 W (i,j) 0,0 W (i,j) 0,1 W (i,j) 0,N 1 W (i,j) 0,N W (i,j) 1,0 W (i,j) 1,1 W (i,j) 1,N 1 W (i,j) 1,N = W (i,j) N 1,0 W (i,j) N 1,1 W (i,j) N 1,N 1 W (i,j) N 1,N W (i,j) N,0 W (i,j) N,1 W (i,j) N,N 1 W (i,j) N,N 8 denote[ ] W (i,j) (i,j) (i,j) W1,1 W = 1,2 (i,j) (i,j) W 2,1 W 2,2 f 1 f (N) i f 1 (N) f i W (1,1) W (1,m) 9 put F =,W = f m W (m,1) W (m,m) f (N) m f m (N) f m 10 solve the following liner system WG = F 11 Estimte g,g by computing Tylor expnsion for G g(t,r) = N g(t,r) = N p=0 1 p! p=0 1 (p) g(t,r) p! (p) g(t,r) t p t p t=z (t z) p, t=z (t z) p, z b,j = 1,,m

12 188 M Amwi, N Qtnni Thus we obtin the following results: W (i,j) (i,j) 1,1 = W 2,2 = [ ] W 1,1 (i,j) = W 2,2 (i,j) = , hence [ ] 1,1 1,1 W1,1 W W = 1,2 1,1 1,1 W 2,1 W 2,2 f(t,r) f (t,r) (r +1)(exp( t)+t sint) f (t,r) (r +1)( exp( t)+1 cost) f (r +1)(exp( t)+sint) (t,r) F = f(t,r) = (r +1)( exp( t)+cost) (3 r)(exp( t)+t sint) f (t,r) (3 r)( exp( t)+1 cost) f (t,r) (3 r)(exp( t)+sint) f (3 r)( exp( t)+cost) (t,r) (r + 1) (r + 1) (r + 1) F = (r + 1) (3 r) (3 r) (3 r) (3 r) Solving the following liner system t= 1 2 WG = F, we obtin

13 NUMERICAL METHODS FOR SOLVING FUZZY 189 G = Now, g(05, r) g (05,r) g (05,r) g (05,r) g(05, r) g (05,r) g (05,r) g (05,r) = (r + 1) (r + 1) (r + 1) (r + 1) (3 r) (3 r) (3 r) (3 r) g(t,r) = N p=0 p=0 1 (p) g(t,r) p! t p t= 1(t )p = (r + 1) (r + 1)t (r +1)t (r +1)t 3, N 1 (p) g(t,r) g(t,r) = p! t p t= 1(t )p = (3 r) (3 r)t (3 r)t (3 r)t 3 (53) Figure 1 compres the exct solution nd the pproximte solution for fixed t = 1 r g exct g exct g pproximte g pproximte error = D(g exct,g pproximte ) Tble 1: The error resulted by lgorithm (51) t t = 1 Exmple 2 (Trpezoidl method) The fuzzy Fredholm integrl equtions (51) hve the exct solution (52) where n = 51, on the intervl [0,1],h = = ,0 = t 0 t 1 t 51 = 1,t i = ih (b ) n

14 190 M Amwi, N Qtnni Figure 1: Exct solution nd pproximte solution for t = 1 The pproximte fuzzy function clculted t the 24 th itertion with n = 51 The following lgorithm implements the trpezoidl rule using the MAPLE softwre Algorithm (52): 1 input,b,λ,k(s,t),f(t,r),f(t,r),n,m 2 h = b n 3 t 0 =,t n = b 4 For i =, to n, compute t i = +ih 5 compute S (0) n (t,r) = f(t,r), [ f(,r)+f(b,r) S n (r) = h 2 + ] n 1 i=1 f(t i,r), S (m) n (t,r) = f(t,r)+λ k(s,t)s n 1 (m 1) ds (0) 6 compute[ S n (t,r) = f(t,r), S n (r) = h f(,r)+f(b,r) 2 + ] n 1 i=1 f(t i,r), S n (m) (t,r) = f(t,r)+λ k(s,t)s n 1(m 1) ds Thus we obtin the following results: (0) (t,r) = (r +1)(exp( t)+t sint) (1) (t,r) = (r +1)[exp( t)+t] (r +1)sint

15 NUMERICAL METHODS FOR SOLVING FUZZY 191 (2) (t,r) = (r +1)[exp( t)+t] (r +1)sint (3) (t,r) = (r +1)[exp( t)+t] ( r )sint (4) (t,r) = (r +1)[exp( t)+t] (r +1)sint (5) (t,r) = (r +1)[exp( t)+t] ( r )sint (6) (t,r) = (r +1)[exp( t)+t] (r +1)sint (7) (t,r) = (r +1)[exp( t)+t] (r +1)sint (8) (t,r) = (r +1)[exp( t)+t] (r +1)sint (9) (t,r) = (r +1)[exp( t)+t] (r +1)sint (10) (t,r) = (r +1)[exp( t)+t] (r +1)sint (11) (t,r) = (r +1)[exp( t)+t] (r +1)sint (12) (t,r) = (r +1)[exp( t)+t] (r +1)sint (13) (t,r) = (r +1)[exp( t)+t] (r +1)sint (14) (t,r) = (r +1)[exp( t)+t] (r +1)sint (15) (t,r) = (r +1)[exp( t)+t] ( r )sint (16) (t,r) = (r +1)[exp( t)+t] ( r )sint (17) (t,r) = (r +1)[exp( t)+t] ( r )sint (18) (t,r) = (r +1)[exp( t)+t] ( r )sint (19) (t,r) = (r +1)[exp( t)+t] ( r )sint (20) (t,r) = (r +1)[exp( t)+t] ( r )sint (21) (t,r) = (r +1)[exp( t)+t] (r +1)sint (22) (t,r) = (r +1)[exp( t)+t] (r +1)sint (23) (t,r) = (r +1)[exp( t)+t] (r +1)sint (24) (t,r) = (r +1)[exp( t)+t] (r +1)sint nd (0) (t,r) = (3 r)(exp( t)+t sint) (1) (t,r) = (3 r)[exp( t)+t] ( r)sint (2) (t,r) = (3 r)[exp( t)+t] ( r)sint (3) (t,r) = (3 r)[exp( t)+t] ( r)sint (4) (t,r) = (3 r)[exp( t)+t] ( r)sint (5) (t,r) = (3 r)[exp( t)+t] ( r)sint (6) (t,r) = (3 r)[exp( t)+t] ( r)sint (7) (t,r) = (3 r)[exp( t)+t] ( r)sint (8) (t,r) = (3 r)[exp( t)+t] ( r)sint (9) (t,r) = (3 r)[exp( t)+t] ( r)sint (10) (t,r) = (3 r)[exp( t)+t] ( r)sint

16 192 M Amwi, N Qtnni (11) (t,r) = (3 r)[exp( t)+t] ( r)sint (12) (t,r) = (3 r)[exp( t)+t] ( r)sint (13) (t,r) = (3 r)[exp( t)+t] ( r)sint (14) (t,r) = (3 r)[exp( t)+t] ( r)sint (15) (t,r) = (3 r)[exp( t)+t] ( r)sint (16) (t,r) = (3 r)[exp( t)+t] ( r)sint (17) (t,r) = (3 r)[exp( t)+t] ( r)sint (18) (t,r) = (3 r)[exp( t)+t] ( r)sint (19) (t,r) = (3 r)[exp( t)+t] ( r)sint (20) (t,r) = (3 r)[exp( t)+t] ( r)sint (21) (t,r) = (3 r)[exp( t)+t] ( r)sint (22) (t,r) = (3 r)[exp( t)+t] ( r)sint (23) (t,r) = (3 r)[exp( t)+t] ( r)sint (24) (t,r) = (3 r)[exp( t)+t] ( r)sint Figure 2 compres the exct solution nd the pproximte solution for fixed t = 1 Figure 2: Exct solution nd pproximte solution for t = 1

17 NUMERICAL METHODS FOR SOLVING FUZZY 193 r g exct g exct g pproximte g pproximte error = D(g exct,g pproximte ) Tble 2: The error resulted by lgorithm (52) t t = 1 6 Conclusions In this rticle, some numericl schemes, nmely: the Tylor expnsion method nd the Trpezoidl method, hve been investigted nd implemented to pproximte the solution of the fuzzy Fredholm integrl eqution of the second kind A comprison between these methods shows the Tylor expnsion method is more efficient thn the Trpezoidl method Moreover, it hs been concluded in [11] tht if the exct solution of the fuzzy Fredholm integrl eqution of the second kind is polynomil then the metric error is zero, nd this hs lso been justified in [2] According to our test numericl exmple one cn observe from Tbles 1 nd 2 nd Figures 1 nd 2 tht the Tylor expnsion method is more ccurte thn the Trpezoidl method References [1] S Altie, Numericl solution of fuzzy integrl equtions of the second kind using bernstein polynomils, Journl of Al-Nhrin University, 15 (2012), [2] M Amwi, Fuzzy Fredholm integrl eqution of the second kind, MSc Thesis, An-Njh Ntionl University (2014) [3] H Attri nd A Yzdni, A computtionl method for fuzzy Voltrr- Fredholm integrl equtions, Fuzzy Inf Eng 2 (2011), [4] W Congxin nd M M, On embedding problem of fuzzy number spces, Fuzzy Sets nd Systems, 44 (1991), 33-38, nd 45 (1995), [5] D Dubois nd H Prde, Towrds fuzzy differentil clculus, Fuzzy Sets nd System 8 No 1-7 (1982), ,

18 194 M Amwi, N Qtnni [6] M Friedmn, M Ming nd A Kndel, Numericl solutions of fuzzy differentil nd integrl equtions, Fuzzy Sets nd Systems 106 (1999), [7] M Friedmn, M Ming nd A Kndel, On fuzzy integrl equtions, Fundmentl Informtice 37 (1999), [8] R Goetschel nd W Voxmn, Elementry clculus, Fuzzy Sets nd Systems 18 (1986), [9] M Ghnbri, R Toushmlni nd E Kmrni, Numericl solution of liner Fredholm fuzzy integrl eqution of the second kind by block-pulse functions, Aust J Bsic nd Appl Sci 3 (2009), [10] H Goghry nd M Goghry, Two computtionl methods for solving liner Fredholm fuzzy integrl equtions of the second kind, Applied Mthemtics nd Computtion, 182 (2006), [11] A Jfrin, S MesoomyNi, S Tvn nd M Bnifzel, Solving liner fredholm fuzzy integrl equtions system by tylor expnsion method, Applied Mthemticl Sciences 6 (2012), [12] Y Jfrzdeh, Numericl solution for fuzzy integrl equtions with upperbound on error by splinders interpoltion, Fuzzy Inf Eng 3 (2012), [13] O Klev, Fuzzy differentil equtions, Fuzzy Sets nd Systems 24 (1987), [14] M Keynpour nd T Akbrin, New pproch for solving of liner Fredholm fuzzy integrl equtions using sinc function, J of Mthemtics nd Computer Science 3 (2011), [15] T Lotfi nd M Mhdini, Fuzzy Glerkin method for solving Fredholm integrl equtions with error nlysis, Int J Industril Mthemtics 3 (2011), [16] Kh Mleknejd, R Mollpoursl, P Torbi nd M Alizdeh, Solution of first kind Fredholm integrl eqution by sinc function, World Acdemy of Science, Engineering nd Technology 66 (2010) [17] M Mtlok, On Fuzzy integrls, In: Proc 2nd Polish Symp on Intervl nd Fuzzy Mthemtics, Politechnik Poznnsk (1987),

19 NUMERICAL METHODS FOR SOLVING FUZZY 195 [18] F Mirzee, M Pripour nd M Yri, Numericl solution of Fredholm fuzzy integrl eqution of the second kind vi direct method using tringulr functions, Journl of Hyperstructures 1 (2012), [19] S Nnd, On integrtion of fuzzy mppings, Fuzzy Sets nd System 32 (1999), [20] N Prndin nd M Arghi, The pproximte solution of liner fuzzy Fredholm integrl equtions of the second kind by using itertive interpoltion, World Acdemy of Science, Engineering nd Technology 49 (2009), [21] N Prndin nd M Arghi, The numericl solution of liner fuzzy Fredholm integrl equtions of the second kind by using finite nd divided differences methods, Soft Comput 15 (2010), [22] J Prk, YC Kwun nd JU Jeong, Existence of solutions of fuzzy integrl equtions in bnch spces, Fuzzy Sets nd Systems 72 (1995), [23] S Ziri, R Ezzti nd S Abbsbndy, Numericl solution of liner fuzzy Fredholm integrl equtions of the second kind using fuzzy hr wvelet, In: Advnces in Computtionl Intelligence, Springer Ser Communictions in Computer nd Informtion Science 299 (2012), 79-89; 9

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