Solving Linear Fredholm Fuzzy Integral Equations System by Taylor Expansion Method

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1 Applied Mthemticl Sciences, Vol 6, 212, no 83, Solving Liner Fredholm Fuzzy Integrl Equtions System y Tylor Expnsion Method A Jfrin 1, S Mesoomy Ni, S Tvn nd M Bnifzel Deprtment of Mthemtics, Urmi Brnch Islmic Azd University, Urmi, Irn Astrct In this pper we intend to offer numericl scheme to solve liner Fredholm fuzzy integrl equtions system of the second kind For this im, we pply Tylor expnsion method to convert the given fuzzy system into liner system in crisp cse Now the solution of this system yields the unknown Tylor coefficients of the solution functions The proposed method is illustrted y n exmple Results re compred with the exct solution y using computer simultions Keywords: Fredholm fuzzy integrl equtions system; Tylor series; Convergence nlysis; Approximte solutions 1 Introduction Integrl equtions re very useful for solving mny prolems in severl pplied fields like mthemticl economics nd optiml control theory Since these equtions usully cn not e solved explicitly, so it is required to otin pproximte solutions There re numerous numericl methods which hve een focusing on the solution of integrl equtions For exmple, Tricomi, in his ook [19], introduced the clssicl method of successive pproximtions for nonliner integrl equtions Vritionl itertion method [12] nd Adomin decomposition method [4] were effective nd convenient for solving integrl equtions Also the Homotopy nlysis method (HAM) ws proposed y Lio [13] nd then hs een pplied in [1] Moreover, some different vlid methods for solving this kind of equtions hve een developed First time, Tylor expnsion pproch ws presented for solution integrl equtions y Knwl nd Liu in [11] nd then hs een extended in [14, 15, 16] In ddition, Bolin et l [3] y using the orthogonl tringulr sis functions, solved some integrl equtions systems Jfri et l [9] pplied Legendre wvelets method to find 1 jfrin5594@yhoocom

2 414 A Jfrin, S Mesoomy Ni, S Tvn nd M Bnifzel numericl solution system of liner integrl equtions In this pper we wnt to propose new numericl pproch to pproximte the solution of liner Fredholm fuzzy integrl equtions system This method converts the given fuzzy system tht supposedly hs n unique fuzzy solution, into crisp liner system For this scope, first the Tylor expnsions of unknown functions re sustituted in prmetric form of the given fuzzy system Then we differentite oth sides of the resulting integrl equtions of the system N times nd lso pproximte the Tylor expnsion y suitle trunction limit This work yields liner system in crisp cse, such tht the solution of the liner system yields the unknown Tylor coefficients of the solution functions An interesting feture of this method is tht we cn get n pproximte of the Tylor expnsion in ritrry point to ny desired degree of ccurcy Here is n outline of the pper In section 2, the sic nottions nd definitions of the integrl eqution nd the Tylor polynomil method re riefly presented Section 3 descries how to find n pproximte solution of the given Fredholm fuzzy integrl equtions system y using proposed pproch Finlly in section 4, we pply the proposed method y n exmple to show the simplicity nd efficiency of the method 2 Preliminries In this section the most sic used nottions in fuzzy clculus nd nd integrl equtions re riefly introduced We strted y defining the fuzzy numer Definition 1 A fuzzy numer is fuzzy set u : R 1 I =[, 1] such tht: i u is upper semi-continuous, ii u(x) = outside some intervl [, d], iii There re rel numers, c : c d, for which: 1 u(x) is monotoniclly incresing on [, ], 2 u(x) is monotoniclly decresing on [c, d], 3 u(x) =1, x c The set of ll fuzzy numers (s given y definition 1 ) is denoted y E 1 [7, 17] Definition 2 A fuzzy numer v is pir (v, v) of functions v(r) nd v(r) : r 1, which stisfy the following requirements: i v(r) is ounded monotoniclly incresing, left continuous function on (, 1] nd right continuous t,

3 Liner Fredholm fuzzy integrl equtions 415 ii v(r) is ounded monotoniclly decresing, left continuous function on (, 1] nd right continuous t, iii v(r) v(r): r 1 A populr fuzzy numer is the tringulr fuzzy numer v =(v m,v l,v u ) where v m denotes the modl vlue nd the rel vlues v l nd v u represent the left nd right fuzziness, respectively The memership function of tringulr fuzzy numer is defined s follows: μ v (x) = x v m v l +1, v m v l x v m, v u +1, v m x v m + v u,, otherwise v m x Its prmetric form is: v(r) =v m + v l (r 1), v(r) =v m + v u (1 r), r 1 Tringulr fuzzy numers re fuzzy numers in LR representtion where the reference functions L nd R re liner 21 Opertion on fuzzy numers We riefly mention fuzzy numer opertions defined y the Zdeh extension principle [21, 22] The following ddition, multipliction nd nonliner mpping of fuzzy numers re necessry to define when deling with fuzzy integrl equtions: μ A+B (z) =mx{μ A (x) μ B (y) z = x + y}, μ f(net) (z) =mx{μ A (x) μ B (y) z = xy}, where A nd B re fuzzy numers, μ () denotes the memership function of ech fuzzy numer, is the minimum opertor, nd f is continuous ctivtion function (such s f(x) = x) of output unit of our fuzzy neurl network The ove opertions on fuzzy numers re numericlly performed on level sets (ie α-cuts) For <α 1, α-level set of fuzzy numer A is defined s: [A] α = {x μ A (x) α, x R}, nd [A] = αɛ(,1] [A]α Since level sets of fuzzy numers ecome closed intervls, we denote [A] α y [A] α =[[A] α l, [A]α u],

4 416 A Jfrin, S Mesoomy Ni, S Tvn nd M Bnifzel where [A] α l nd [A] α u re the lower nd the upper limits of the α-level set [A] α, respectively From intervl rithmetic [2], the ove opertions on fuzzy numers re written for the α-level sets s follows: [A] α +[B] α =[[A] α l, [A]α u ]+[[B]α l, [B]α u ]=[[A]α l +[B] α l, [A]α u +[B]α u ], (1) f([net] α )=f([net] α l, [Net]α u ]) = [f([net]α l ),f([net]α u )], k[a] α = k[[a] α l, [A] α u]=[k[a] α l,k[a] α u], if k, (2) k[a] α = k[[a] α l, [A] α u]=[k[a] α u,k[a] α l ], if k < For ritrry u =(u, u) nd v =(v, v) we define ddition (u + v) nd multipliction y k s [7, 17]: (u + v)(r) =u(r)+v(r), (u + v)(r) =u(r)+v(r), (ku)(r) =ku(r), (kv)(r) =ku(r), if k, (ku)(r) =ku(r), (kv)(r) =ku(r), if k < Definition 3 For ritrry fuzzy numers u, v ɛ E 1 the quntity D(u, v) = sup {mx[ u(r) v(r), u(r) v(r) ]} r 1 is the distnce etween u nd v It is shown tht (E 1,D) is complete metric spce [18] Definition 4 Let f :[, ] E 1 For ech prtition P = {t,t 1,, t n } of [, ] nd for ritrry ξ i ɛ [t i 1,t i ](1 i n), suppose R P = n i=1 f(ξ i )(t i t i 1 ), Δ:=mx{ t i t i 1,i=1,, n} The definite integrl of f(t) over [, ] is f(t)dt = lim Δ R P

5 Liner Fredholm fuzzy integrl equtions 417 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 [7] Also, ( ( f(t, r) dt) = f(t, r) dt) = f(t, r)dt, f(t, r)dt More detils out properties of the fuzzy integrl re given in [7, 1] 22 System of integrl equtions The sic definition of integrl eqution is given in [8] Definition 5 The Fredholm integrl eqution of the second kind is where F (t) =f(t)+λ(ku)(t), (3) (ku)(t) = K(s, t)f (s)ds In Eq (3), K(s, t) is n ritrry kernel function over the squre s, t nd f(t) is function of t : t If the kernel function stisfies K(s, t) =, s>t,we otin the Volterr integrl eqution F (t) =f(t)+λ t K(s, t)f (s)ds (4) In ddition, if f(t) e crisp function then the solution of ove eqution is crisp s well Also if f(t) e fuzzy function we hve Fredholm fuzzy integrl eqution of the second kind which my only process fuzzy solutions Sufficient conditions for the existence eqution of the second kind where f(t) is fuzzy function, re given in [5, 6] Definition 6 The second kind fuzzy Fredhoolm integrl equtions system is in the form F 1 (t) =f 1 (t)+ ( m ) j=1 λ 1j K 1j(s, t)f j (s)ds F i (t) =f i (t)+ ( m ) j=1 λ ij K ij(s, t)f j (s)ds, (5) F m (t) =f m (t)+ m j=1 ( ) λ mj K mj(s, t)f j (s)ds

6 418 A Jfrin, S Mesoomy Ni, S Tvn nd M Bnifzel where t, s nd λ ij (for i,j =1,, m) re rel constnts Moreover, in system (5), the fuzzy function f i (t) nd kernel K i,j (s, t) re given nd ssumed to e sufficiently differentile with respect to ll their rguments on the intervl t, s Also F (t) =[F 1 (t),, F m (t)] T is the solution to e determined Now let (f i (t, r), f i (t, r)) nd (F i (t, r), F i (t, r)) ( r 1; t ) e prmetric form of f i (t) nd F i (t), respectively In order to design numericl scheme for solving (5), we write the prmetric form of the given fuzzy integrl equtions system s follows: F i (t, r) =f i (t, r)+ j=1 ( ) λ ij U i,j(s, r)ds, (6) F i (t, r) =f i (t, r)+ j=1 ( ) λ ij U i,j(s, r)ds,i=1,, m, where nd K i,j (s, t)f j (s, r),k i,j (s, t) U i,j (s, r) = K i,j (s, t)f j (s, r),k i,j (s, t) < K i,j (s, t)f j (s, r),k i,j (s, t) U i,j (s, r) = K i,j (s, t)f j (s, r),k i,j (s, t) <, 23 Tylor series Let us first recll the sic principles of the Tylor polynomil method for solving Fredholm fuzzy integrl equtions system (5) Becuse these results re the key for our prolems therefore we explin them Without loss of generlity, we ssume tht λ i,j K i,j (s, t) λ i,j K i,j (s, t) <, s c i,j,c i,j s

7 Liner Fredholm fuzzy integrl equtions 419 With ove supposition, the system (6) is trnsformed to following form: F i (t, r) =f i (t, r)+ ( m j=1 λ ci,j ij K i,j (s, t)f j (s, r)ds + ) c i,j K i,j (s, t)f j (s, r)ds,i=1,, m F i (t, r) =f i (t, r)+ ( m j=1 λ ci,j ij K i,j (s, t)f j (s, r)ds + ) c i,j K i,j (s, t)f j (s, r)ds (7) Now we wnt to otin the solution of the ove system in the form of F j,n (t, r) = N i= ( 1 i! (i) F j (t, r) t i t=z (t z) i ), t, z, r 1, (8) F j,n (t, r) = N i= ( 1 i! (i) F j (t, r) t i t=z (t z) i ), t, z, r 1, (for j =1,, m) which re the Tylor expnsions of degree N t t = z for the unknown functions F j (t, r) nd F j (t, r), respectively For this scope we differentite ech eqution of system (7), (N +1) times (for p =,, N) with respect to t nd get (p) F i (t,r) = (p) f i (t,r) + t p t ( p m j=1 λ ci,j (p) K i,j (s,t) ij t p (p) F i (t,r) = (p) f i (t,r) + t p t ( p m j=1 λ ci,j (p) K i,j (s,t) ij t p F j (s, r)ds + c i,j (p) K i,j (s,t) t p F j (s, r)ds + c i,j (p) K i,j (s,t) t p where (i =1,, m) For revity, we define elow symols s: F (p) ) F j (s, r)ds ) F j (s, r)ds, (9) j (z, r) := (p) F j (t, r) t p t=z nd F (p) j (z, r) := (p) F j (t, r) t p t=z,j=1,, m The im of this study is determining of the coefficients F (p) j (z, r) nd F (p) j (z, r), (for p =,, N; j =1,, m) in system (9) For this intent, we expnded F j (s, r) nd F j (s, r) in Tylor series t ritrry point z : z for

8 411 A Jfrin, S Mesoomy Ni, S Tvn nd M Bnifzel exmple, z = nd sustituted it s N-th trunction in (9) Now we cn write: F (p) i (, r) = (p) f i (t,r) t p t= + ( m N j=1 q= w(i,j) p,q F (q) j (, r)+ ) N q= w (i,j) p,q F (q) j (, r), (1) F (p) i (, r) = (p) f i (t,r) t p t= + ( m N j=1 q= w(i,j) p,q F (q) j (, r)+ ) N q= w (i,j) p,q F (q) j (, r) where p,q = λ i,j q! ci,j (p) K i,j (s, t) t p t= (s ) q ds, p, q =,, N, nd w (i,j) p,q = λ i,j q! c i,j (p) K i,j(s, t) t p t= (s ) q ds, i, j =1,, m Consequently, the mtrix form of expression (1) cn e written s follows: where Y =[F 1 (, r),, F (N) 1 (, r), F 1 (, r),, F (N) 1 (, r), WY = E, (11),F m (, r),, F m (N) (, r), F m (, r),, F (N) m (, r)], E =[ f 1 (, r),, (N) f 1 (t, r) t N t=, f 1 (, r),, (N) f 1 (t, r) t N t=, f m (, r),, (N) f m (t, r) t N t=, f m (, r),, (N) f m (t, r) t N t= ] nd W (1,1) W (1,m) W = W (m,1) W (m,m)

9 Liner Fredholm fuzzy integrl equtions 4111 Prochil mtrices W (i,j), (for i,j =1,, m) re defined with following elements: W (i,j) 1,1 W (i,j) 1,2 W (i,j) =, W (i,j) 2,1 W (i,j) 2,2 where W (i,j) 1,1 = W (i,j) 2,2 =, 1,1,N 1 1, 1,1 1 1,N 1 N 1,1 N 1,N 1 1 N 1, N, N,1 N,N 1,N 1,N w(i,j) N 1,N N,N 1, W (i,j) 1,2 = W (i,j) 2,1 =,,1,N 1 1, 1,1 N 1, 1,N 1 N 1,1 N 1,N 1 N, N,1 N,N 1,N 1,N N 1,N N,N 3 Convergence nlysis In this section we proved tht the ove numericl method convergence to the exct solution of fuzzy system (5) Theorem 1 Let F j,n (t) nd F j,n (t) (for j =1,, m) e Tylor polynomils of degree n tht their coefficients hve een produced y solving the liner system (11) Then these polynomils converge to the exct solution of the fuzzy Fredholm integrl equtions system (5), when N + Proof Consider the system (5) Since, the series (8) converge to F j (t, r) nd F j (t, r) (for j =1,, m) respectively, then we conclude tht:

10 4112 A Jfrin, S Mesoomy Ni, S Tvn nd M Bnifzel F in (t, r) =f i (t, r)+ ( m j=1 λ ci,j ij K i,j (s, t)f jn (s, r)ds + ) c i,j K i,j (s, t)f jn (s, r)ds F in (t, r) =f i (t, r)+ ( m j=1 λ ci,j ij K i,j (s, t)f jn (s, r)ds + ) c i,j K i,j (s, t)f jn (s, r)ds, (12) where (i =1,, m) nd it holds tht F j (t, r) = lim F jn(t, r), nd F j (t) = lim F jn(t, r) N N We defined the error function e N (t, r) y sutrcting Eqs (12)-(7) s follows: where e N (t, r) = e in (t, r) = ( F i (t, r) F in (t, r) ) + e i,n (t, r), (13) i=1 e i,n (t, r) =e i,n (t, r)+e i,n (t, r), j=1 λ ij ( ci,j K i,j (s, t)(f j (s, r) F jn (s, r))ds ) nd + j=1 e in (t, r) = ( F i (t, r) F in (t, r) ) + ( ) λ ij c i,j K i,j (s, t)(f j (s, r) F jn (s, r))ds, j=1 λ ij ( ci,j K i,j (s, t)(f j (s, r) F jn (s, r))ds ) + j=1 ( ) λ ij c i,j K i,j (s, t)(f j (s, r) F jn (s, r))ds, We must prove when N +, the error function e N (t) ecomes to zero Hence we proceed s follows: ( e N e in = e in + e in ein + e in ) i=1 i=1 i=1

11 Liner Fredholm fuzzy integrl equtions 4113 ( (F i (t, r) F in (t, r)) + (F i (t, r) F in (t, r)) ) + i=1 ( λ i,j i=1 j=1 k i,j ( F j (s, r) F jn (s, r) + F j (s, r) F jn (s, r) )ds) Since k i,j is ounded, therefore (F j (s, r) F jn (s, r)) nd (F j (s, r) F jn (s, r)) imply tht e N nd proof is completed 4 Numericl exmples In this section, we present n exmple of liner Fredholm fuzzy integrl equtions system nd results will e compred with the exct solutions Exmple 41 Consider the system of Fredholm fuzzy integrl equtions with: f 1 (t, r) = 14t2 (r 2) 3 + 3t2 (r 3 2) 4 t(r 2) + 9rt2 (r 4 +2), 4 f 1 (t, r) =rt 27t2 (r 3 2) 4 14rt2 3 rt2 (r 4 +2), 4 f 2 (t, r) = 8(t2 + 1)(r 2) 3 t(3r 3 6) + 9(r3 2)(t 2) r(r4 + 2)(t 2) 2, 1 f 2 (t, r) =t(r 5 +2r) 141(r3 2)(t 2) 2 1 8r(t2 +1) 3 3r(r4 + 2)(t 2) 2, 1 kernel functions K 1,1 (s, t) =t 2 (1 + s), K 1,2 (s, t) =t 2 (1 s 2 ), K 2,1 (s, t) =s(1 + t 2 ) nd K 2,2 (s, t) =(t 2) 2 (1 s 3 ), s, t 2,

12 4114 A Jfrin, S Mesoomy Ni, S Tvn nd M Bnifzel nd =, =2, λ i,j =1(for i,j =1, 2) The exct solution in this cse is given y F 1 (t, r) =t(2 r), F 1 (t, r) =tr, F 2 (t, r) =t(6 3r 3 ) nd F 2 (t, r) =t(r 5 +2r) In this exmple we ssume tht z = Using Eqs (1)-(11), the coefficients mtrix W is clculted s following: W 1,1 W 1,2 W =, W 2,1 W 2,2 where 1 W 1,1 = 1 1,W1,2 =, W 2,1 = 2 8 nd W 2, = With using of ove mtrices, we cn rewrite the liner system (11) s follows: F 1 (,r) F 1(,r) F 1 (,r) r F 1 W (,r) = r 2 F 2 (,r) 6 F 2 (,r) 5 r r3 + 76r F 2 (,r) r r3 22r F 5 r r3 64 r (,r) 94 5 r r r

13 Liner Fredholm fuzzy integrl equtions 4115 The vector solution of ove liner system is: F 1 (,r) F 1 (,r) F 1 (,r) r F 1(,r) = 2 r F 2 (,r) F 2 (,r) r 5 +2r F 2 (,r) F 2 (,r) 6 3r 3 As showing in Figs 1 nd 2, fter propgting this solution in Eq (8) the clculted solution is equl to exct solution In other words, with using of this method we cn find the nlyticl solution for this kind of equtions system, if the exct solution of given prolem e polynomil r 1 8 F 1 (s,r) F 1 (s,r) t Fig 1 F 1 (s, r) nd F 1 (s, r) for Exmple F 2 (s,r) F 2 (s,r) r t Fig 2 F 2 (s, r) nd F 2 (s, r) for Exmple 41

14 4116 A Jfrin, S Mesoomy Ni, S Tvn nd M Bnifzel 5 Conclusions In some cses, n nlyticl solution cn not e found for integrl equtions system Therefore, numericl methods hve een pplied In this pper we hve worked out computtionl method to pproximting solution of Fredholm fuzzy integrl equtions system of the second kind In this study the present course is method for computing unknown Tylor coefficients of the solution functions Consider tht to get the est pproximting solutions of the given fuzzy equtions, the trunction limit N must e chosen lrge enough An interesting feture of this method is finding the nlyticl solution for given system, if the exct solution e polynomils of degree N or less thn N The nlyzed exmple illustrted the ility nd reliility of the present method References [1] S Asndy, Numericl solution of integrl eqution: Homotopy perturtion method nd Adomin s decomposition method, Applied Mthemtics nd Computtion, 173 (26), [2] G Alefeld nd J Herzerger, Introduction to Intervl Computtions, Acdemic Press, New York, 1983 [3] E Bolin, Z Msouri nd S Htmzdeh-Vrmzyr, A direct method for numericlly solving integrl equtions system using orthogonl tringulr functions, Int J Industril Mth 2 (29), [4] E Bolin, H Sdeghi Goghry nd S Asndy, Numericl solution of liner Fredholm fuzzy integrl equtions of the second kind y Adomin method, Applied Mthemtics nd Computtion, 161 (25), [5] W Congxin nd M Ming, On the integrls, series nd integrl equtions of fuzzy set-vlued functions J Hrin Inst Technol 21 (199), 9-11 [6] M Friedmn, M M nd A Kndel, Numericl solutions of fuzzy differentil nd integrl equtions, Fuzzy Sets nd Systems 16 (1999), [7] R Goetschel nd W Voxmn, Elementry clculus, Fuzzy Sets nd Systems, 18 (1986), [8] H Hochstdt, Integrl equtions, New York, Wiley, 1973 [9] H Jfri, H Hosseinzdeh nd S Mohmdzdeh, Numericl solution of system of liner integrl equtions y using Legendre wvelets, Int J Open Prolems Compt Mth 5 (21), 63-71

15 Liner Fredholm fuzzy integrl equtions 4117 [1] O Klev, Fuzzy differentil equtions, Fuzzy Sets nd Systems, 24 (1987), [11] RP Knwl nd KC Liu, A Tylor expnsion pproch for solving integrl equtions, Int J Mth Educ Sci Technol 2 (1989), [12] X Ln, Vritionl itertion method for solving integrl equtions, Comput Mth Appl 54 (27), [13] SJ Lio, Beyond Perturtion: Introduction to the Homotopy Anlysis Method, Chpmn Hll/CRC Press, Boc Rton, 23 [14] K Mleknejd nd N Aghzdeh, Numericl solution of Volterr integrl equtions of the second kind with convolution kernel y using Tylor-series expnsion method, Applied Mthemtics nd Computtion, 161 (25), [15] S Ns, S Ylcins nd M Sezer, A Tylor polynomil pproch for solving high-order liner Fredholm integrodifferentil equtions, Int J Mth Educ Sci Technol 31 (2), [16] S Ns, S Ylcins nd M Sezer, A Tylor polynomil pproch for solving high-order liner Fredholm integrodifferentil equtions, Int J Mth Educ Sci Technol 31 (2), [17] HT Nguyen, A note on the extension principle for fuzzy sets, J Mth Anl Appl 64 (1978), [18] ML Puri nd D Rlescu, Fuzzy rndom vriles, J Mth Anl Appl 114 (1986), [19] FG Tricomi, Integrl equtions, Dover Pulictions, New York, 1982 [2] S Ylcins nd M Sezer, The pproximte solution of high-order liner Volterr Fredholm integro-differentil equtions in terms of Tylor polynomils, Applied Mthemtics nd Computtion, 112 (2), [21] LA Zdeh, Towrd generlized theory of uncertinty (GTU) n outline, Informtion Sciences, 172 (25), 1-4 [22] LA Zdeh, The concept of liguistic vrile nd its ppliction to pproximte resoning: Prts 1-3, Informtion Sciences, 8 (1975), , ; 9 (1975) 43-8 Received: Mrch, 212