Research Article Cardinal Basis Piecewise Hermite Interpolation on Fuzzy Data

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1 Advaces Fuzzy Systems Volume 2016, Artcle ID , 8 pages Research Artcle Cardal Bass Pecewse Hermte Iterpolato o Fuzzy Data H. Vosough ad S. Abbasbady Departmet of Mathematcs, Islamc Azad Uversty, Scece ad Research Brach, Tehra, Ira Correspodece should be addressed to S. Abbasbady; abbasbady@yahoo.com Receved 6 Jue 2016; Accepted 7 December 2016 Academc Edtor: Katsuhro Hoda Copyrght 2016 H. Vosough ad S. Abbasbady. Ths s a ope access artcle dstrbuted uder the Creatve Commos Attrbuto Lcese, whch permts urestrcted use, dstrbuto, ad reproducto ay medum, provded the orgal work s properly cted. A umercal method alog wth explct costructo to terpolato of fuzzy data through the exteso prcple results by wdely used fuzzy-valued pecewse Hermte polyomal geeral case based o the cardal bass fuctos, whch satsfy a vashg property o the successve tervals, has bee troduced here. We have provded a umercal method full detal usg the lear space otos for calculatg the preseted method. I order to llustrate the method computatoal examples, we take recourse to three prme cases: lear, cubc, ad qutc. 1. Itroducto Fuzzy terpolato problem was posed by Zadeh [1]. Lowe preseted a soluto to ths problem, based o the fudametal polyomal terpolato theorem of Lagrage (see, e.g., [2]). Computatoal ad umercal methods for calculatg thefuzzylagrageterpolatewereproposedbykaleva [3]. He troduced a terpolatg fuzzy sple of order l. Importat specal cases were l = 2,thepecewselear terpolat, ad l=4,afuzzycubcsple.moreover,kaleva obtaed a terpolatg fuzzy cubc sple wth the ota-kot codto. Iterpolatg of fuzzy data was developed to smple Hermte or osculatory terpolato, E(3) cubc sples, fuzzy sples, complete sples, ad atural sples, respectvely,[4 8]byAbbasbadyetal.Later,Lodwckad Satos preseted the Lagrage fuzzy terpolatg fucto that loses smoothess at the kots at every α-cut; also every α-cut (α =1)offuzzy sple wth the ot-α-kot boudary codtos of order k has dscotuous frst dervatves othekotsadbasedotheseterpolatssomefuzzy surfaces were costructed [9]. Zeal et al. [10] preseted a method of terpolato of fuzzy data by Hermte ad pecewse cubc Hermte that was smpler ad cosstet ad also herted smoothess propertes of the geerator terpolato. However, probably due to the swtchg pots dffcultes, the method was expressed a very specal case ad oe of three remag mportat cases was ot vestgated ad ths s a fudametal reaso for the method weakess. I total, low order versos of pecewse Hermte terpolato are wdely used ad whe we take more kots, the error breaks dow uformly to zero. Usg pecewsepolyomal terpolats stead of hgh order polyomal terpolats o the same materal ad spaced kots s a useful way to dmsh the wgglg ad to mprove the terpolato. These facts, as well as cardal bass fuctos perspectve, motvated us [11] to patch cubc Hermte polyomals together to costruct pecewse cubc fuzzy Hermtepolyomaladprovdeaexplctformulaa succct algorthm to calculate the fuzzy terpolat cubc case as a ew replacemet method for [4, 10]. Now, ths paper, lght of our prevous work, we wat to troduce a wde geeral class of fuzzy-valued terpolato polyomals by extedg the same approach [11] applyg a very specal case of whch geeral class of fuzzy polyomals could be a alteratve to fuzzy osculatory terpolato [4] ad so ts lowest order case (m = 1), amely, the pecewse lear polyomal, s a aalogy of fuzzy lear sple [3]. Meawhle, whe m = 2 wth exactly the same data, we wll smply produce the secod lower order form of metoed geeral class that was troduced [11] ad the terpolato of fuzzy data [10]. The paper s orgazed fve sectos. I Secto 2, we have revewed deftos ad prelmary results of several

2 2 Advaces Fuzzy Systems basc cocepts ad fdgs; ext, we costruct pecewse fuzzy Hermte polyomal detal based o cardal bass fuctos ad prove some ew propertes of the troduced geeral terpolat (Secto 3). I Secto 4, we have produced three tal, lear, cubc [11], ad qutc cases ad show the relatoshp betwee some of the metoed cases ad the ewly preseted terpolats [3, 4, 10]. Furthermore,tollustratethemethod,somecomputatoalexamples are provded. Fally, the coclusos of ths terpolato are Secto Prelmares To beg, let us troduce some bref accout of otos used throughout the paper. We shall deote the set of fuzzy umbers by R F the famly of all oempty covex, ormal, upper semcotuous, ad compactly supported fuzzy subsets defed o the real axs R.Obvously,R R F. If u R F s a fuzzy umber, the u α =x R u) α}, 0<α 1,showstheα-cut of u. Forα=0by the closure of the support, u 0 = clx x R, u) > 0}.Itswell kow the α-cuts of u R F are closed bouded tervals R adwewlldeotethembyu α = [u α, u α ];fuctos u ( ), u ( ) aretheloweradupperbrachesofu.thecoreofu s u 1 =x x R, u) = 1}.Itermsofα-cuts, we have the addto ad the scalar multplcato: (u+v) α =u α + V α =x+y x u α, y V α } (λu) α =λu α = λx x u α } (0) α = 0} for all 0 α 1, u, V R F,adλ R. u = a,b,c,d specfes a trapezodal fuzzy umber, where a b c dad f b=cwe obta a tragular fuzzy umber. For α [0,1],wehaveu α = [a+α(b a),d α(d c)]. I the rest of ths paper, we wll assume that u s a tragular fuzzy umber. Defto 1 (see, e.g., [5]). A L-R fuzzy umber u = (m, l, r) LR s a fucto from the real umbers to the terval [0, 1] satsfyg R( x m ) for m x m+r, r u ) = L( m x ) for m l x m, (2) l 0 otherwse, where R ad L are cotuous ad decreasg fuctos from [0, 1] to [0, 1] fulfllg the codtos R(0) = L(0) = 1 ad R(1) = L(1) = 0.WheR) = L) = 1 x,wewllhavel L fuzzy umbers that volve the tragular fuzzy umbers. For a L Lfuzzy umber u=(m,l,r), the support s the closed terval [m l,m+r](see, e.g., [6]). The lear space of all polyomals of degree at most N wll be desgated by P N. Full Hermte terpolato problem defes a uque polyomal, called p N ), whchsolvesthe followg problem. (1) Theorem 2 (see [12] (exstece ad uqueess)). Let x 0,x 1,...,x be +1dstct pots, α 0,α 1,...,α be postve tegers, k = 0,1,...,α,adN=α 0 +α 1 + +α +.Set w) = =0 x ) α +1 ad l k ) x ) k α =w) k! x ) α +1 k dx [ x ) (α k) w ) p N ) = =0 r l 0 ) + =0 d (α k) r l 1 ) + + s a uque member of P N for whch r (α ) =0 α +1 ] x=x l α ) p N 0 )=r 0, p N ) =r 0,..., p(α 0 1) N 0 )=r (α 0) 0.. p N )=r, p N )=r,..., p(α 1) N )=r (α ). Whe α 0 =α 1 = = α =1,thefullHermteterpolato smplfes to smple Hermte or osculatory terpolato. Defto 3. Gve dstct kots x 0,x 1,...,x,assocated fucto values f 0,f 1,...,f,adalearspaceΦ of specfc real fuctos geerated by cotuous cardal bass fuctos φ j : R R, (j = 0,1,...,), φ j ) = δ j, ( = 0,1,...,), we say that the fucto F orgazed the shape F) = j=0 f jφ j ) s a terpolat based o cardal bass ad such a procedure s the cardal bass fuctos method. 3. Pecewse Fuzzy Hermte Iterpolato Polyomal A specal case of full Hermte terpolato s pecewse Hermte terpolato (see, e.g., [13, 14]). Let us assume throughout the paper that Δ : a = x 0 < x 1 < < x = b s a grd of I = [a,b] wth kots x ad m s a postve teger. All pecewse Hermte polyomals form a certa fte dmesoal smooth lear space whch we ame H 2m 1 (Δ; I). Defto 4. H 2m 1 (Δ; I) s a collecto of all real-valued pecewse-polyomal fuctos s) of degree at most 2m 1, defed o I, suchthats) C m 1 (I). Theassocated fucto to s) o successve tervals [x 1,x ], 1, wth kots from Δ, s defed by s ),thats,a(m 1)-tmes cotuously dfferetable pecewse Hermte polyomal of degree 2m 1,oI. Defto 5 (see [15]). Gve ay real-valued fucto, f) C m 1 (I). LettsuqueH 2m 1 (Δ; I)-terpolate, for (3) (4)

3 Advaces Fuzzy Systems 3 m ad grd Δ of I, betheelemets) of degree 2m 1 o each terval [x 1,x ], 1,suchthat D k s )=D k f ) 0 k m 1, 0, D k = dk dx k. Exstece ad uqueess of full Hermte terpolato s provded [12]. Because of ths, presetato (5) s actually a specal case of such terpolato o a grdded terval ad t follows that each fucto belogg to C m 1 (I) has a uque terpolate H 2m 1 (Δ; I). A partcular cardal bass for lear space H 2m 1 (Δ; I) of dmeso m( + 1) s B = φ k )},m 1 =0,k=0, (see, e.g., [16]), where the bass fucto φ k ) s defed by D l φ k j )=δ kl δ j, 0 k,l m 1, 0,j, D l = dl dx l. Some mportat results based o (6) are smple to see the sequel, as φ 0 ) = 1 ad φ 0 j ) = 0 at all kots x j ad sce s) outsde [x 1,x ], 1, satsfes zero data, the φ 0 0 for all x 0 x x 1 ad x +1 x x. φ 1 ) s of degree 2m 1, adφ 1 ) = 1 but t s zero at all other kots. Moreover, because outsde the terval [x 1,x +1 ]φ 1 ) terpolates zero data, the φ 1 ) must be vashed detcally for all x x +1 ad x x 1 (see, e.g., [13, 14, 17]). Aalogous reasog apples to φ 1 ) x 1 ) m m 2 x ) a j x j, x 1 x x, j=0 = x +1 ) m m 2 x ) b j x j, x x x +1, j=0 0, otherwse. I the followg theorem, we wll use the recet features. Theorem 6. Assume that φ k B ad satsfes the pecewse Hermte polyomal cardal bass fucto costrats (6). The, (5) (6) () φ 0 ) + φ +1 0 ) 1, forallx,x +1 ), = 0,1,..., 1. () For all =1,2,..., 1, φ 1 chages the sg at x.the sg of φ 1 sotpostveoaysubterval[x 1,x ], ad that s ot egatve o [x,x +1 ]. () The sg of all other elemets of B sotegatveoi. Proof. Wth the assumpto of (6), let φ 0 ) + φ +1 0 ) be polyomal of degree 2m 1 o the terval [x 1,x +2 ] ad terpolate the data j,f j ),wheref j =1for j=,+1ad zero o the other kots of partto Δ. Supposethat0<< 1ad φ 0 ) + φ +1 0 ) < 1 for some x,x +1 ).By (7) the mea value theorem, ts dervatve has a zero o,x +1 ). The dervatve has two (m 2)th order zeros at x ad x +1 ad ts two other zeros are x 1, x +2. The, t has at least 2m 1 zeros o the terval [x 1,x +2 ], whch s a cotradcto. The cases =0ad = 1are treated smlarly. I lght of represetato (7) ad codto (6), the polyomal φ 1 ) s of degree 2m 1. It has oly oe mmum pot o [x 1,x ] ad a sgle maxmum o the subterval [x,x +1 ]. Suppose that each of the above pots are oe more. The, by the mea value theorem, frst dervatve of φ 1 ) has at least three zeros o 1,x ) ad three zeros o,x +1 ). Also, the dervatve has two (m 2)th order zeros at x 1, x +1. The, t has at least 2m + 2 zeros, whch s a cotradcto. Hece, φ 1 ) has oly oe zero o each of the tervals 1,x ) ad,x +1 ).Now,recall φ 1 ) = 1;tfollowsthatφ 1 ) 0, o[x 1,x ] ad φ 1 ) 0,o[x,x +1 ]. Ths gves (). A smlar proof va defto of bass fuctos ad (6) follows the clam (). For a gve f) C m 1 (I) ad ts pecewse Hermte terpolate s) H 2m 1 (Δ; I), a equvalet explct represetato of s) (5) ca be uquely expressed (see, e.g., [13, 17]); amely, s ) = m 1 k=0 =0 f (k) ) φ k ). (8) Now, we wat to costruct a fuzzy-valued fucto as s: I R F such that s (k) )=f (k) )=u k R F, 0 k m 1, 0.Also,fforall0 k m 1, 0, u k = y (k) are crsp umbers R ad f (k) ) = χ y (k) (see, e.g., [2]), the there s a polyomal of degree 2m 1 o successve tervals [x 1,x ], 0,wths (k) )=y (k), 0 k m 1, 0 such that s) = χ f) for all x R, where,f,f,...,f(m 1) ) f (k) R F, 0 k m 1, 0 } s gve. We suppose that such a fuzzy fucto exsts ad we attempt to fd ad compute t wth respect to terpolato polyomal preseted by Lowe [2]. Let, for each x [x 0,x ], s) be a fuzzy pecewse Hermte polyomal ad Λ = y (k) },m 1 =0,k=0 ; the, from Kaleva [3] ad Nguye [18], we obta the α-cuts of s) a succctly algorthm as follows: s α ) =t R μ (t) s) α}=t R y(k) where :μ (y(k) ) u k α, 0 k m 1, 0, s Λ ) =t}=t R t=s Λ ), y (k) }= m 1 k=0 =0 u α k φ k ), s Λ ) = u α k, 0 k m 1, 0 m 1 y (k) k=0 =0 (9) φ k ) (10)

4 4 Advaces Fuzzy Systems s a pecewse Hermte polyomal crsp case ad by defto s α ) = m 1 k=0 =0 u α k φ k ) (11) we obta a formula that comprses a smple practcal way for calculatg s): s ) = m 1 k=0 =0 u k φ k ). (12) Sce, for each 0 k m 1, 0, u α k =[uα k, uα k ], the we wll have s α ) by solvg the followg optmzato problems: max & m subject to m 1 y (k) k=0 =0 φ k ) u α k y(k) u α k, 0 k m 1, 0. (13) From the φ j s sg that we represeted Theorem 6, these problems have the followg optmal solutos: y (k) Maxmzato s as follows: = u α k f φ k ) 0 u α k f φ k ) <0, y (k) Mmzato s as follows: = u α k f φ k ) 0 u α k f φ k ) <0, 0 k m 1, 0. 0 k m 1, 0. (14) (15) Theorem 7. If s) = m 1 k=0 =0 u kφ k ) s a terpolatg pecewse fuzzy Hermte polyomal, the for all α [0,1], 0,1,..., 1}, x [x,x +1 ], where le s α ) m le s α ),les α +1 )}, (16) s α ) =[s α ), s α )], le s α ) = s α ) s α ). (17) Proof. By usg Theorem 6 ad (11), we have s α ) = u α 0, s α +1 ) = u α 0+1 ad le sα ) = le u α 0,lesα +1 ) = le u α 0+1. Sce the addto does ot decrease the legth of a terval from (11), we ca wrte s α ) = m 1 k=0 j=0 uα k j φ jk );the, le s α ) m 1 k=0 φ j0 ) le u 0j j=0 φ jk ) le uα kj j=0 φ 0 ) le u 0 + φ +1 0 ) le u 0+1 m le u 0, le u 0+1 }( φ 0 ) + φ +1 0 ) ) m le u 0, le u 0+1 } = m le s α ),le s α +1 )}. (18) Theorem 8. Let u k = (m k,l k,r k ), 0 k m 1, 0,beatragularL Lfuzzy umber; the, also s), the pecewse fuzzy Hermte polyomal terpolato, s such a fuzzy umber for each x. Proof. The closed terval [m l,m+r]s the support of u= (m, l, r),atragularl Lfuzzy umber; the for each x ad u k,wehave m 1 s ) =( k=0 =0 u k φ k )) = [ (m k l k )φ k ) φ [ k 0 + (m k +r k )φ k ), φ k <0 (m k +r k )φ k ) φ k 0 + (m k l k )φ k )] φ k <0 ] m 1 = [ k=0 [ =0 (l k φ k ) r k φ k )) φ k 0 φ k <0 m 1 + k=0 =0 m k φ k ) m k φ k ) +(r k ) φ k ) l k φ k ))] φ k 0 φ k <0 ] = (m ) l),m) +r)). (19)

5 Advaces Fuzzy Systems 5 It follows that f s) = (m), l), r)), satragular L Lfuzzy umber for each x,the m ) = m 1 k=0 =0 m k φ k, l ) = l k φ k ) r k φ k ), φ k 0 φ k <0 r ) = r k φ k l k φ k ). φ k 0 φ k <0 4. Pecewse-Polyomal Lear, Cubc, ad Qutc Fuzzy Hermte Iterpolato (20) We cosder m=1ad compute the pecewse fuzzy lear terpolat as the tal case of the preseted method based o (12) ad for a gve set of fuzzy data,f ) f R F, 0 }, as follows: s ) = =0 u 0 φ 0 ), (21) where u 0 =f, 0, ad subject to codtos (6), Fgure 1: Graph of Example Fgure 2: Graph of Example 10. 0, x x 1 φ 00 ) = ( 1 x ), x 1 x 0 x 0 x x 1 0, x x 1, x x +1 φ 0 ) = ( x x 1 ), x x 1 ( x +1 x ), x +1 x x 1 x x x x x , x x 1 φ 0 ) = ( 1 ), x x 1 x 1 x x. (22) The obtaed s) s the same as fuzzy sple of order l=2 thathadbeetroduced[3]becausethebascsplesad the cardal bass fuctos two terpolats are equal. Example 9 (see [4]). Suppose the data (1, (0, 2, 1), (1, 0, 3)), (1.3, (5, 1, 2), (0, 2, 1)), (2.2, (1, 0, 3), (4, 4, 3)), (3, (4, 4, 3), (5, 1, 2)), (3.5, (0, 3, 2), (1, 1, 1)), (4, (1, 1, 1), ad (0, 3, 2)). I Fgure 1, the dashed le s the 0.5-cut set of pecewse cubc fuzzy terpolato s), x [1,4]adthesoldles represet the support ad the core of s). Whe m = 2, we get the pecewse cubc fuzzy Hermte polyomal terpolat [11] for a gve set of data,f,f ) f,f R F, 0 }, s ) = =0 u 0 φ 0 ) + =0 u 1 φ 1 ), (23) where u k =f (k), k=0,1, 0. A outstadg feature of ths study s that, by smply applyg the secod case of the preseted geeral method ad exactly the same data, we have produced a alteratve to smple fuzzy Hermte polyomal terpolato [4]. Heretofore, the metoed cubc case (23) was depedetly troduced [10] but oly very weak codtos ad wthout usg the exteso prcple. The cardal bass fuctos φ k ), k=0,1, 0, were computed [17]. Example 10. Suppose the data (1, (0, 2, 1), (1, 0, 3)), (1.5, (5, 1, 2), (0, 2, 1)), (2.7, (1, 0, 3), (4, 4, 3)), (3, (4, 4, 3), (5, 1, 2)), (3.7, (0, 3, 2), (1, 1, 1)), ad (4, (1, 1, 1), (0, 3, 2)). I Fgure 2, thedashedlesthe0.5-cut set of pecewse cubc fuzzy terpolatos s), x [1,4] ad the sold les represet thesupportadthecoreofs). Let m = 3; from (6), we shall costruct the cardal bass for H 5 (Δ; I). The qutc Hermte polyomals

6 6 Advaces Fuzzy Systems φ 0 ), φ 1 ), ad φ 2 ) are solvg the terpolato problem D l φ k j ) =δ kl δ j, 0 k,l 2, 0,j. (24) To ths ed, we determe uquely all the pervous φ j s by the (24). For 1 1,let 1 x) 3 1 x ) 5 [ 1 +3x) 1 5x )+6x 2 +10x 2 ], x 1 x x, φ 0 ) = +1 x) 3 +1 x ) 5 [ +1 +3x) +1 5x )+6x 2 +10x 2 ], x x x +1, 0, otherwse, 3 ) x 1 x ( x 1 x φ 1 ) = ( x +1 x x )[1+3 x x x 1 x ], x 1 x x, 3 ) x x +1 0, otherwse, x)[1+3 x x x x +1 ], x x x +1, ( x 1 x ) x) 2, x 1 x 2 x 1 x x, φ 2 ) = ( x x +1 ) x) 2, x x +1 2 x x x +1, 0, otherwse. (25) The sx ext fuctos are smlarly defed. I partcular, 1 x) 3 φ 00 ) = 1 x 0 ) 5 [ 1 +3x) 1 5x 0 )+6x 2 +10x 3 0 ], x 0 x x 1, 0, x 1 x x, 1 x) 3 φ 0 ) = 1 x ) 5 [ 1 +3x) 1 5x )+6x x 2 ], x 1 x x, 0, x 0 x x 1, ( 3 1 x ) φ 01 ) = x 0 x 0 x)[1+3 x 0 x ], 1 x 0 x 1 x 0 x x 1, 0, x 1 x x, ( 3 1 x ) x φ 1 ) = x 1 x )[1+3 x x ], x 1 x x 1 x x, 0, x 0 x x 1, ( 1 ) 0 x) 2, φ 02 ) = x 0 x 1 2 x 0 x x 1, 0, x 1 x x,

7 Advaces Fuzzy Systems Fgure 3: Graph of Example 11. ( 1 x ) x) 2, φ 2 ) = x 1 x 2 x 1 x x, 0, x 0 x x 1. (26) Thus, we ca mmedately wrte dow pecewse qutc fuzzy Hermte terpolato polyomal s) usg s ) = =0 u 0 φ 0 ) + =0 u 1 φ 1 ) + =0 u 2 φ 2 ), (27) where,f,f,f ) f (k) B F, 0 k 2, 0 },s gve ad u k =f (k). Example 11. Suppose that (0, (0, 1, 3), (0, 2, 2), (1, 4, 4)), (1.3, (0.05, 1.9, 3.5), (0.3, 3.2, 0.8), (1, 3.1, 3)), (2, (2, 6.7, 5.3), (2, 0.5, 3.5), (1, 2.6, 2.4)), (4, (8, 10.1, 9.9), (4, 4, 0), (1, 0.6, 0.5)), (5.3, (14, 13, 12), (5.3, 0.2, 3.8), (1, 1.5, 1.7)), (6, (18, 13.2, 14.8), (6, 0.9, 3), ad (1, 3.4, 3.2)) are the terpolato data. I Fgure 3, thesoldlesdeotethesupportadthecoreofpecewse qutc fuzzy Hermte terpolato s), x [0,6],adthe dashed le s the 0.5-cut set of s). 5. Coclusos ad Further Work Based o the cardal bass fuctos for m( + 1) dmeso H 2m 1 (Δ, I) lear space, terpolato of fuzzy data by the fuzzy-valued pecewse Hermte polyomal as the exteso of same approach [11] has bee successfully troduced geeral case ad provded a succct formula for calculatg the ew fuzzy terpolat. Moreover, two frst cases of the preseted method have bee appled as a aalogy to fuzzy sple of order two [3] ad a alteratve to fuzzy osculatory terpolato [4], respectvely. I the guse of a remarkable achevemet, the pecewse fuzzy cubc Hermte polyomal terpolato that was costructed wth poor codtos ad wthout usg the exteso prcple [10] hasbeeproducedtheroleofaveryspecalsubdvsofor the preseted geeral method ths study. Fally, the thrd tal case, pecewse fuzzy qutc Hermte polyomal, has bee descrbed detal. The ext step to mprove ths method s terpolato of fuzzy data cludg swtchg pots by a fuzzy dfferetable pecewse terpolat. Competg Iterests The authors declare that they have o competg terests. Refereces [1] L. A. Zadeh, Fuzzy sets, Iformato ad Computato, vol. 8, pp , [2] R. Lowe, A fuzzy Lagrage terpolato theorem, Fuzzy Sets ad Systems,vol.34,o.1,pp.33 38,1990. [3] O. Kaleva, Iterpolato of fuzzy data, Fuzzy Sets ad Systems. A Iteratoal Joural Iformato Scece ad Egeerg, vol.61,o.1,pp.63 70,1994. [4] H. S. Goghary ad S. Abbasbady, Iterpolato of fuzzy data by Hermte polyomal, Iteratoal Joural of Computer Mathematcs,vol.82,o.5,pp ,2005. [5] H.Behforooz,R.Ezzat,adS.Abbasbady, Iterpolatoof fuzzy data by usg E(3) cubc sples, Iteratoal Joural of Pure ad Appled Mathematcs,vol.60,o.4,pp ,2010. [6] S. Abbasbady, R. Ezzat, ad H. Behforooz, Iterpolato of fuzzy data by usg fuzzy sples, Iteratoal Joural of Ucertaty, Fuzzess ad Kowledge-Based Systems,vol.16,o. 1, pp , [7] S. Abbasbady, Iterpolato of fuzzy data by complete sples, Korea Joural of Computatoal & Appled Mathematcs,vol.8,o.3,pp ,2001. [8] S. Abbasbady ad E. Babola, Iterpolato of fuzzy data by atural sples, TheKoreaJouralofComputatoal&Appled Mathematcs,vol.5,o.2,pp ,1998.

8 8 Advaces Fuzzy Systems [9]W.A.LodwckadJ.Satos, Costructgcosstetfuzzy surfaces from fuzzy data, Fuzzy Sets ad Systems. A Iteratoal Joural Iformato Scece ad Egeerg, vol.135, o. 2, pp , [10] M. Zeal, S. Shahmorad, ad K. Mra, Hermte ad pecewse cubc Hermte terpolato of fuzzy data, Joural of Itellget & Fuzzy Systems,vol.26,o.6,pp ,2014. [11] H. Vosough ad S. Abbasbady, Iterpolato of fuzzy data by cubc ad pecewse-polyomal cubc hermtes, Ida Joural of Scece ad Techology,vol.9,o.8,2016. [12] P. J. Davs, Iterpolato ad Approxmato,Dover,NewYork, NY, USA, [13] B. Wedroff, Theoretcal Numercal Aalyss, Academc Press, New York, NY, USA, [14] P. G. Carlet, M. H. Schultz, ad R. S. Varga, Numercal methods of hgh-order accuracy for olear boudary value problems, Numersche Mathematk, vol. 9, pp , [15] G.Brkhoff,M.H.Schultz,adR.S.Varga, PecewseHermte terpolatooeadtwovarableswthapplcatosto partal dfferetal equatos, Numersche Mathematk, vol. 11, pp , [16] R. S. Varga, Hermte terpolato-type Rtz methods for twopot boudary value problems, Numercal Soluto of Partal Dfferetal Equatos, J.H.Bramble,Ed.,pp , Academc Press, New York, NY, USA, [17] P. M. Preter, Sples ad Varatoal Methods, Wley-Iterscece, [18] H. T. Nguye, A ote o the exteso prcple for fuzzy sets, Joural of Mathematcal Aalyss ad Applcatos, vol.64,o. 2, pp , 1978.

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