Solving Fuzzy Linear Volterra Intergro- Differential Equation Using Fuzzy Sumudu Transform

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1 Vlme 9 N. 5 8, ISSN: (n-line versin) rl: p:// Slving Fzzy Liner Vlerr Inergr- Differenil Eqin Using Fzzy Smd Trnsfrm p:// Rjkmr.A Assisn Prfessr (SG), Dep. f Memics Hindsn Insie f Tecnlgy nd Science Cenni, Indi Jesrj.C, Mmmed Spiqe.A 3,3 Asscie Prfessr, Dep. f Memics I.F.E.T Cllege f Engineering, Gngrmplym, Tmilnd, Indi Absrc Tis pper prpses med fr slving vris ypes f fzzy liner Vlerr Inegr- Differenil eqin cnsidering e iniil cndiin s generlized ringlr fzzy nmber. Te prpsed med is bsed n e Fzzy Smd rnsfrm (FST).Sme resls bsed n e prperies f FST re ls prpsed. Nmericl exmples nd grps re presened fr ec ype illsre e vlidiy f e prpsed med. Keywrds Fzzy liner Inegr differenil eqin; Generlized ringlr fzzy nmber;fzzy differenil eqin,;nd fzzy Smd rnsfrm.. INTRODUCTION Te Fzzy Differenil eqins nd e fzzy inegrl eqins ply vil rle in mdeling e engineering nd bilgicl prblems. Wile slving prblems reled differenil eqin mdels ne nrmlly cnsider e iniil cndiins given in e mdel re well-defined, b in prcice de eqivcl r pril infrmin b e vribles invlved in e prblems we need pply fzzy cnceps differenil nd inegrl eqins rrive e slins. Te cncep f fzzy in inegrin ws firs inrdced by Dbis nd Prde[-3]nd ey mesred cerin ype f fzzy-vled fncin nd defined e inegrl fncin sing e exensin principle.fzzy inegrl eqins f e secnd kind sdied by Jn mrdesn nd Willim Newmn [4] nd is nmericl wy f bining e slin ws discssed by Bblin, Sdegi, Abbsbndy[5].Te exisence erems fr cerin Vlerr inegrl eqins nd Fredlm inegrl eqin fr e fzzy se vled mppings discssed by Jng Yel Prk nd Je Jeng [6]. Exisence f slins f fzzy inegrl eqins in Bnc spces sdied by Jng Yel Prk, Yng Cel Kwn nd Je Ug Jeng[7].Te exisence nd niqeness f slins f fzzy inegrl eqin ws discssed by vris rs [8-]. Inegrl Trnsfrm meds sc s Frier rnsfrm nd Lplce rnsfrm plys n essenil l in perinl clcls slve vris prblems in e disciplines f engineering nd pysics [-3]. As we knw e Frier Trnsfrm cnges se f ime dmin d vecrs in se f freqency dmin vecrs nd e Lplce rnsfrm wic rnsfrms e differenil eqin in n lgebric expressin. Tese re f clssicl ype f inegrl rnsfrm. In recen imes, e ide f rnsfrm n differenil eqin ws expnded sc s Mellin rnsfrm, Hnkel rnsfrm, Differenil rnsfrm meds, Smd rnsfrm nd N- rnsfrm [4-6]. Wgl [7-8] s fmilirized e cncepin f Smd rnsfrm wic is lrgely pplied in differenil eqins nd in prblems priclrly reled Cnrl Engineering.Te cncep f Smd rnsfrm ws frer exended pril differenil eqin by Weerkn. Belgcem e l. Weerkn [9-] presened pplicins cnvlin ype inegrl eqins. Cnvlin erem in Smd rnsfrm discssed by Asir[],Belgcem nd Krblli,Kll pplied Smd rnsfrm inegrl prdcin eqins [- 4].Applicin f fzzy Smd rnsfrm n fzzy frcinl differenil eqin sdied by Abdl Rmn nd Zini Amd [5].In generl Smd rnsfrm is cnsidered s pwerfl rnsfrm fr slving differenil eqins becse f e niqe prpery nd erefre is ese e prcess f finding e slin. Te rrngemen f is pper is s fllws, e firs secin dels wi e inrdcin cnceps reled fzzy liner Inegr- Differenil eqin nd fzzy Smd rnsfrm. In e secnd nd ird secin dels wi e bsic definiins reled fzzy nmbers, fzzy differenils, nd fzzy Smd rnsfrm, e fr nd fif secin deils e slin prcedre nd slved fzzy liner inegr differenil eqin bsed n fzzy Smd rnsfrm. PRELIMINARIES Fzzy ses definiins In is secin e bsic definiins f fzzy ses nd fzzy nmbers re reviewed frm [6]. Definiin Fzzy se. A fzzy se is crcerized by membersip fncin mpping f dmin spce (i.e) mpping beween niverse f discrse U e ni inervl [,]given by 373

2 , ( ) / A Here A : U, A x x x U is clled e degree f membersip fncin f e fzzy se A.. Definiin Nrml fzzy se A fzzy se A f e niverse f discrse U is clled nrml fzzy se if ere exis - les ne xu sc ( x ) = A.3 Definiin Heig f fzzy se Te lrges membersip grde bined by ny elemen in e fzzy se nd i is given by ( A) spp ( x).4 Definiin Cnvex fzzy se A fzzy se A is sid be cnvex if nd nly if fr ny x, x U, e membersip fncin f A sisfies e cndiin x x x x ( ) min ( ), ( ), [,] A A A.5 Definiin Exensin principle. Le X Xx X x... x X n be Cresin prdc f e niverse X nd A, A... An be e n fzzy nmbers. Le f : X Y be mpping en e exensin principle is defined by e fzzy se B iny by A B y, ( y) / y f(x, x... xn), (x, x... xn) X B x x, erwise i is wek slin. On sc cses sp min ( x), ( x)... ( xn ), f ( y) A A A n ( y) x, x... xn f ( y) B x( ) min x, x,mx x, x erwise. Definiin : Fzzy derivive [3] Fr n=, e bve exensin principle redced B y, ( y) / y f(x),x) X ( y) B B sp ( ), x f ( y ) A x f ( y), erwise.6 Definiin Tringlr Fzzy Nmber [7] Le AF(R), A (,, 3 ) is clled ringlr fzzy nmber if is membersip fncin is given by x, x ( ) 3 x ( x), A x 3 ( 3 ), x, 3 x.7 Definiin Generlized ringlr fzzy nmber [8] Le AF(R), A (,, 3; ) is clled Generlized ringlr fzzy nmber if is membersip fncin is given by x, x ( ) 3 x ( x), A x 3 ( 3 ), x, 3 x.8 Definiin Fzzy rdinry differenil eqin [9] Cnsider n rdinry differenil eqin wse iniil cndiin is described in fzzy nmbers dy f (, y), T d y( ) y(), y() (,b,c; ),, Ten we sy e given differenil eqin is fzzy differenil eqin..9 Definiin Srng nd wek slin f fzzy differenil eqin [3] Cnsider firs rder nn mgenes liner fzzy rdinry differenil eqin x ()=f(,x()), x( )=x wi x s GTFN. Le () x( ) x, x be e slin nd α- cs x nd f e bve differenil eqin respecively. We sy e slin x () is srng slin if i sisfies e cndiin Le f : (, b) E nd x (, b). we sy f is srngly generlized differenil x (Bede-Gl differenil) if ere exiss n elemen f ( x ) E, sc (i) Sfficienly smll, f ( x ) f ( x ), f ( x ) f ( x ) nd e limis is given by f ( x ) f ( x ) f ( x ) f ( x ) lim lim f( x ) (r) (ii) Sfficienly smll, f ( x ) f ( x ), f ( x ) f ( x ) nd e limis is given by f ( x ) ( ) ( ) ( ) lim f x f x f x lim f( x ) (r) (iii) Sfficienly smll, f ( x ) f ( x ), f ( x ) f ( x ) nd e limis is given by f ( x ) ( ) ( ) ( ) lim f x f x f x lim f( x ) (r) 374

3 (iv) Sfficienly smll, f ( x ) f ( x ), f ( x ) f ( x ) nd e limis is given by f ( x ) ( ) ( ) ( ) lim f x f x f x lim f( x ). Definiin [3] Le f : (, b) E fr, en be fncin nd le f f, f ()If f is (i) differenible, en f& f re differenible fncin nd f f, f ()If f is (ii) differenible, en f & f re differenible fncin nd f f, f.3 Terem: [33] Le f:r F(R)nd f f, f.fr ny fixed,, ssme fnd f re Riemnn- Inergrble n [,b] bnd le s ssme b b f d C nd f d C were C nd C re ny w psiive qniy.en,f() is imprper fzzy Riemnn- Inergrble n [, ) nd e imprper fzzy Riemnn inergrble is fzzy nmber.ts f () d f() d, f() d Prpsiin f ( x) nd f ( x ) is fzzy-vled fncin nd fzzy If Riemnn-inergrble n [, ),en f( x) f( x) is ls fzzy Riemnn-inergrble n [, ). f f d f d f d I I I 3. FUZZY SUMUDU TRANSFORM. 3. Definiin: f: R F(R) be cnins fzzy vled fncin. Sppse f ( ) e is n imprper fzzy Riemnn-inegrble n [, ), en f ( ) e d e fzzy Smd rnsfrm nd i is dened by F( ) S f f ( ) e d,, Ts f ( ). e d, f( ). e d is clled S f s f, s f IV.FUNDAMENTAL THEOREMS AND PROPERTIES OF FST 3. Terem. [34] If f: R F(R) be cnins fzzy vled fncin nd if F( ) S f en F( ) f () if f is(i) differenible,> S f () f () F( ) if f is(ii) differenible,> Prf: cse (i) le s ssme f is (i) Differenible, en F( ) () s f f () s f f () f, s f, s f F( ) f () Hence S f () Cse (ii) Assme f is (ii) Differenible, en f () ( ) f () s f f () s f F, ( ), ( ) s f s f f () F( ) Hence S f () 3.3 Terem. [34] If f: R F(R) be cnins fzzy vled fncin nd if F( ) S f en x S e f F, nd Prf: by definiin. ( ) ( ) S e f f( )e e d, f( )e e d ( ) ( ) ( )e f d, f( )e d v v v v e, e f dv f dv bysbsiin were v = v f e v dv x Hence S e f () F x Similrly we cn prve S e f () F 375

4 3.4 Terem If f: R F(R) be cnins fzzy vled fncin nd if F( ) S f en S f d F( ) Prf: Assme e fncin g is (i) Differenible, en Le g f d, g ( ), g ( ), g f G( ) g() Sg s( g) g () s( g) g (), s( g ) s( g ), s f d, s f d S f d F( ) Hence we rrive 3.5 Terem If f: R F(R) be cnins fzzy vled fncin nd if F( ) S f en v S f ( ) d F( ) d v Prf: By definiin f FST, F( ) S f f ( ) e d en by cse (i) definiin. 4. SOLUTION PROCEDURE Cnsider e fllwing fzzy liner Vlerr inegrdifferenil eqin ( ) ( ) F f e d, f( ) e d v v v F( ) d f( ) e d d, f( ) e d d y x v v v k(, ). y( ) d,, b v v y( ) ( y (), y ()), e f( ) dv d, e f( ) dv d ; w v v Were k (, ) is n rbirry rel vled kernel fncin ver (, ) / [,b],, f (w) dw e d, f(w) dw e d v v v v v v S f () d f (w) dwe d v Hence S f ( ) d F( ) d v 3.6 Terem If f: R F(R) be cnins fzzy vled fncin nd if F( ) S f en / S f ( ) H( ) e F( ), H( )is evisidefncin Prf: By definiin. f ( ) H() e d, S f ( ) H( ) ( ) H() f e d T / T / f(t )H(T ) e dt, f(t )H(T ) e dt T/ T/ f(t ) H(T ) e dt f(t ) H(T ) e dt T / T /, f(t ) H(T ) e dt f(t ) H(T ) e dt By Heviside fncin [ (T ) T /, (T ) T / f e dt f e dt ] T T [ f(t ) e dt, f(t) e dt ] / T e f (T) e dt / S f ( )H( ) e F( ) Sppse e bve iniil cndiin nd x () re mdeled wi generlized ringlr fzzy nmber nd generlized ringlr fzzy fncin respecively en e fzzy vlerr inegrl differenil eqin cn be expressed in fr differen ypes. Type : Cnsidering e iniil cndiin s generlized ringlr fzzy nmber 376

5 y k y( ) d, T y( ) ( y(), y()), Type : Cnsidering cnsn k s generlized ringlr fzzy nmber y k y( ) d, T k k, k, Type 3: Cnsidering b cnsn nd iniil cndiin s generlized ringlr fzzy nmber y k y( ) d, T k k, k nd y( ) y (), y (), Type 4: Cnsidering x () s fzzy fncin nd iniil cndiin s generlized ringlr fzzy nmber y x y( ) d y( ) ( y(), y()), Applying FST n b sides f e bve eqins, we bin S y S x k(, ). y( ) d Tw cses re discssed, cse () if e derivive is (i) differenible en by definiin. dy y, y d Were ( ) ( ) ( ) s y y s y y ( ), ( ) ( ) s y s y Te Smd rnsfrm f vlerr -Inegr-differenil eqin x k(, ). y( ) d cn be bined by sing e FST sing erem 7. Le s ssme e slin fer king FST be L R s y G ( ) nd s y G ( ) By sing inverse FST we cn bin y, y nd i is given by L R y s G ( ) nd y s G ( ) Cse () if e derivive is (ii) differenible en by definiin. dy y, y d y ( ) ( ) s y () s y Were y( ) ( ) () s y s y Assme e slin fer king FST be L R s y H ( ) nd s y H ( ) By sing inverse FST we cn bin y, y nd i is given by L R y s H ( ) nd y s H ( ) 5. NUMERICAL EXAMPLE. Type-I. Cnsider e fllwing fzzy liner Vlerr inegrdifferenil eqin y k y( ) d, T y( ) ( y(), y()), Here we cnsider fzzy iniil cndiins s generlized y( ) (,b,c, )were b c ringlr fzzy nmber y( ),c, were b ; c b Le e slin f e bve differenil eqin be y () nd is α-c is given by yˆ y, y Cse () le s cnsider y() is (i) differenible en y k y( ) d () y k y( ) d () Tking FST n b sides f e eqin () s y y() k s y () k s y () Tking inverse FST n b sides y () s k s y cs k sin Similrly by pplying FST nd inverse FST n () we bin y c cs k sin (4) (3) 377

6 We ne y cs nd y cs Hence e slin bined is srng slin, e α cs f e slin is given by yˆ ke ke, (5) c c k e k e Cse () le s cnsider y() is (ii) differenible en y k y( ) d (6) y k y( ) d (7) Tking FST n b sides f e eqin (6) y () s y k s y () ( ) ( ) s y s y k (8) Tking FST n b sides f e eqin (7) y() s y k s y () s y s y k c (9) Slving (8) nd (9) we ge 3 k k s y c k k s y c Tking inverse FST n b sides y k sin cs c () cs c Similrly we cn bin y k sin cs c () cs c Hence e slin fr cse (ii) is given by yˆ y, y k sin cs c cs c, yˆ () k sin cs c cs c Fig (5.) Fig (5.) () Figre 5. nd 5.: Grpicl represenin f e slin f eqin (5) nd () cnsidering y ( ) (,,3;), We ne e slin y () increses fr [,] nd y () decreses. Terefre y nd y Hence e slin bined is srng slin Type II. Cnsider e fllwing fzzy liner Vlerr-inegrdifferenil eqin f e ype y k y( ) d, T k k, k, Here we cnsider k s generlized ringlr fzzy nmber, k (,b,c, ) were b c nd we cnsider e iniil cndiin y y () y () ( cnsn) k,c, were b ; c b 378

7 Le e slin f e bve inegrl eqin y () nd is α- c is given by yˆ y, y Cse () cnsider y() is (i) differenible en y k y ( ) d (5) y k y ( ) d (6) Tking FST n b sides f e eqin (5) s y y() k s y() s y () Tking inverse FST we ge y cs sin (7) Similrly we cn bin y cs c sin (8) We ne y nd y, Hence e slin is given by yˆ cs sin, cs c sin (9) is srng slin. Cse () Cnsider y() is (ii) differenible en y k y ( ) d () y k y ( ) d () Tking FST n b sides f e eqin () y() s y k s y() ( ) ( ) s y s y k () Tking FST n eqin f e eqin () ( ) ( ) s y s y k (3) Slving we ge (5.) nd (5.3) 3 k k s y () 3 k k s y () Tking inverse FST n (4) nd (5) we ge (4) (5) y cs sin c (6) sin c y cs sin c (7) sin c Hence e slin is given by cs sin c sin c, yˆ () (8) cs sin c sin c Fig 5.3 Fig 5.4 Figre 5.3 nd 5.4: Grpicl represenin f e slin f eqin (9) nd (8) cnsidering y ( ) (,,3;),, Frm e figre 5.3 nd 5.4 we ne e slin y () increses fr [,] nd y () decreses. Terefre y nd y Hence e slin bined is srng slin. 379

8 Type III Cnsider e fllwing fzzy liner Vlerr inegr-differenil eqin f e ype y k y( ) d, T k k, k nd y( ) y (), y (), Here we cnsider k s generlized ringlr fzzy nmber, k (, b, c, ) were b c, nd cnsider e iniil cndiin y ( ) (,b,c, ); Were y( ),c, were b ; c b k k, k,c, were b ; c b Le e slin f e bve eqin be y () nd is α-c is given by yˆ y, y Cse (). Cnsider y() is (i) differenible en y k y ( ) d (9) y k y ( ) d (3) Tking FST n b sides f e eqin (9) s y y() k s y() s y () Tking inverse FST we ge ( ) y cs sin Similrly we cn bin y c cs c sin We ne y nd y Hence e slin is given by cs sin, ˆ y () c cs c sin Cse () Cnsider y() is (ii) differenible en (3) (3) (33) y k y ( ) d (34) y k y ( ) d (35) Tking FST n b sides f e eqin (34) y() s y k s y() ( ) ( ) s y s y k (36) Tking FST n eqin f e eqin (35) s y s y k c (37) Slving we ge (36) nd (37) 3 k k s y c (38) 3 k k s y c (39) Tking inverse FST n (4) nd (5) we ge sin c sin c () y cs c cs c (4) sin c sin c () y (4) cs c cs c Hence e slin is given by yˆ y, y (4) 38

9 Fig 5.5 Fig 5.6 Figre 5.5 nd 5.6 represens f e slin f eqin (33) nd (4) cnsidering y ( ) (,,3;), Frm e figre 5.5 nd 5.6 we ne e slin y () increses fr [,] nd y () decreses. Terefre y nd y.hence e slin bined is srng slin. Type IV: Cnsider e fllwing fzzy liner vlerr inegr differenil eqin y x y( ) d y( ) ( y(), y()), Here we cnsider x () s fzzy fncin x() sin, c sin, were b ; c b And iniil cndiin s generlized ringlr fzzy nmber y( ),c, were b ; c b Le e slin f e bve differenil eqin be y ˆ( ) nd is α-c is given by yˆ y, y Cse () le s cnsider y() is (i) differenible en y x y ( ) d (43) y x y ( ) d (44) Tking FST n b sides f e eqin (43) s y y() s y () 4 s y () Tking inverse FST n b sides f e bve eqin y (sin sin ) cs Similrly we cn bin fr e pper bnd y c (sin sin ) c cs Hence e slin is given by y (sin sin ) cs, c (sin sin ) c cs (45) Cse () le s cnsider y() is (i) differenible en y x y ( ) d. (46) y x y ( ) d (47) Tking FST n eqin (5.46) nd (5.47) we ge s y sy c 4 s y s y c 4 Slving e bve w eqin we rrive s y ( ) c 4 ( )( ) ( )( ) 4 4 c (48) 4 4 ( ) ( ) s y () 4 ( )( ) c ( )( ) 4 4 c 4 4 ( ) ( ) Tking inverse FST n eqin (5.48) nd (5.49) we ge 49 38

10 s y ( ) c sin cs cs sin cs cs cs cs c cs cs 5 s y sin cs cs c sin cs cs Hence e slin is given by yˆ y, y 5 c cs cs cs cs Fig 5.9 Fig 5. (5) Figre 5.9 nd 5. represenin f e slin f eqin (45) nd (5) cnsidering y ( ) (,,3;), / Hence we cnclde e slin bined is srng slin. 6. CONCLUSION. Tis pper cegrized e fzzy liner Vlerr Inegr Differenil eqin in fr ypes bsed n e iniil cndiin nd prpsed med rrive slin fr e sme nd relevn exmples were discssed. Sme imprn prperies reled Fzzy Smd rnsfrm were ls discssed. REFERENCES []D. Dbis nd H. Prde, Twrds fzzy differenil clcls. I. Inegrin f fzzy mppings, Fzzy Ses nd Sysems, 8() (98), 7. []D. Dbis nd H. Prde, Twrds fzzy differenil clcls. II. Inegrin n fzzy inervls, Fzzy Ses nd Sysems, 8() (98), 5 6. [3]D. Dbis nd H. Prde, Twrds fzzy differenil clcls: III, differeniin, Fzzy Ses nd Sysems, 8 (98), [4] Jn mrdesn, Willim Newmn, Fzzy Inegrl Eqins, infrmin sciences 87, 5 9 (995) [5]E.Bblin, H.Sdegi, Abbsbndy,Nmericl slin f liner Fredlm fzzy inegrl eqins f e secnd kind, Applied Memics nd Cmpin 6 (5) [6]Jng Yel Prk, Je Ug Jeng, A ne n fzzy inegrl eqins, Fzzy Ses nd Sysems 8 (999) 93 [7]Jng Yel Prk, Yng Cel Kwn, Je Ug Jeng, Exisence f slins f fzzy inegrl eqins in Bnc spces, Elsevier Science B.V. 65-4/95/, 995 [8] S. Slsr nd T. Allvirnl, Applicin f fzzy differenil rnsfrm med fr slving fzzy Vlerr inegrl eqins, Applied Memicl Mdeling, 37(3) (3), 6 7. [9] K. Blcndrn nd P. Prks, On fzzy vlerr inegrl eqins wi deving rgmens, Jrnl f Applied Memics nd Scsic Anlysis, (4), [] S. J. Sng, Q.Y. Li nd Q. C X, Exisence nd cmprisn erems Vlerr fzzy inegrl eqin in (En, D), Fzzy Ses nd Sysems, 4 (999), []Nmis, V. Te frcinl rder Frier rnsfrm nd is pplicin qnm mecnics. IMA J.Appl. M. 98, 5, []Si, S. Te Weiersrss rnsfrm nd n ismery in e e eqin. Appl. Anl. 983, 6, 6. [3]Gemi, F.; Yns, R.; Amdin, A.; Slsr, S.; Sleimn, M.; Sle, S.F. Applicin f Fzzy Frcinl Kineic Eqins Mdeling f e Acid Hydrlysis Recin. Absr. Appl.Anl. 3, 3, 634. [4]Spinelli, R. Nmericl inversin f Lplce rnsfrm. SIAM J. Nmer. Anl. 966, 3, [5]Lymn, J.W. Te Hnkel rnsfrm nd sme f is prperies. J. Ineger Seq., 4,. [6] Rner, C. Te se f e Mellin rnsfrm in finding e sress disribin in n infinie wedge.q. J. Mec. Appl. M. 948,, 5 3. [7]Wgl, G.K. Smd rnsfrms A new inegrl rnsfrm slve differenil eqins nd cnrl engineering prblems. In. J. M. Edc. Sci. Tecnl. 993, 4, [8]Wgl, G.K. Smd rnsfrms- new inegrl rnsfrm slve differenil eqins nd cnrl engineering prblems. M. Eng. Ind. 998, 6, [9]Weerkn, S. Applicin f Smd rnsfrm pril differenil eqins. In. J. M.Edc. Sci. Tecnl. 994, 5, []Weerkn, S. Cmplex inversin frml fr Smd rnsfrm. In. J. M. Edc. Sci. Tecnl.998, 9, []Asir, M.A. Smd rnsfrm nd e slin f inegrl eqins f cnvlin ype. In. J.M. Edc. Sci. Tecnl., 3, []Belgcem, F.B.M.; Krblli, A.A.; Kll, S.L. Anlyicl invesigins f e Smd rnsfrm nd pplicins inegrl prdcin eqins. M. Prbl. Eng. 3, 3, 3 8. [3]Belgcem, F.B.M., Krblli, A.A. Smd rnsfrm fndmenl prperies invesigins nd Applicins. In. J. Sc. Anl. 6, 6, di:.55/jamsa/6/983. [4]Belgcem, F.B.M. Smd rnsfrm pplicins Bessel fncins nd eqins. Appl. M.Sci., 4, [5]Nrzrizl Abdl Rmn nd Mmmd Zini Amd, slving fzzy frcinl differenil eqins sing fzzy Smd rnsfrm, J.Nnliner Sci.Appl.Vl.(X),-. [6]Dbis D,Prde H(978) Operins n fzzy nmbers,inerninl jrnl f sysems science,9:63-66 [7]A kfmnn nd M M Gp, Inrdcin fzzy rimeic :Tery nd pplicin,vn Nsrnd Reinld,New yrk, [8]C.C.Lee,Fzzy lgic in cnrl sysems:fzzy lgic cnrller,vlme (),44-48,99 [9]Pri, M.L.; Rlesc, D.A. Differenils f fzzy fncins. J. M. Anl. Appl. 983, 9, [3]Z Ding,Ming M,Abrm Kndel Exisence f e slins f fzzy differenil eqins wi prmeers, infrmin sciences,997,vl99(3):5-7 [3] Bede B., Gl S.G.: Generlizins f e differenibiliy f fzzynmber-vled fncins 38

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