A New Method for Solving Fuzzy Volterra Integro-Differential Equations

Transcription

1 Astrln Jornl of Bsc n Apple Scences, 5(4: 54-64, 2 ISS A ew Metho for Solvng Fzzy Volterr Integro-Dfferentl Eqtons T. Allhvrnloo, 2 A. Amrtemoor, M. Khezerloo, S. Khezerloo Deprtment of Mthemtcs, Arbl Brnch, Islmc Az Unversty, Arbl, Irn. 2 Deprtment of Mthemtcs, Rsht Brnch, Islmc Az Unversty, Rsht, Irn. Islmc Az Unversty, Yong Resercher Clb, Arbl Brnch, Arbl, Irn. Abstrct: In ths pper, the LU-representton of fzzy nmber s consere n bse on the LUrepresentton of fzzy volterr ntegro-fferentl eqton (FVIDE s scsse. The estence of solton of FVIDE s broght n etls. Then, the solton s fon n three cses of FVIDE tht s concle from the kernel. Fnlly, the metho s llstrte by solvng two emples. Key wors: Fzzy ntegro-fferentl eqtons; Fzzy solton; Fzzy vle fnctons; LU-fzzy representton; ITRODUCTIO The fzzy fferentl n ntegrl eqtons re mportnt prt of the fzzy nlyss theory n they hve the mportnt vle of theory n pplcton n control theory. Sekkl (987 hs efne the fzzy ervtve whch s the generlzton of the Hkhr ervtve n (Pr, M.L., D.A. Rlesc, 98, the fzzy ntegrl whch s the sme s tht of Dbos n Pre (982, n by mens of the etenson prncple of Zeh, showe tht the fzzy ntl vle problem ( t = f ( tt, (, ( = hs nqe fzzy solton when f stsfes the generlze Lpschtz conton whch grntees nqe solton of the etermnstc ntl vle problem. Klev (99 ste the Cchy problem of fzzy fferentl eqton, chrcterze those sbsets of fzzy sets n whch the peno theorem s vl. Prk et l. (995 hve consere the estence of solton of fzzy ntegrl eqton n Bnch spce n Sbrhmnm n Srsnm (994 hve prove the estence of solton of fzzy fnctonl eqtons. Rohprvr et l. (2 hve scsse the estence n nqeness of solton of the Cchy rectonffson eqton by Aomn ecomposton metho n Abbsbny n Allhvrnloo (26 hve pple Aomn ecomposton metho for solvng fzzy systems of the socon kn. Bee et l (27 hve ntroce more generl efnton of the ervtve for fzzy mppngs, enlrgng the clss of fferentble. Prk n Jeong (999, 2 ste estence of solton of fzzy ntegrl eqtons of the form t (= t f( t f(, t s, ( s s, t where f n re fzzy fnctons n k s crsp fncton on rel nmbers. Ths pper s orgnze s followng: In Secton 2, the bsc concept of fzzy nmber operton s broght. In Secton, the mn secton of the pper, s scsse. The propose e re llstrte by some emples n the Secton 4. Fnlly conclson s rwn n Secton 5. 2 Bsc Concepts: There re vros efntons for the concept of fzzy nmbers ([, 4]. Defnton 2.: An rbtrry fzzy nmber n the prmetrc form s represente by n orere pr of Corresponng Athor: T. Allhvrnloo, Deprtment of Mthemtcs, Arbl Brnch, Islmc Az Unversty, Arbl, Irn. 54

2 Ast. J. Bsc & Appl. Sc., 5(4: 54-64, 2 fnctons (, whch stsfy the followng reqrements: s bone left-contnos non-ecresng fncton over [, ]. s bone left-contnos non-ncresng fncton over [, ].,. A crsp nmber r s smply represente by = = r,. If <, we hve fzzy ntervl n f =, we hve fzzy nmber. In ths pper, we o not stngsh between nmbers or ntervls n for smplcty we refer to fzzy nmbers s ntervl. We lso se the notton =[, ] to enote the α-ct of rbtrry fzzy nmber. If =(, n v=( v, v re two rbtrry fzzy nmbers, the rthmetc opertons re efne s follows: Defnton 2.2 (Aton v=( v, v ( n n the terms of α-cts ( v =[ v, v ], [,] (2 Defnton 2. (Sbtrcton v=( v, v ( n n the terms of α-cts ( v = [ v, v ], [,] (4 Defnton 2.4 (Sclr mltplcton For gven k ( k, k, k > k = ( k, k, k < (5 n ( k =[mn{ k, k },m{ k, k }] (6 In prtclr, f k =, we hve =(, n wth α-cts ( = [, ], [,] Defnton 2.5 (Mltplcton v =(( v,( v (7 55

3 Ast. J. Bsc & Appl. Sc., 5(4: 54-64, 2 n ( v =mn{ v, v, v, v} ( v =m{ v, v, v, v}, [,] (8 Defnton 2.6 (Dvson [ v, v ] If =((,( v v v n ( =mn{,,, } v v v v v ( =m{,,, }, [,] v v v v v (9 ( Defnton 2.7 (Stefnn, L., 26. LU-representton of n rbtrry fzzy nmber by vector of 8 component of the ntervl [, ], wth = (wthot nternl ponts n = n =, s s follows: ( =(,,,,,,, where,,, re se for the lower brnch, n,,, re se for the pper brnch, by pplcton of monotonc nterpoltor on the whole ntervl [,]. In prtclr, the slops corresponng to re enote by, etc. By efnton (2., t s cler tht,, n. For n rbtrry trpezol fzzy nmber, we hve =, = n f =, then s n trnglr fzzy nmber. As reporte n (Stefnn, L., 26, sng LU-representton (, the membershp fncton LU-fzzy nmber s obtne by ( =sp{ [, ]} ( of the (2 In prtclr, corresponng to the noes of the α-cts, we hve ( = ( =, =,,, n, n the fferentble cse ' ' ( =, ( =, =,,, 56

4 The membershp fncton ( Ast. J. Bsc & Appl. Sc., 5(4: 54-64, 2 cn be ppromte by the se of monotonc splne nterpolton (see (Gerr, M.L., L. Stefnn, 25. In ths pper, we wll sppose the fferentble cse, for whch we se the representton =(,,, =,,, ( wth the t (4 n the slops, (5 Stefnn et l. (26 efne corresponng spces of fzzy nmbers by the LU-representton. In the fferentble cse, they enote by Fˆ ={ =(,,, } =,,, the set of LU-fzzy nmbers. F s 4( menson spce. Let n v re two LU-fzzy nmbers n form =(,,,, v=( v, v, v, v ˆ =,,, =,,, An Eclen-lke stnce on Fˆ ws efne by where (, v= v = ( ( ( ( = v v v v v (6 It s worthwhle to observe tht lso the one-se fzzy nmbers cn be esly represente by form smlr to the LU-representton: left-se fzzy nmber hs α-cts of the form [, [ n rght- se fzzy nmber hs α-cts of the form ], ]. Then, left-se fzzy nmber cn be wrtten s =(, =,,..., or rght-se fzzy nmber s =(, (see (Stefnn, L., 26. =,,..., The rthmetc opertors ssocte to the LU-representton cn be obtne s follows. Defnton 2.8 (Aton, (Stefnn, L., 26. v=( v, v, v, v =,,..., (7 Defnton 2.9 (Sclr mltplcton, (Stefnn, L., 26. For gven k k ( k, k, k, k, k > =,,..., = (,, k k k, k =,,...,, k < (8 57

5 Ast. J. Bsc & Appl. Sc., 5(4: 54-64, 2 In prtclr, f k = then =(,,, =,,..., n sbtrcton s efne by v= ( v ote tht the sclr mltplcton s lwys reproce ectly n ll the moels for ll [,] bt, n generl, ths s not tre for the ton s the sm of rtonl or me fnctons s not lwys rtonl or me fncton of the sme orers (see (Stefnn, L., 26. Integrls n ervtves of fzzy-vle fnctons hve scsse by Dbos n Pre (982, Klev (987, 99 n Pr n Rlesc (98; see lso (Km, Y.L., 997 for some recent reslts. Let :[, b] Fˆ s fncton where t (=( (, t ( t for t [, b ] s n LU fzzy nmber of the form (=( t (, t (, t (, t ( t =,,, the ntegrl of t ( wth respect to t[, b]. s gven by v := b tt ( = b (, b tt ( tt, [,] Defnton 2. (Integrton, (Stefnn, L., 26. In the LU-fzzy representton, the ntegrl of efne s t ( s v=( v, v, v, v where =,,, b b v = ( t t, v = ( t t, =,,, The (Hkhr ervtve of the fzzy-vle fncton t ( t the pont ˆt s s follows: ˆ ' ( ( t = ( t, ( t t t= tˆ t t= tˆ prove tht the followng contons hol: ech ( t s nonecresng wth respect to t[, b]. t ech ( t s nonecresng wth respect to t[, b]. t ( t (, t [,] t t Defnton 2. (Dervton, (Stefnn, L., 26. In the LU-fzzy representton, the ervtve of t ( t the pont ˆt s efne s 58

6 Ast. J. Bsc & Appl. Sc., 5(4: 54-64, 2 (=( tˆ (, tˆ (, tˆ (, tˆ ( tˆ ' ' ' ' ' =,,, n the contons for vl fzzy ervtve re, for =,,, ( tˆ ( tˆ ( tˆ ( tˆ ( tˆ (, tˆ ' ' ' ' ' ' ' ( ˆ t ' ( ˆ t Fzzy Volterr Integro-fferentl Eqtons: In ths secton, we conser the fzzy Volterr ntegro-fferentl eqton ( = f( k(, t ( t t, ( = c Fˆ (9 where f :[, ] Fˆ n k :, where ={(, t: t } n :[, ] Fˆ re contnos. Let c=( c, c, c, c =,,, be the LU-fzzy ntl conton n let ( n f( hve the followng forms, respectvely, ( =( (, (, (, (, =,,, =,,, f( =( f (, f (, f (, f ( then Eq. (9 cn replce by ( (, (, (, ( =( f (, f (, f (, f ( kt (, ( (, (, (, ( t t t t t ( (, (, (, ( = ( c, c, c, c (2 Cse. Sppose kt (, then kt (, ( (, t (, t (, t ( t t = ( kt (, ( t, kt (, ( t, kt (, ( t, kt (, ( t t =( kt (, ( tt, kt (, ( tt, kt (, ( tt, kt (, ( tt we cn clclte ( by solvng the followng 2( ntegro-fferentl eqtons (IDEs, for : =,,, (2 59

7 Ast. J. Bsc & Appl. Sc., 5(4: 54-64, 2 ( = f ( k (, t ( t t ( = f ( k (, t ( t t ( = c (22 To etermne the corresponng slopes we to (22 the followng 2( IDEs, for =,,, : ( = f ( k(, t ( t t ( = f ( k(, t ( t t ( = c Cse 2. Sppose kt (, < then kt (, ( (, t (, t (, t ( t t kt t kt t kt t kt t t = ( (, (, (, (, (, (, (, ( =( kt (, ( tt, kt (, ( tt, kt (, ( tt, kt (, ( tt (2 (24 we cn clclte ( by solvng the followng 2( ntegro-fferentl eqtons (IDEs, for =,,, : ( = f ( k (, t ( t t ( = f ( k (, t ( t t ( = c (25 n to etermne the corresponng slopes we to (25 the followng 2( IDEs, for =,,, : ( = f ( k(, t ( t t ( = f ( k(, t ( t t ( = c (26 Cse. Sppose kt (, hs not constnt sgn on [, ], then ( (, (, (, ( = ( f (, f (, f (, f ( K(, t, ( t t ( (, (, (, ( = ( c, c, c, c (27 6

8 Ast. J. Bsc & Appl. Sc., 5(4: 54-64, 2 where ((, kt (, t kt (, (, t kt (, (, t kt (, (, t kt (, K(, t, ( t= ((, kt (, t kt (, (, t kt (, (, t kt (, (, t kt (,< (28 So, f kt (, on [, ] n kt (, < on [, ], then n ( = f ( k (, t ( (, ( t t k t t t ( = f ( k (, t ( (, ( t t k t t t ( = c ( = f ( k(, t ( (, ( t t k t t t ( = f ( k(, t ( (, ( t t k t t t ( = c (29 ( Theorem. The solton of (2, (, s fzzy nmber for ll. Proof: In Eq. (9, we sppose F(, ( = f( G (, ( where G (, ( = kttt (, ( then we cn rewrte Eq. (9 s follows: ( = F(, (, ( = c Fˆ where F s contnos mppng. Usng Theorem (. n (Sekkl, 987, t s cler tht Eq. (9 hs nqe fzzy solton ( then Eq. (2 hs nqe fzzy solton (,.e. ( Fˆ,. 4 mercl Emples: Emple 4.:Conser FVIDE ( (, (, (, (, (, (, (, ( =, = ( e e (,,,,,,, =, ( (, (, ( t e ( (, (, (, (, (, (, (, ( =,t, (, (, (, (, ( = (,,,,,,, =, 6

9 Then, Ast. J. Bsc & Appl. Sc., 5(4: 54-64, 2 ( = ( e e e e t ( t t, (= ( t =( e e e e ( t t, (= t ( = e e ( t t, (= t ( = e e ( t t, (= n t ( =( e e e e ( t t, (= t ( = ( e e e e ( t t, (= t ( =( e e e e ( t t, (= t ( = ( e e e e ( t t, (= By solvng bove systems, we obtn ( = e (,,,,,,, Emple 4.2 Conser FVIDE =, ( (, (, (, (, (, (, (, ( =, = (,2,2,2,4, 2,2, 2 (,,,,2,,, 4 =, =, ( ( (, (, (, (, (, (, (, ( =,t =, ( (, (, (, (, (, (, (, ( = (,,,,2,,, Then, ( = ( t t, (= 2 4 ( =4 ( t t, (=2 4 ( =2 ( t t, (= 4 ( =2 ( t t, (= 62

10 n Ast. J. Bsc & Appl. Sc., 5(4: 54-64, 2 4 ( = 2 ( t t, (= 4 ( = 2 ( t t, (= 4 ( = 2 ( t t, (= 4 ( = 2 e t ( t t, (= By solvng bove systems, we obtn 2 ( = (,,,,2,,, =, 5 Conclson: In ths work, the LU-representton of FVIDE ws scsse n lso the estence of soltons s prove s theorem. The strctre of LU-representton of FVIDE s comprson wth the others for emple α-ct n prmetrc forms s smple. REFERECES Abbsbny, S., T. Allhvrnloo, 26. The Aomn ecomposton metho pple to the fzzy system of the secon kn, Interntonl Jornl of Uncertnty, Fzzness, n Knowlege Bse Systems (IJUFKS, 4(: -. Bee, B., I.J. Rs, L. Attl, 27. Frst orer lner fzzy fferentl eqtons ner generlze fferentblty, Informton Scences, 77: Dbos, D., H. Pre, 982. Towrs fzzy fferentl clcls: Prt, fferentton, Fzzy Sets n Systems, 8: Gl, S.G., 2. Appromton theory n fzzy settng, n: G.A. Anstsso (E., Hnbook of Anlytc- Compttonl Methos n Apple Mthemtcs, Chpmn Hll CRC Press, pp: Gerr, M.L., L. Stefnn, 25. " Appromte fzzy rthmetc operton sng monotonc nterpolton", Fzzy Sets n Systems, 5: 5-. Klev, O., 987. Fzzy fferentl eqtons, Fzzy Sets n Systems, 24: -7. Klev, O., 99. The clcls of fzzy vle fnctons, Appl. Mth. Lett., : Klev, O., 99. The Cchy problem for fzzy fferentl eqtons, Fzzy sets n Systems, 5: Km, Y.L., B.M. Ghl, 997. Integrls of fzzy-nmber-vle fnctons, Fzzy Sets n Systems, 86: Prk, J.Y., J.U. Jeong, 999. A note on fzzy fnctonl eqtons, Fzzy Sets n Systems, 8: 9-2. Prk, J.Y., Y.C. Kwn, J.U. Jeong, 995. Estence of soltons of fzzy ntegrl eqtons n Bnch spces, Fzzy Sets n Systems, 72: Prk, J.Y., S.Y. Lee, J.U. Jeong, 2. On the estence n nqeness of soltons of fzzy Volterr- Freholm ntegrl eqtons, Fzzy Sets n Systems, 5: Pr, M., D. Rlesc, 986. Fzzy rnom vrbles, J. Mth. Anl. Appl., 4: Pr, M.L., D.A. Rlesc, 98. Dfferentls of fzzy fnctons, J. Mth. Anl. Appl., 9: Rohprvr, H., S. Abbsbny, T. Allhvrnloo, 2. Estence n nqeness of solton n ncertn chrcterstc Cchy recton-ffson eqton by Aomn ecomposton metho, Jornl of Mthemtcl & Compttonl Applctons(MCA, 5(: Sekkl, S., 987. On the fzzy ntl vle problem, Fzzy Sets n Systems, 24: 9-. Stefnn, L., L. Sorn, L.M. Gerr, 26. " Prmetrc representton of fzzy nmbers n pplcton to fzzy clcls", Fzzy Sets n Systems, 57:

11 Ast. J. Bsc & Appl. Sc., 5(4: 54-64, 2 Sbrhmnm, P.V., S.K. Srsnm, 994. On some fzzy fnctonl eqtons, Fzzy Sets n Systems, 64: -8. Zeh, L.A., 975. " The concept of lngstc vrble n ts pplcton to ppromte resonng", Informton Scences, 8: