Method For Solving Fuzzy Integro-Differential Equation By Using Fuzzy Laplace Transformation

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1 INERNAIONAL JOURNAL OF SCIENIFIC & ECHNOLOGY RESEARCH VOLUME 3 ISSUE 5 May 4 ISSN Meod For Solving Fuzzy Inegro-Differenial Equaion By Using Fuzzy Laplace ransformaion Manmoan Das Danji alukdar Absrac: In is aricle we sudy e meod of solving fuzzy inegro-differenial equaion under cerain condiion by using fuzzy Laplace ransformaion. Finally we give some illusraive examples. Keywords: Fuzzy numbers Volerra inegro-differenial equaion fuzzy Laplace ransformaion Rimann-inegrable Hausdorff disance.. Inroducion e concep of fuzzy ses se operaions was firs inroduced by Zade [9] subsequenly several auors ave sudied various aspecs of e eory applicaions of fuzzy ses. Abbasby e. al [ ] inroduced a numerical algorim for solving linear Fredolm fuzzy inegral equaions of second kind by using parameric form of fuzzy number convering a linear fuzzy Fredolm inegral equaions o wo linear sysems of inegral equaion of e second kind in crisp case Badolian e. al sudied anoer numerical meod for solving linear fuzzy Fredolm inegral equaion of second kind by using Adomian meod. Moreover Friedman e. al [8] invesigaed an embendding meod o solve fuzzy Volera Fredolm inegral equaions. However many oer auors obained e numerical inegraion of fuzzy valued funcions solving fuzzy Volera Fredolm equaions. e concep of fuzzy Laplace ransformaion was inroduced by Allaviranloo Amadi [6]. Afer many oer researcer use i o solve fuzzy differenial equaions fuzzy inegral equaions ec. Some auors discussed e soluion of fuzzy inegro-differenial equaion by fuzzy differenial ransform meod in eir researc paper [3].. Definiion Background A fuzzy number is a fuzzy subse of e real line R i.e a wic is bounded convex funcion u: R[] normal. Le E denoe e se of all fuzzy numbers wic are upper semi coninuous ave compac suppor. e - level se [u] α se of a fuzzy real number u for < defined as [u] α = { R : u() }. A fuzzy real number u is called convex if u() u(s) u(r) = min ( u(s) u(r) ) were s < < r. If ere exiss R suc a u( ) = en e fuzzy real number u is called normal. A fuzzy real number u is said o be upper semi- coninuous if for eac > u ([ a + )) for all a [] is open in e usual opology of R. e absolue value u of u E is defined as (see for insance Kaleva Seikkala u () = max { u() u(-) } if = if <. I is clear a a e - level se of e fuzzy number u is a closed bounded inerval u α u α were u( ) denoes e lef- end poin u α denoes e rig- end poin of u α. wo arbirary fuzzy numbers u = u α u α v = v α v α are said o be equal i.e u = v if only if u α = v α u α = v α. Since eac y R can be regarded as a fuzzy number y defined by : y = if = y if y e Hausdorff disance beween fuzzy numbers is a mapping d : L( R) L( R) R defined by d( u v) sup max u( ) v( ) u( ) v( ) Were u = u α u α = v α v α. I can be easily sown a d is a meric on LR ( ) wi e following properies :. d( u w v w) d( u v) for all u v w L( R).. d( ku kv) k d( u v) for all u v L( R) 3. d( u v w e) d( u w) d( v e) for all u v w e L( R) 4. d L( R) is a complee meric space. Manmoan Das Dep. Of Maemaics Bajali College Assam India. mdas.bajali@gmail.com Danji alukdar Dep. Of Pysics Bajali College Assam India. danji@gmail.com Definiion.: Le f : R L( R) funcion. f is said o be coninuous a ere exiss a suc a d( f ( ) f ( )) wenever R if for eac. IJSR 4 9

2 INERNAIONAL JOURNAL OF SCIENIFIC & ECHNOLOGY RESEARCH VOLUME 3 ISSUE 5 May 4 ISSN Definiion.: Le f : R L( R) funcion x x if ere exiss R. We say a f is differeniable a f ( x ) L( R) suc a f ( x ) f ( x ) f ( x ) f ( x ) ( a) lim lim f( x ) Or f ( x ) f ( x ) f ( x ) f ( x ) ( b) lim lim f( x ) eorem.: Le f : R L( R) funcion denoe f ( ) f ( ) f ( ) for eac followings are old (a) If (b) If f is differeniable in e firs form (a) in definiion. en f( ) f( ) are differeniable f ( ) f ( ) f ( ). f is differeniable in e nd form (b) in definiion. en f( ) f( ) are differeniable f ( ) f ( ) f ( ). eorem.: Le f : R L( R) funcion denoe f ( ) f ( ) f ( ) for eac en (a) If f f are differeniable in e firs form (a) in definiion. or if f f are differeniable in e nd form (b) in definiion. en f ( ) f ( ) differeniable are f ( ) f ( ) f ( ). (b) If f is differeniable in e firs form (a) f is differeniable in e nd form (b) or if f is differeniable in e nd form (b) f is differeniable in e firs form (a) in definiion. en en f ( ) f ( ) are differeniable f ( ) f ( ) f ( ). eorem.3: Le f( x ) be a fuzzy real valued funcion on [ a ) i is represened by f ( x ) f ( x ). For any fixed [] r assume f ( x ) f ( x ) are Rimann-inegrable on [ ab ] for every b a le ere exiss wo posiive M ( ) M ( ) suc a b f ( x ) dx M ( ) f ( x ) dx M ( ) a for every b a. en f( x ) is improper fuzzy Rimanninegrable on [ a ) e improper fuzzy Rimanninegral is a fuzzy number. Furer f ( x) dx f x dx f x dx a a a b a more we ave Proposiion.: Le f( x ) gx ( ) be a fuzzy real valued funcion also fuzzy Rimann-inegrable on I [ a ) en f x g x is Rimann-inegrable on I [ a ) f ( x) g( x) dx f ( x) dx g( x) dx I I I Definiion.3: Fuzzy Laplace ransformaion e fuzzy Laplace ransform of a fuzzy real valued funcion f() is defined as follows: s s F( s) L f ( ) e f ( ) d lim e f ( ) d wenever e limi exis. e symbol L is fuzzy Laplace ransformaion wic acs on fuzzy real valued funcion f f () generaes F( s) L f ( ). e lower upper Laplace ransform of a fuzzy real valued funcion f() are given as follows: F( s ) L f ( ) l f ( ) l f ( ) Were s s l f ( ) e f ( ) d lim e f ( ) d IJSR 4 9

3 INERNAIONAL JOURNAL OF SCIENIFIC & ECHNOLOGY RESEARCH VOLUME 3 ISSUE 5 May 4 ISSN s s l f ( ) e f ( ) d lim e f ( ) d Definiion.3: Fuzzy Convoluion eorem : e convoluion of wo fuzzy real valued funcions defined for by ( f g)( ) f ( ) g( ) d f g eorem.4 : If f g are piecewise coninuous fuzzy real valued funcion on [ ) wi exponenial order p en- L ( f g)( ) L f ( ). L g( ) F( s). G( s) s p. (). ;... n ; ; l u ; s l u s a s a a l f l k (3)... ;... n ; ; l u ; s l u s a s a a l f l k Now we discuss e following cases : Case : If u; k ( ; ) are bo posiive en l k ; l u ; l k( ; ) l u( ; ) 3. Main Meod Consider e following Fuzzy Volerra Inegro-Differenial equaion : ()... n wi u f k s u s d s k k k u a a a ; k n. Case : If ; l k ; l u ; l k( ; ) l u( ; ) en u is negaive ( ; ) k is posiive l k ; l u ; l k( ; ) l u( ; ) Were n u denoes e n derivaive of u. aking Fuzzy Laplace ransformaion on bo sides of () we ge n L[ u ] L[ f ] L[ k s u s ] ds By definiion of Fuzzy Laplace ransformaion using Fuzzy Convoluion eorem we ave n s L u s u s u... u L[ f ] L[ k s u s ] ds is implies a l k ; l u ; l k( ; ) l u( ; ) Case 3: If ; en u is posiive ( ; ) k is negaive l k ; l u ; l k( ; ) l u( ; ) l k ; l u ; l k( ; ) l u( ; ) Case 4: If ; en u ( ; ) k are bo negaive l k ; l u ; l k( ; ) l u( ; ) l k ; l u ; l k( ; ) l u( ; ) We inroduce explici formula for Case oers are similar. So we ave from () (3) IJSR

4 INERNAIONAL JOURNAL OF SCIENIFIC & ECHNOLOGY RESEARCH VOLUME 3 ISSUE 5 May 4 ISSN ;... ; ( ; ) ( ; ) s l u s a s a a n l f l k l u ;... ; ( ; ) ( ; ) s l u s a s a a n l f l k l u In compac form: l u ; l u ; l f ( ; ) s a s a.... a n s l k( ; ) n n n l f ( ; ) s a s a.... a n s l k( ; ) n n n By aking e inverse of fuzzy Laplace ransformaion on bo sides of above relaions we can easily obain e value of u; ; u were. Example 3.: o solve e following Fuzzy Volerra Inegro-Differenial equaion 4 ( ) u x u x dx ; u() ( 3 ); u() ( ) By using fuzzy Laplace ransformaion on bo sides of e equaion we ge L[ u ] L[ 4 ] L[ ] L[ u( )] is implies s L u su u L ( ) () () { 4 } L{ } L{ u( )} i.e ; ( ) ; s l u l l l u s( ) ( ) ; (4 ) ; s l u l l l u s(3 ) ( ) i.e 3 s lu ; ( ) ( ) 4 4 s s s ( ) 4 s 3 s l u ; (4 ) (3 ) 4 4 s s s ( ) 4 s By aking e inverse of fuzzy Laplace ransformaion on bo sides of above relaions we ave u ; ( ). sin sin ( ). cos cos ( ) sin sin u ; (4 ). sin sin (3 ). cos cos ( ) sin sin. Example 3.: Consider e Fuzzy Volerra Inegro- Differenial equaion IJSR

5 INERNAIONAL JOURNAL OF SCIENIFIC & ECHNOLOGY RESEARCH VOLUME 3 ISSUE 5 May 4 ISSN sin( ) u e x u x dx u() u() u () () By using fuzzy Laplace ransformaion on bo sides of e equaion we ge s lu ; ( ) 5 3 ( s s )( s ) s 5 3 l u ; (3 ) ( s s )( s ) By aking e inverse of fuzzy Laplace ransformaion on bo sides of above relaions we can easily obain e value of u; ; u were. References []. B.BedeQuadraure rules for inegrals of fuzzynumber-valued funcion. Fuzzy Ses Sysems 45 (4) []. B.BedeS.G. Gal Generalizaions of e differeniabiliy of fuzzy-number-valued funcions wi applicaions o fuzzy differenial equaions Fuzzy Ses Sysems 5(5) [3]. H.C. Wu e improper fuzzy Riemann inegral is numerical inegraion Informaion Science (999)9-37. [9]. M. Friedman M.Ma A.Kel Numerical meods for calculaing e fuzzy inegrals Fuzzy Ses Sysem 83(996) []. M. Malok On fuzzy inegrals Proce.ndPolis Symp. On Inerval Fuzzy Maemaics Poliecnika Poznansk987pp []. R.GoescelW. Voxman Elemenery calculus Fuzzy Ses Sysems 8 (986) []. S. Abbasby E. Babolian M. Alavi Numerical meod for solving linear Fredolm fuzzy inegral equaions of nd kind Caos Soluion Fracals 3(7) [3]. S. Salasour e al. IJIM Vol.4 No. () -9. [4]. S.S.L.CangL. Zade On fuzzy mapping conrol. IEEE rans Sysem Cyberne(97)3-34. [5]. S.NaOn inegraion of fuzzy mappings Fuzzy Ses Sysem 3(989) 95-. [6].. Allavirenloo M. Barkordari Amadi Fuzzy Laplace ransforms Sof Compuing 4 () [7]. V.Laksmikanam R.N. Moapara eory of Fuzzy Differenial Equaions inclusionsaylor Francis 3. [8]. W.Congxin M.Ma On e inegrals series inegral equaions of fuzzy se-valued funcions J. Harbin Ins. ecnol.(99) -9. [9]. Zade L.A. Zade (965) Fuzzy ses Inform Conrol 8 (965) [4]. H.C. Wu e fuzzy Riemann inegral is numerical inegraion Fuzzy Se Sysems ()-5. [5]. I.Perfilieva Fuzzy ransforms :eory Applicaions Fuzzy Ses Sysems 57(6) [6]. I. Perfilieva H. De Meyer B. De Baes Caucy problem wi fuzzy iniial condiion is approximae soluion wi e elp of fuzzy ransform.wcci 8Proceedings Hong Kong IEEE Compuaional Inelligence Sociey (8)85-9. [7]. J.Y.Park.C.KwanJ. V.Jeong Exisence of soluions of fuzzy inegral equaions In Branc spaces Fuzzy Ses Sysem 7(995) [8]. M.Friedman M.MaA.Kel Numerical soluion of fuzzy differenial inegral equaionsfuzzy Ses Sysem 6 (996) IJSR