Fuzzy Laplace Transforms for Derivatives of Higher Orders

Transcription

1 Maemaical Teory and Modeling ISSN -58 (Paper) ISSN 5-5 (Online) Vol, No, 1 wwwiiseorg Fuzzy Laplace Transforms for Derivaives of Higer Orders Absrac Amal K Haydar 1 *and Hawrra F Moammad Ali 1 College of Educaion for Girls, Universiy of Kufa, Najaf, Iraq *amalkayder@uokufaeduiq In is paper, we find e formula of fuzzy derivaive of e ird order and four order and find e fuzzy Laplace ransforms for e fuzzy derivaive of e above menioned orders by using generalized H-differeniabiliy Keywords: Fuzzy numbers, generalized H-differeniabiliy, Fuzzy Laplace ransform 1 Inroducion Te concep of e fuzzy derivaive was firs inroduced by Cang and Zade (197), i was followed up by Dubois and Prade (198), Puri and Ralescu (198) Recenly, Allaviranloo and Amadi (1) ave proposed e fuzzy Laplace ransforms for solving firs order FIVP under generalized H-differeniabiliy Salasour and Allaviranloo (1) ave proposed fuzzy Laplace ransform and is inverse, proof and discussion abou some useful resuls and proposed equivalen inegral forms for solving second order FIVP Tis paper is arranged as follows: Basic conceps are given in Secion In Secions and, we find e formula, also fuzzy Laplace ransforms for e fuzzy derivaive of e ird order and four order respecively In Secions 5, we solve an example of e four order In Secions 6, conclusions are drawn Basic Conceps In is secion, some necessary definiions and conceps are inroduced: Definiion 1 (Allaviranloo e al 11) A fuzzy number u in parameric form is a pair ( uu, ) of funcions u ( ) and u ( ), 1 wic saisfy e following requiremens: 1 u ( ) is a bounded non-decreasing lef coninuous funcion in (,1], and rig coninuous a, u ( ) is a bounded non-increasing lef coninuous funcion in (,1], and rig coninuous a, u( ) u( ), 1 A crisp number is simply represened by u( ) u( ), 1 We recall a for a b c wic a, b, c R, e riangular fuzzy number u ( a, b, c) deermined by a, b, c is given suc a u ( ) a ( ba) and a u ( ) c ( c b) are e endpoins of e level ses, for all α [,1] Te Hausdorff disance beween fuzzy numbers is given by d : E E R {}, d ( u, v ) sup max{ u ( ) v ( ), u ( ) v ( )}, [,1] were u ( u( ), u ( )), v ( v ( ), v ( )) R is uilized Ten, i is easy o see a d is a meric in E and as e following properies: 1 d ( u w, v w ) d ( u, v ), u, v, w E d ( ku, kv ) k d ( u, v ), k R, u, v E, d ( u v, w e) d ( u, w ) d ( v, e), u, v, w, e E ( Ed, ) is a complee meric space x, ye z E x y z z e H-difference of x and y, and i is denoed by x Ө y In is paper, e sign Ө sands always for H-difference, and le us remark a x Ө y x ( 1) y Definiion (Bede e al 6) Le If ere exiss suc a, en is called

2 Maemaical Teory and Modeling ISSN -58 (Paper) ISSN 5-5 (Online) Vol, No, 1 wwwiiseorg Definiion (Allaviranloo e al 11) Le f :( a, b) E and x ( a, b) We say a f is srongly generalized differenial a x if ere exiss an elemen f ( x ) E, suc a i For all f x f x, f x f x and e limis (in e meric sufficienly small, Ө Ө d) lim[( f ( x ) Ө f x ))/ ] lim[( f ( ) Ө f ( x )) / ] f ( x) or ii For all ( x sufficienly small, Ө f x, f x Ө f x lim[( f( x ) Ө f x ))/ ] lim[( f ( x ) or iii For all f ( x )) / ] f ( x ) ( Ө sufficienly small, f x Ө f x, f x Ө ( lim[( f ( x ) Ө f x ))/ ] lim[( f ( x ) Ө f ( x)) / ] f ( x) f x and e limis (in e meric d) f x and e limis (in e meric d) or iv For all sufficienly small, f x Ө f x, f x Ө f x and e limis (in e meric d) lim[( f( x ) Ө f x ))/ ] lim[( f ( ) Ө f ( x )) / ] f ( x ) ( x Definiion (Allaviranloo and Amadi 1) Le () s f e is improper fuzzy Rimann-inegrable on,, en a ( ) Laplace ransforms and is denoed as s L( f ( )) f ( ) e d, s -s -s -s f ( ) e d ( f (, r) e d, f (, r) e d ) Also by using e definiion of classical Laplace ransform: s s l[ f (, r)] f (, r) e d and l[ f (, r)] f (, r) e d Ten, we follow: L ( f ( )) ( l ( f (, r)), l ( f (, r))) f be coninuous fuzzy-valued funcion Suppose s f ( ) e d is called fuzzy We ave Definiion 5 (Salasour and Allaviranloo 1) A fuzzy-valued funcion f as exponenial order p if ere exis consans M and p suc a for some, f ( ) Me p 1, Fuzzy Laplace Transforms for e Tird Order Derivaive In is secion, we ave e following resuls for ird order derivaive under generalized H-differeniabiliy: Teorem 1 Le and be differeniable fuzzy-valued funcions, and if -cu represenaion of is denoed by en (a) Le and be (i)-differeniable, or F() be (i)-differeniable and F () and F () be (ii)-differeniable, or F () be (i)-differeniable and F () and F() be (ii)-differeniable, or F () be (i)-differeniable and F () and F() be (ii)-differeniable; en f ( ) and g () ave firs order, second order and ird order derivaives and (b) Le F() and F () be (i)-differeniable and F () be (ii)-differeniable, or F ( ) and F () be (i)-differeniable and F() be (ii)-differeniable, or be (i)-differeniable and be (ii)-differeniable, or F ( ) and F () F( ), F ( ) F() F () [ F( )] [ f ( ), g( )], F( ), F ( ) F() [ F ( )] [ f ( ), g ( )] F () 5

3 Maemaical Teory and Modeling ISSN -58 (Paper) ISSN 5-5 (Online) Vol, No, 1 wwwiiseorg F ( ), F () and F() be (ii)-differeniable; en f () and ird order derivaives and [ F ( )] [ g ( ), f ( )] Proof Le () F () F() we ge (Salasour and Allaviranloo 1): and g () ave firs order, second order F, and be (ii)-differeniable Since F () and F () [ F ( )] [ f ( ), g ( )] Since F() is (ii)-differeniable en by ii of definiion we ge: [ F( ) Ө F( )] =[ f ( ) f ( ), g ( ) g ( )], be (ii)-differeniable, en [ F( ) Ө F ( )] =[ f ( ), f ( ), g ( ) g ( )] 1 and, muliplying by, we ge: 1 g ( ) g ( ) f ( ) f ( ) [ F ( ) Ө F( )] [, ], and 1 g ( ) g ( ) f ( ) f ( ) [ F ( ) Ө F( )] [, ] Finally, using on bo sides of aforemenioned relaion we ge: [ F ( )] [ g ( ), f ( )] Te oer proofs are similar Teorem Suppose a g( ), g ( ) and g () are coninuous fuzzy-valued funcions on [, ) and of exponenial order and a g () is piecewise coninuous fuzzy-valued funcion on [, ) wi g( ) ( g(, ), g(, )), en: (1) If gg, and g be (i)-differeniable, en: L( g( )) s L( g( )) Ө sg () Ө sg() Ө g () () If g and g be (i)-differeniable and g be (ii)-differeniable, en: L( g( )) s g() Ө ( s ) L( g( )) Ө sg () Ө g() () If g and g be (i)-differeniable and g is (ii)-differeniable, en: () L( g( )) s g() ( s ) L( g( )) sg() g() Ө Ө (5) If g and g are (i)-differeniable and g is (ii)-differeniable, en: L( g( )) s g() Ө ( s ) L( g( )) sg () g () (6) If g are (i)-differeniable and g and g be (ii)-differeniable, en: L( g( )) s L( g( )) Ө s g() sg() Ө g() (7) If g be (i)-differeniable and g and g be (ii)-differeniable, en: L( g( )) s L( g( )) Ө s g() sg() g() (8) If g is (i)-differeniable and g and g be (ii)-differeniable, en: L( g( )) s L( g( )) Ө () Ө sg (9) If gg, and g be (ii)-differeniable, en: L( g( )) s g() sg () g () Ө ( s ) L( g( )) Ө sg() g () Proof: Firs, we sae e noaions carefully as follows: g, g and g are e lower endpoins funcion s g g g g, g g derivaives,, and are e upper endpoins funcion s derivaives Also and are e 6

4 Maemaical Teory and Modeling ISSN -58 (Paper) ISSN 5-5 (Online) Vol, No, 1 wwwiiseorg lower endpoins of e derivaives, and are e upper endpoins of e derivaives: For arbirary fixed [,1] we ave g( ) ( g(, ), g(, )) are (i)-differeniable and () Now, we prove () as follows: Since g (), g () g is (ii)-differeniable Since is (i)-differeniable and g() is (ii)-differeniable en by eorem 1(b) we ge: g () g ( ) ( g (, ), g (, )) Terefore, we ge:, Ten, from (1) we ge: L ( g ( )) L ( g (, ), g (, )) =( l ( g (, )), l ( g (, ))) Since is (i)-differeniable and is (ii)-differeniable, we ge: g (, ) g (, ), g (, ) g (, ), g (, ) g (, ), g (, ) g (, ) () We know from e ordinary differenial equaions a: l( g (, )) s l( g(, )) s g(, ) sg(, ) g (, ), l g s l g s g sg g ( (, )) ( (, )) (, ) (, ) (, ) By using () and (), equaion () becomes: L( g ( )) ( s l( g(, ) s g(, ) sg(, ) g(, ), s l( g(, ) s g(, ) sg(, ) g(, )) g, g g g (, ) g (, ) g (, ) g (, ) g () g() sg() Ө ( s ) L( g( )) sg() Ө g () Te oer proofs are similar (1) () () Fuzzy Laplace Transforms for e Four Order Derivaive In is secion, we ave e following resuls for four order derivaive under generalized H-differeniabiliy: be differeniable fuzzy-valued funcions, and if -cu Teorem 1 Le F( ), F( ), F ( ) and F () F () [ F( )] [ f ( ), g( )], (a) Le F( ), F( ), F ( ) and F () represenaion of is denoed by en: be (i)-differeniable, or F() and F() be (i)-differeniable and F () and F() F() and F() be (i)-differeniable and F () and F() F() and F() be (i)-differeniable and F () and F() F () and F() be (i)-differeniable and F() and F() F () and F() be (i)-differeniable and F() and F() F () and F() be (i)-differeniable and F() and F() be (ii)-differeniable, or be (ii)-differeniable, or be (ii)-differeniable, or be (ii)-differeniable, or be (ii)-differeniable, or be (ii)-differeniable, or 7

5 Maemaical Teory and Modeling ISSN -58 (Paper) ISSN 5-5 (Online) Vol, No, 1 wwwiiseorg F( ), F( ), F ( ) and F() be (ii)-differeniable; en f () ird order and four order derivaives and (b)le F( ), F ( ) and F () F( ), F( ) and F () F( ), F( ) and F () F( ), F ( ) () () () [ F ( )] [ f ( ), g ( )] be(i)-differeniable and be(ii)-differeniable, or F () F() be (i)-differeniable and be (ii)-differeniable, or F() be (i)-differeniable and be (ii)-differeniable, or F () F() and be(i)-differeniable and be (ii)-differeniable, or F () be (i)-differeniable and F( ), F ( ) F() and g () ave firs order, second order, and be (ii)-differeniable, or F() be (i)-differeniable and F( ), F( ) and F () be (ii)-differeniable, or F() be (i)-differeniable and F( ), F( ) and F () be (ii)-differeniable, or F () be (i)-differeniable and F( ), F ( ) and F () be (ii)-differeniable; en f () and g () ave firs order, second order, ird order and four order derivaives and () () () [ F ( )] [ g ( ), f ( )] Proof: Te proofs as in proof of eorem 1 Teorem Suppose a g( ), g( ), g ( ) and g () and of exponenial order and a g ( ) ( g (, ), g (, )), en (1) If g, g, g and g be (i)- differeniable, en: are coninuous fuzzy-valued funcions on g () () is piecewise coninuous fuzzy-valued funcion on wi L( g ( )) s L( g ( )) Ө s g () Ө s g () Ө sg () Ө g () () If g, g and g be (i)-differeniable and g is (ii)-differeniable, en: () L ( g ( )) s g () Ө ( s ) L ( g ( )) Ө s g () Ө sg () Ө g() () If, g g and g be (i)-differeniable and g is (ii)-differeniable, en: () L ( g ( )) s g () Ө ( s ) L ( g ( )) s g () Ө sg () Ө g() () If, gg and g be (i)-differeniable and g is (ii)-differeniable, en: () L ( g ( )) s g () Ө ( s ) L ( g ( )) s g() sg () Ө g() g (5) If gg, and be (i)-differeniable and is (ii)-differeniable, en: () L ( g ( )) s g () Ө ( s ) L ( g ( )) s g() sg () g() (6) If g and g be (i)-differeniable and g and g be (ii)-differeniable, en: g [, ) [, ) 8

6 Maemaical Teory and Modeling ISSN -58 (Paper) ISSN 5-5 (Online) Vol, No, 1 wwwiiseorg L g s L g ( ( )) ( ( )) Ө s g () s g () Ө Ө sg () g() (7) If g and g be (i)-differeniable and g and g be (ii)-differeniable, en: L g s L g ( ( )) ( ( )) Ө s g () s g () sg () Ө g() (8) If g and g be (i)-differeniable and g and g be (ii)-differeniable, en: L g s L g ( ( )) ( ( )) Ө s g () s g () sg () g () (9) If g and g be (i)-differeniable and g and g be (ii)-differeniable, en: L g s L g ( ( )) ( ( )) Ө s g() Ө s g () sg () Ө g() (1)If g and g be (i)-differeniable and g and g be (ii)-differeniable, en: L g s L g ( ( )) ( ( )) Ө s g() Ө s g () sg () g () (11) If g and g be (i)-differeniable and g and g be (ii)-differeniable, en: L g s L g ( ( )) ( ( )) s g sg () g () Ө s g() Ө () Ө g (1) If be (i)-differeniable and, gg and g be (ii)-differeniable, en: () L ( g ( )) s g () Ө ( s ) L ( g ( )) Ө s g () sg () Ө g() g (1) If be (i)-differeniable and gg, and be (ii)-differeniable, en: () L ( g ( )) s g () Ө ( s ) L ( g ( )) Ө s g () sg () g () g (1) If be (i)-differeniable and, g g and g be (ii)-differeniable, en: () L ( g ( )) s g () Ө ( s ) L ( g ( )) Ө s g () Ө sg () g () g (15) If be (i)-differeniable and, g g and g be (ii)-differeniable, en: () L ( g ( )) s g () Ө ( s ) L ( g ( )) s g () Ө sg () g () g (16) If g, g, g and g be (ii)-differeniable, en: L g s L g ( ( )) ( ( )) Ө s g () s g () Ө sg () g() Proof: Te proofs as in proof of eorem 5An Illusraive Example Consider e following four order FIVP: y () ( ) y ( ), y () y () y () y () ( r 1,1 r) We sall consider e same cases given in eorem respecively, as follows: 9

7 Maemaical Teory and Modeling ISSN -58 (Paper) ISSN 5-5 (Online) Vol, No, 1 wwwiiseorg Case 1 By using relaion (1) of eorem, en we ge e r -cu represenaion of soluion as following: y (, r) ( r 1) e, y (, r) (1 r) e Tis is e same resul wic was found by Tapaswini and Cakravery (1) Case By using relaion () of eorem, en we ge e r -cu represenaion of soluion as following: y (, r) ( r 1)(cos cos sin cos sin sin ), y (, r) (1 r)(cos cos sin cos sin sin ) Case By using relaion () of eorem, en we ge e r -cu represenaion of soluion as following: y (, r) ( r 1)(cos cos sin sin cos sin ), y (, r) (1 r)(cos cos sin sin cos sin ) Case By using relaion () of eorem, en we ge e r -cu represenaion of soluion as following: y (, r) ( r 1)(cos cos cos sin sin sin ), y (, r) (1 r)(cos cos cos sin sin sin ) Case 5 By using relaion (5) of eorem, en we ge e r -cu represenaion of soluion as following: y (, r) ( r 1)(cos cos sin sin sin cos ), y (, r) (1 r)(cos cos sin sin sin cos ) Case 6 By using relaion (6) of eorem, en we ge e r -cu represenaion of soluion as following: y (, r) ( r 1)(cos sin ), y (, r) (1 r)(cos sin ) Case 7 By using relaion (7) of eorem, en we ge e r -cu represenaion of soluion as following: y (, r ) ( r 1)(cos sin ), y (, r) (1 r)(cos sin ) Case 8 By using relaion (8) of eorem, en we ge e r -cu represenaion of soluion as following: y (, r ) ( r 1)(cos sin ), y (, r) (1 r)(cos sin ) Case 9 By using relaion (9) of eorem, en we ge e r -cu represenaion of soluion as following: y (, r) ( r 1)(sin cos ), y (, r) (1 r)(sin cos ) Case 1 By using relaion (1) of eorem, en we ge e r -cu represenaion of soluion as following: y (, r ) ( r 1)(cos sin ), y (, r) (1 r)(cos sin ) Case 11 By using relaion (11) of eorem, en we ge e r -cu represenaion of soluion as following: y (, r ) ( r 1)(cos sin ), y (, r) (1 r)(cos sin ) Case 1 By using relaion (1) of eorem, en we ge e r -cu represenaion of soluion as following: y (, r) ( r 1)(cos cos sin sin sin cos ),

8 Maemaical Teory and Modeling ISSN -58 (Paper) ISSN 5-5 (Online) Vol, No, 1 wwwiiseorg y (, r) (1 r)(cos cos sin sin sin cos ) Case 1 By using relaion (1) of eorem, en we ge e r -cu represenaion of soluion as following: y (, r) ( r 1)(cos cos cos sin sin sin ), y (, r) (1 r)(cos cos cos sin sin sin ) Case 1 By using relaion (1) of eorem, en we ge e r -cu represenaion of soluion as following: y (, r) ( r 1)(cos cos cos sin sin sin ), y (, r) (1 r)(cos cos cos sin sin sin ) Case 15 By using relaion (15) of eorem, en we ge e r -cu represenaion of soluion as following: y (, r) ( r 1)(cos cos sin cos sin sin ), y (, r) (1 r)(cos cos sin cos sin sin ) Case 16 By using relaion (16) of eorem, en we ge e r -cu represenaion of soluion as following: y (, r) ( r 1) e, y (, r ) (1 r ) e 6 Conclusions Te formula of fuzzy derivaive of e ird and four orders are found In addiion, fuzzy Laplace ransforms for e fuzzy derivaives of e same orders are found We used fuzzy Laplace ransforms in solving FIVP of e four order, and muliple soluions are provided for is FIVP References Allaviranloo T, Amadi MB( 1), "Fuzzy Laplace Transforms", Sof Compu, 1, 5- Allaviranloo T, Abbasbandy S, Salasour S, Hakimzade A (11), "A New Meod for Solving Fuzzy Linear Differenial Equaions", Compuing, 9, Bede B, Rudas IJ, Bencsik AL(6), "Firs Order Linear Fuzzy Differenial Equaions Under Generalized Differeniabiliy" Inf Sci, 177, Cang SSL, Zade L (197), "On Fuzzy Mapping and Conrol" IEEE Trans Sys Cybern,, - Dubios D, Prade H (198), "To Wards Fuzzy Differenial Calculus", Fuzzy Se Sys, 8, (1-7):15-116, 5- Puri ML, Ralescu DA, "Differenials for Fuzzy Funcions", Journal of Maemaical Applicaions, (198), 91, Analysis and Salasour S, Allaviranloo T (1), "Applicaions of Fuzzy Laplace ransforms", Sof compu, 17, Tapaswini S, Cakravery S (1), "Numerical Soluion of n- Order Fuzzy Linear Differenial Equaions by Homoopy Perurbaion Meod", Inernaional Journal of Compuer Applicaions, 6, 5-9 1