Homotopy Analysis Method for Solving Fractional Sturm-Liouville Problems

Transcription

1 Ausralia Joural of Basic ad Applied Scieces, 4(1): , 1 ISSN Homoopy Aalysis Mehod for Solvig Fracioal Surm-Liouville Problems 1 A Neamay, R Darzi, A Dabbaghia 1 Deparme of Mahemaics, Uiversiy of Mazadara, Babolsar, Ira Islamic Azad Uiversiy Neka Brach, Neka, Ira Absrac: The aim of his paper is o prese a efficie ad reliable reame of he homoopy aalysis mehod (HAM) for solvig fracioal Surm-Liouville differeial equaio D y() q() y() () y(), 1, [,1] which he secod order derivaives is replaced by a fracioal derivaive y() is posiive ad smooh fucio ad D a ha deoes he fracioal differeial operaor of order a, is described i he Capuo sese Several illusraive examples are give o demosrae effeciveess ad reliabiliy of HAM for solvig fracioal Surm-Liouville problems Key words: Fracioal derivaive ad iegral Capuo derivaive Homoopy aalysis mehod Diffusio ad Wave equaio INTRODUCTION The aalyic mehod, homoopy aalysis mehod (HAM), proposed by Liao (Brow ad Churchill, 199) is a powerful mehod o solve o-liear problems Several aalyical ad umerical mehod have bee proposed o solve FDEs Adomia decomposiio mehod (ADM) (Al-Mdallal, ;Momai, 6; Momai, 5; 6; 6; Odiba ad Momai, 6), Variaioal ieraio mehod (VIM) (He, 1998; 1997) ad oher mehods (Momai ad Odiba, 7; 6; 8) I rece years; his mehod (HAM) employed o solve may ypes of differeial equaio of fracioal order (Momai ad Odiba, 6) For example, o solve fracioal parial differeial equaio I his paper, we solve fracioal Surm-Liouville problem D y() q() y() () y(), 1, (,1) subec o ay() by(), cy(1) dy(1) a,b,c,d R This paper has bee divided as follows: several defiiio of fracioal calculus, aalysis of (HAM), applyig his mehod for solvig fracioal Surm-Liouville differeial equaio ad fracioal diffusio-wave equaio Fracioal Calculus: Defiiio1: A real fucio f(), >, is said o be i he space C, R, if here exiss a real umber > such ha f() (), f () C [, k k ) ad i is said o be i he if ad oly if f, k N f1 1 C ( ) Defiiio: The Riema-Liouville fracioal iegral operaor of order or > of a fucio f, of a fucio f C,, 1 is defied as (Miller, 199) C Correspodig Auhor: A Neamay, Deparme of Mahemaics, Uiversiy of Mazadara, Babolsar, Ira amay@umzacir 518

2 Aus J Basic & Appl Sci, 4(1): , 1 1 Ia f r f r dr ( ) I f() I f() I f() f() () x ( 1 ) ( ) a Remark: The Riema-Liouville iegral operaor has he followig properies: i) I ai a I a ii) ( 1) Ia ( xb) ( xb) ( 1) ad f C, 1,, 1 Defiiio: The Capuo's fracioal derivaive of f is give by 1 D f r f r dr ( k ) m1 ( m) () ( ) ( ) k k D f() I D f() k f C, 1,, 1 k k kn k Noig o above defiiios, for f C, 1, k 1 k, k N ad >, we have i) DaIa f() f() ii) m1 ( ) ( a) IaDa f() f() f ( )! Defiiio4: The Miage-Leffler fucio E ( z) for > ad zc is defied by he series represeaio [6,14] k z E ( z), ( k 1) v, zc k Aalysis of he Mehod: To describe he basic idea of he sadard HAM, we cosider differeial equaio N[ y( )] g( ) whi N[ y( )] D y( ) ( q( ) ( )) y( ), N is suggess ha we defie a o-liear operaor, deoes he idepede variable, y() is ukow fucio ad g() is kow aalyic fucio represeig he ohomogeeous erm By meas of geeralizig he radiioal homoopy mehod Liao (199), cosrucs he socalled zeroh-order deformaio equaio: (1 ql ) [ ( q, ) y( )] qhhn ( ) [ ( q, )] (1) q [,1] is a embeddig parameer, h is a o-zero auxiliary parameer, H () is a o-zero auxiliary fucio, L( D,1 ) is a auxiliary liear operaor, y () is he iiial guesses of y() ad (, q)is ukow fucios I is impora o oe ha oe has gree freedom o choose he auxiliary obecs i HAM (, q) y() (, ) () For q ad q 1 ad q y hold, respecively Therefore, as icrease from 519

3 Aus J Basic & Appl Sci, 4(1): , 1 o 1 he soluios (, q) vary from he iiial guesses y () o he soluios y() Expadig (, q) i Taylor series wih respec o q, oe has (, q) y() y() 1 1 ( q, ) y() q If he auxiliary liear operaor, he iiial guesses, he auxiliary parameer! q h ad he auxiliary fucio are properly chose he series () coverges a q=1, he we have (,1) y () y () 1 which is oe of he soluios o-liear equaio as proved by Liao (He, 1998) Defie he vecors y { y ( ), y ( ),, y ( )} 1 Differeiaig equaio (1) imes wih respec o he embeddig parameer q ad he seig q=, ad fially dividig hem by!, we coclude h-order deformaio equaio Ly [ ( ) y ( )] hhr ( ) ( y ) { N[ (, q)]} R( y 1) 1 q ( 1)! q ad Operaig he Riema-Liouville iegral operaor I (1 ) o boh side of Eq() y () y () { y ( ) y ( )} hh() I R ( y ) I should be emphasized ha y i (x) is govered by he Eq() wih he liear boudary codiios ha come from he origial problem, which ca be easily solved by symbolic compuaio sofwares such as Maple ad Mahemaica 4 Examples: I his secio, we offer hree examples by usig HAM Example1: Firs, we cosider a fracioal differeial equaio () () 4 [ ()] () (), (,1) D y y y (4) subec o y(), y() A Where A is cosa To solve Eq(4) by meas of HAM, by usig iiial codiios, i is a sraighforward ha he iiial approximaio should be i he form y () A We choose he liear operaor 5

4 Aus J Basic & Appl Sci, 4(1): , 1 4 [ (, )] [ (, )] (, ) (, ) N q D q q q Accordig o above defiiio, assumpio H () 1, we cosruc he zeroh-order deformaio equaio, (1 q) L[ (, q)] y ( )] qhn[ (, q)] Thus, we obai he - h- order deformaio equaio Ly [ ( ) y ( )] hr ( y ) 1 1 R [( y )] D [ y ( )] y ( ) y ( ) (5) (6) 4 I Operaig he Riema-Liouville iegral operaor o boh side of Eq(5) we obai y () ( h) y () ( h) y () ( h) y () hi y () hi y () Fially, we have y() y () y () So, From (6) ad iiial codiio y () A, we ge i1 i 7 1 h() h() y1 () 1 1 ( ) ( ) 1 16 h () ( ) h () ( ) h() h() h () h () y () ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) h () ( ) 17 h () ( ) 1 1 ( ) ( ) ( ) ( ) So, whe h = -1, he series soluio expressio by he HAM is 1 16 () ( ) () ( ) () () y () 1 ( ) ( ( ) ( ) ( ) ( ) ( ) ( ) () ( ) 17 () ( ) ) 1 1 ( ) ( ) ( ) ( ) Example: We cosider he regular fracioal eigevalue problem D y () y (), (,1) (7) 51

5 alog wih he boudary codiios Aus J Basic & Appl Sci, 4(1): , 1 y (), y (1) To solve Eq (7) by meas of sadard HAM, i is aural o choose iiial approximaio y () A, A is cosa We choose he liear operaor L[ ( q, )] D[ ( q, )] wih he propery L(c)= c is cosa Furhermore, suggess ha we defie a oliear operaor Also, we defie a oliear operaor N[ (, q)] D[ (, q)] (, q) Noig above, wih H() = 1, we cosruc he zeroh-order deformaio (1 ql ) [ ( q, ) y( )] qhn[ ( q, )] Thus, we obai he h-order deformaio equaio L[ y ( ) y ( )] hr [ y ] 1 1 R [ y ] D [ y ( )] y ( ) Operaig he Riema-Liouville iegral operaor i y ( ) ( h) y ( ) ( h) y () ( h) y () hi y ( ) y() y () y () 4 I o boh side of Eq(8) we obai () Fially, From (9) ad iiial codiio y A we have h 4h y1() ( ) 5 ( ) i1 i h(1 h) h 4h 4h h y() ( ) 5 ( ) (4) 6 h(1 h) h (1 h) h 4h 4h h 4h y() ( 5 (4) 11 ( ) ( ) 6 9 4h h h h h ) (8) (9) Hece y () y() y() y() 1 Now, if h 1, we have 5

6 Aus J Basic & Appl Sci, 4(1): , 1 9 ( ) (4) y ( ) 1 ( ) ( ) E( ) 5 11 ( ) ( ) (1 ) E ( ) deoes he Miag-Leffler fucio Now, usig he boudary codiio, we explore he firs hree eigevalues (, ad ) umerically i followig able represes he umber of erms used 1, i, i, i i he followig series, ie i y () y() The umericl evidece i able suggess ha he firs hree eigevalue are 11778, 17658, Table 1: The approximaio o he firs hree eigevalues i 1,i,i,i Example: We cosider he followig ohomogeeous fracioal Diffusio-Wave equaio ux (, ) D [ u( x, )] k, 1 x subec o: u(, ) u(, ), 1 ux (,) x, x u ( x,), x u, k is cosa ad he capuo fracioal derivaive of order, for ( x, ), is defied as: m u ( (, )) 1 1 xs u m1 D [ ux (, )] D ( ( x, )) ( s) s ds, m1 m, mn ( ) m m s By separaio of variables mehod, we suppose ux (, ) XxT ( ) ( ) is a soluio of homogeeous par (1), ogeher wih he boudary codiios(11) So, we have (1) (11) X( x) X( x), X(), X( ) D [ T( )] kt( ), T() B, T(), 1 B is cosa The Surm-Liouville problem (1), has eigevalues ad he correspodig (1) (1) 5

7 Aus J Basic & Appl Sci, 4(1): , 1 eigefucios X ( x) si x,( 1,), Now, for he case, by he HAM, we solve fracioal Surm-Liouville equaio D T k T T B T [ ( )] ( ), (), (), 1 (14) To solve Eq(14), by HAM, accordig o iiial codiios, i isaural o choose iiial approximaio T () B B is cosa We choose he liear operaor L[ ( q, )] D [ ( q, )], (i here (, q) T() q i ) wih he propery Lc (), c is cosa Furhermore, N suggess ha we i defie a oliear operaor i N q D q k q [ (, )] [ (, )] (, ) Accordig o above defiiio, assumpio H()=1, we cosruc he zeroh-order deformaio (1 ql ) [ ( q, ) T ( )] qhn[ ( q, )] Thus, we obai he -h-order deformaio equaio [ ( ) 1( )] [ 1] LT T hr T (15) R [ T ( )] D [ T ( )] k T ( ) Operaig he Riema-Liouville iegral operaor I o boh side of Eq(15), we obai T h T h T h T k hi T () ( ) 1() ( ) 1() ( ) 1() () (16) From (16)ad iiial codiio T () B, we ge () BK T 1 ( 1) hbk h B( k ) T () (1 h) ( 1) ( 1) Now, if h=-1, we have () K T1 ( 1) B ( k ) T () ( 1) B ( k ) ( 1) T () So, he soluio of (14) is 54

8 Aus J Basic & Appl Sci, 4(1): , 1 i ( K ) i () ( ), i ( i 1) T B B E k E deoes he Miag-Leffler fucio Now, we look for he soluio of he ohomogeeous problem (1-11), which is of he form U( x, ) H( ) Six (17) 1 From (Brow ad Churchill, 199) we have [1 ( 1) ] 1 Si x 1 x (18) Therefore, from (1) we coclude [1 ( 1) ] D H () K H () Si x Si x 1 1 (19) Hece, we ge by ideifyig he coefficies wo side of he Si series (19) [1 ( 1) ] D H() K H() 1,, () ha is a o-homogeeous fracioal Surm-Liouville equaio wih [1 ( 1) ] g () Usig (11) ad ux (,) ie H () Six x (1) 1 So, we ge () 1 Six Cosx h ) H () ha is idepede of To solve Eq (), by HAM, accordig o (), i is aural o choose iiial approximaio H h We choose he liear operaor L[ ( q, )] D [ ( q, )], (i here () (, q) H () q i ) wih he propery Lc (), c is cosa Furhermore, N suggess ha we i defie a oliear operaor i N q D q k q [ (, )] [ (, )] (, ) Accordig o above defiiio, assumpio H () 1, we cosruc he zeroh-order deformaio (1 ql ) [ ( q, ) H ( )] qhn[ ( q, ) M] Where [1 ( 1) ] M [ ( ) 1( )] [ 1] LH H hr H Thus, we obai he -h-order deformaio equaio () R[ H ( )] D[ H ( )] kh ( ) ( M M) Operaig he Riema-Liouville iegral operaor I o boh side of Eq(), we obai 55

9 Aus J Basic & Appl Sci, 4(1): , 1 H () ( hh ) () ( hh ) () ( hh ) () khi H () I ( M M) From (16)ad iiial codiio H () h, we ge () hk M H 1 h ( 1) ( ) hh( K) M H() (1 h) H 1 h ( 1) ( ) Now, if h=-1, we have () K M H 1 ( 1) ( ) h( k ) M 1 H () ( 1) ( ) So, he soluio of (14) is i i1 i () i ( i 1) i1 ( i ) 1 ( K ) ( K ) H h M or i1 M M H( ) he( K ) E,( K ) K K Subsiuig () ad (4) i (17), we ge Six Cosx M M u( x, ) E ( K )( ) Six E ( k ) Six Six, 1 1 k 1 k Coclusios: I his work, he homoopy aalysis mehod has bee applied o sudyhe fracioal Surm-Liouville problems I HAM we ca choose h i such away ha we ge mos accurae soluio ad his is he mosimpora feaure of his echique I he exdample, we have proposed he umerical soluio by usig mahemaica These soluios, as h=-1, are he same resuls give by Adomia decomposiio mehod (Al-Mdallal,) ACKNOWLEDGMENTS The auhors would like o hak he reviewers The work was Suppored by Mazadara ad Islamic Azad Uiversiies REFERENCES Al-Mdallal, QM, A efficie mehod for solvig fracioal surm-liouville problems, Chaos Soluios ad Fracals, i press Brow, JW, RV Churchill, 199 Furier series ad boudary value problems, Mc Grow Hill [5h ediio,] He, JH, 1998 Approximaio soluio of oliear differeial equaio wih covoluio produc olieariies, Compu MehodsAppl Mech Eg, 167(1-): 69-7 He, JH, 1997 AVariaioal ieraio mehod for delay differeial, CommuNoliear Sci Numer Simul, (4): 5-6 (4) 56

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