Spectral method for fractional quadratic Riccati differential equation

Transcription

1 Journal of Aled Matheatcs & Bonforatcs vol2 no ISSN: (rnt) (onlne) Scenress Ltd 212 Sectral ethod for fractonal quadratc Rccat dfferental equaton Rostay 1 K Kar 2 L Gharacheh 3 and M Khasarfard 4 Abstract Fractonal dfferentals rovde ore accurate odels of systes under consderaton In ths aer aroaton technques based on the shfted Legendre sectral ethod s resented to solve fractonal Rccat dfferental equatons The fractonal dervatves are descrbed n the Cauto sense The technque s derved by eandng the requred aroate soluton as the eleents of shfted Legendre olynoals Usng the oeratonal atr of the fractonal dervatve the roble can be reduced to a set of nonlnear algebrac equatons Fro the coutatonal ont of vew the soluton obtaned by ths ethod s n ecellent agreeent wth those obtaned by revous wor n the lterature and also t s effcent to use 1 eartent of Matheatcs Ia Khoen Internatonal Unversty Qazvn Iran e-al: rostay@hotalco 2 Meber of young research club Islac Azad Unversty Kara branch Iran e-al: Kobraar@yahooco 3 Kosar Unversty Qazvn Iran e-al: Lelagharacheh@yahooco 4 eartent of Matheatcs Ia Khoen Internatonal Unversty Qazvn Iran e-al: MKhasarfard@yahooco Artcle Info: Receved : October Revsed : Noveber Publshed onlne : eceber 3 212

2 86 Sectral ethod for fractonal quadratc Rccat dfferental equaton Matheatcs Subect Classfcaton: 65N36 Keywords: Fractonal Rccat dfferental equaton Legendre olynoals Cauto dervatve 1 Introducton Ordnary and artal fractonal dfferental equatons have been the focus of any studes due to ther frequent aearance n varous alcatons n flud echancs vscoelastcty bology hyscs and engneerng [1] Consequently consderable attenton has been gven to the solutons of fractonal dfferental equatons of hyscal nterest Most fractonal dfferental equatons do not have eact solutons so aroaton and nuercal technques [2][3][4][5] ust be used Recently several nuercal ethods to solve the fractonal dfferental equatons have been gven such as varatonal teraton ethod[6] hootoy erturbaton ethod[8] Adoan s decooston ethod [7] hootoy analyss ethod [9] and collocaton ethod[1] We descrbe soe necessary defntons and atheatcal relnares of the fractonal calculus theory requred for our subsequent develoent efnton 1 Cauto s defnton of the fractonal-order dervatve s defned as [1] f 1 ( n ) ( n) f ( t) ( t) ( 1n dt n 1 < n n N where s the order of the dervatve and n s the sallest nteger greater than For the Cauto s dervatve we have: C C s a constant ( 1) ( 1) for for N N and and <

3 Rostay K Kar L Gharacheh and M Khasarfard 87 We use the celng functon to denote the sallest nteger greater than or equal to Also N 12 and N 12 Recall that for N the Cauto dfferental oerator concdes wth the usual dfferental oerator of nteger orderthe an goal n ths artcle s concerned wth the alcaton of Legendre sectral ethod to obtan the nuercal soluton of fractonal Rccat dfferental equaton [11] [12] [13] 2 u( a( b( u g( u 1 1 < (1) wth ntal condtons u ( ) () d 1 1 (2) where the fractonal dfferental oerator s defned as n defnton 1 and where a( b( and g( are gven functons d 1-1 are arbtrary constants and s a araeter descrbng the order of the fractonal dervatve The general resonse eresson contans a araeter descrbng the order of the fractonal dervatve that can be vared to obtan varous resonses In the case of the fractonal equaton reduces to the classcal Rccat dfferental equaton In the resent aer we ntend to etend the alcaton of Legendre olynoals to solve fractonal dfferental equatons Our an a s to generalze Legendre oeratonal atr to fractonal calculus The organzaton of ths aer s as follows In the net secton we descrbe the basc forulaton of shfted Legendre olynoals Secton 3 suarzes the alcaton of Legendre sectral ethod to solve Eqs (1 2) As a result a syste of nonlnear ordnary dfferental equatons s fored and the soluton of the consdered roble s ntroduced In Secton 4 soe coarsons and nuercal results are gven to clarfy the ethod Fgures and Tables are resented n secton 5 And also a concluson s gven n Secton 6

4 88 Sectral ethod for fractonal quadratc Rccat dfferental equaton 2 Shfted Legendre olynoals The well-nown Legendre olynoals are defned on the nterval [ 11] and can be deterned wth the ad of the followng recurrence forulas: ( z) 1 1( z) z 2 1 1( z) z ( z) 1( z) In order to use these olynoals on the nterval[1] we defne the so called shfted Legendre olynoals by ntroducng the change of varable z The shfted Legendre olynoals n are then obtaned as follows: ( 1 1( 2 1 (2 1)(2 1) 1( ( 1( The analytc for of the shfted Legendre olynoal ( of degree gven by ( ) ( )! ( 1) (3) ( )!! ( 2 Note that 1) () ( and (1) 1 The orthogonalty condton s for 1 1 ( ( d for 2 1 A functon y ( square ntegrable n [1] ay be eressed n ters of the shfted Legendre olynoals as y( where the coeffcents c are gven by c (2 1) 1 y( c ( ( d 12 In ractce only the frst ( + 1)-ters shfted Legendre olynoals are

5 Rostay K Kar L Gharacheh and M Khasarfard 89 consdered Then we have y T ( c ( C ( (4) where the shfted Legendre coeffcent vector C and the shfted Legendre vector ( are gven by T T C [ c c ] ( [ ( ( ( ] (5) 1 The dervatve of the vector ( can be eressed by where (1) s the ( 1)( 1) d (1) ( d oeratonal atr of dervatve and for odd gven as and for even gven as

6 9 Sectral ethod for fractonal quadratc Rccat dfferental equaton It s clear that ) ( ) ( (1) d d n n n where N n and the suerscrt n 1 denotes atr owers Then 12 ) ( (1) n n n (6) Theore 1 Let ) ( be the shfted Legendre vector defned n (5) and also suose > then ) ( ) ( ) ( (7) where ( ) s the 1) 1)( ( oeratonal atr of fractonal dervatve of order n Cauto sense and s defned as follows: Where s gven by

7 Rostay K Kar L Gharacheh and M Khasarfard 91 ( l) ( 1) ( )!( l )! ( )!! ( 1)( l)!( l!) ( l 1) l Proof The roof s n [16] ( ) the frst rows are all zero and f Note that n 1 gves the sae result as (6) n N then Theore 3 Alcatons of the oeratonal atr of fractonal dervatve In ths secton we Consder the Eqs (1 2) In order to use Legendre collocaton ethod we frst aroate u ( as T u( c ( C ( (8) where vector C [ c c ] s an unnown vector By usng oeratonal atr of fractonal dervatve we have: C T ( ) ( we now collocate Eq (9) at ( 1 ) onts as: T T 2 a( b( C ( g( ( C ( )) (9) C ( ) a( ) b( ) C ( ) g( )( C ( )) 1 T ( ) T T 2 (1) For sutable collocaton onts we use roots of shfted Legendre ( 1 Eq (1) together wth equatons of the boundary condtons gve ( + 1) equatons whch can be solved for the unnown u

8 92 Sectral ethod for fractonal quadratc Rccat dfferental equaton 4 Nuercal results In ths secton we llustrate effcency and accuracy of the resented ethod by the followng nuercal eales Eale 1 Consder the followng fractonal Rccat equaton: subect to the ntal condton The eact soluton when 1 s d u u dt 2 ( t) 1 < 1 (11) u () (12) and we can observe thatas 2t e 1 u ( t) (13) 2t e 1 t u ( t) 1 The obtaned nuercal results by eans of the roosed ethod are shown n Table 1 and Fgure 1 In Table 1 coarson between the eact soluton the nuercal soluton usng [12] and the aroate soluton usng our roosed ethod for 1 are resented Note that as aroaches 1 the nuercal soluton converges to the analytcal soluton e n the lt the soluton of the fractonal dfferental equatons aroaches to that of the nteger-order dfferental equatons that t s shown n Fgure 1 Eale 2 Consder the followng fractonal Rccat equaton: d u 2u( t) u dt subect to the ntal condton The eact solutonwhen 1s 2 ( t) 1 < 1 (14) u () (15)

9 Rostay K Kar L Gharacheh and M Khasarfard 93 1 (2) 1 u ( t) 1 (2) tanh( (2) log( )) (16) 2 (2) 1 and we can observe that as t u ( t) 1 ( 2) In Table 2 we coare the eact solutonaroate soluton by our ethod and soluton n[12] Also values of u ( for 98 and 98 Fro Fgure 2 we see that as aroaches 1 the nuercal soluton converges to that of nteger-order dfferental equaton 5 Fgures and Tables Fgure 1: Coarson of u( for 1 and wth for Eale 1

10 94 Sectral ethod for fractonal quadratc Rccat dfferental equaton Table 1: Coarson between the nuercal soluton usng [12] and the aroate soluton usng our roosed ethod at 1 for Eale 1 X eact Present ethod ethod n Fgure 2: Coarson of u( for 1 and wth for Eale 2

11 Rostay K Kar L Gharacheh and M Khasarfard 95 Table 2: Coarson between the nuercal soluton usng [12] and the aroate soluton usng our roosed ethod at 1 for Eale 2 X eact Present ethod ethod n [12] a 98 a Conclusons The roertes of the Legendre olynoals are used to reduce the fractonal dffuson equaton to the soluton of syste of nonlnear equatons Fro the solutons obtaned usng the suggested ethod we can conclude that these solutons are n ecellent agreeent wth the already estng ones ([12] [14] [15] [7]) Acnowledgeents The authors are grateful to the referees for ther suggestons whch heled to rove the aer

12 96 Sectral ethod for fractonal quadratc Rccat dfferental equaton References [1] RL Bagley and PJ Torv On the aearance of the fractonal dervatve n the behavor of real aterals J Al Mech (1984) [2] S as Functonal fractonal calculus for syste dentfcaton and controls New Yor Srnger 28 [3] K ethel An algorth for the nuercal soluton of dfferental equatons of fractonal order Electron Trans Nuer Anal (1997) 1-6 [4] JH He Aroate analytcal soluton for seeage flow wth fractonal dervatves n orous eda Cout Meth Al Mech Eng 167(1 2) (1998) [5] K Parand M ehghan AR Rezae and SM Ghader An aroaton algorth for the soluton of the nonlnear Lane-Eden tye equatons arsng n astrohyscs usng Herte functons collocaton ethod Cout Phys Coun [6] M Inc The aroate and eact solutons of the sace- and te-fractonal Burger s equatons wth ntal condtons by varatonal teraton ethod J Math Anal Al (28) [7] S Moan and N Shawagfeh ecooston ethod for solvng fractonal Rccat dfferental equatons Al Math Cout n ress [8] NH Swela MM Khader and RF Al-Bar Nuercal studes for a ultorder fractonal dfferental equaton Phys Lett A (27) [9] NT Shawagfeh Analytcal aroate solutons for nonlnear fractonal dfferental equatons Al Math Cout 131 (22) [1] EA Rawashdeh Nuercal soluton of fractonal ntegro-dfferental equatons by collocaton ethod Al Math Cout (26) 1-6 [11] S Abbasbandy Hootoy erturbaton ethod for quadratc Rccat dfferental equaton and coarson wth Adoan s decooston ethod Al Math Cout 172 (26) [12] Z Odbat and S Moan Modfed hootoy erturbaton ethod:

13 Rostay K Kar L Gharacheh and M Khasarfard 97 Alcaton to quadratc Rccat dfferental equaton of fractonal order Chaos Soltons and Fractals 36 (28) [13] S Abbasbandy Iterated He s hootoy erturbaton ethod for quadratc Rccat dfferental equaton Al Math Cout 175 (26) [14] J Cang Y Tan H Xu S Lao Seres solutons of non-lnear Rccat dfferental equatons wth fractonal order Chaos Soltons and Fractals 4 (29) 1-9 [15] F Geng A odfed varatonal teraton ethod for solvng Rccat dfferental equatons Cout Math Al 6 (21) [16] A Saadatanda M ehghan A new oeratonal atr for solvng fractonal-order dfferental equatons Al Math Cout 59 (21)