THE APPROXIMATE AND EXACT SOLUTIONS OF THE SPACE- AND TIME-FRACTIONAL BURGERS EQUATIONS

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1 IJRRAS () June Kurulay Soluions of he Space & Tie-Fracional Burgers Equaions THE APPROXIMATE AND EXACT SOLUTIONS OF THE SPACE- AND TIME-FRACTIONAL BURGERS EQUATIONS Muhae Kurulay Yildiz Technical Uniersiy Faculy of Ar and Sciences Deparen of Maheaics 4-Daupasa-İsanbul Turey Eail: ABSTRACT In he paper we exend he differenial ransfor ehod o sole nonlinear fracional parial differenial equaions. The ie- and space-fracional Burgers equaions wih iniial condiions are chosen o illusrae our ehod. As a resul we successfully obain soe aailable approxiae soluions of he. The resuls reeal ha he proposed ehod is ery effecie and siple for obaining approxiae soluions of nonlinear fracional parial differenial equaions. The fracional deriaies are considered in he Capuo sense. Keywords: Tie- and space-fracional Burgers equaions; Fracional deriaie; Differenial ransfor ehod PACS codes:..jr; 5.45.Df MSC: 65M; 65M5; 65M. INTRODUCTION In he las pas decades nonlinear fracional parial differenial equaions are widely used o describe any iporan phenoena and dynaic processes in physics such as engineering elecroagneics acousics iscoelasiciy elecrocheisryand aerial science [ 4]. For beer undersanding he phenoena ha a gien nonlinear fracional parial differenial equaion describes he soluions of differenial equaions of fracional order is uch inoled. In general here exiss no ehod ha yields an exac soluion for nonlinear fracional parial differenial equaions. The fracional differenial equaions (FDE) appear ore and ore frequenly in differen research areas and engineering applicaions. Mos recenly Moani [5] has presened nonperurbaie analyical soluions of he spaceand ie-fracional Burgers equaions by Adoian decoposiion ehod. Inc [6] used ariaional ieraion ehod for soling space- and ie-fracional Burgers equaions. Wang [78] exend he applicaion of he hooopy perurbaion and Adoian decoposiion ehods o consruc approxiae soluions for he nonlinear fracional KdV-Burgers equaion. The space-fracional Burgers equaion describes he physical processes of unidirecional propagaion of wealy nonlinear acousic waes hrough a gas-filled pipe. The fracional deriaie resuls fro he eory effec of he wall fricion hrough he boundary layer. The sae for can be found in oher syses such as shallow-waer waes and waes in bubbly liquids. For ore deails on he applicaions associaed wih of he spacefracional Burgers equaion [9]. We consider non-perurbaion analyical soluions of he generalized Burgers equaion wih ie- and spacefracional deriaies of he for[5]: u u u u u x x x (.) where and are paraeers and and are paraeers describing he order of he fracional ie- and space-deriaies respeciely. The funcion u( x ) is assued o be a causal funcion of ie and space i.e. anishing for and x. The fracional deriaies are considered in he Capuo sense. The general response expression conains a paraeer describing he order of he fracional deriaie ha can be aried o obain arious responses. We refer o Eq. (.) as o he ie-fracional Burgers and o he space-fracional Burgers equaion in he cases { } and { } respeciely. 57

2 IJRRAS () June Kurulay Soluions of he Space & Tie-Fracional Burgers Equaions The DTM was firs applied in he engineering doain in []. Hashi [] deonsraed he applicaion of hooopy-perurbaion ehod for soling fkdv. Kurulay [] applied he applicaion of DTM ehod for soling fkdv. The paper is organized as follows. A brief reiew of he fracional calculus heory is gien. We use he Differenial ransfor ehod o consruc our exac soluions of he space- and ie-fracional Burgers equaions. We presen wo exaples o show he efficiency and sipliciy of he proposed ehod. Conclusions will be presened in final.. BASIC DEFINITIONS In his secion le us recall essenials of fracional calculus firs. The fracional calculus is a nae for he heory of inegrals and deriaies of arbirary order which unifies and generalizes he noions of ineger-order differeniaion and n-fold inegraion. We hae well nown definiions of a fracional deriaie of order such as Rieann Liouille Grunwald Leniow Capuo and Generalized Funcions Approach [ 4]. The os coonly used definiions are he Rieann Liouille and Capuo. For he purpose of his paper he Capuo s definiion of fracional differeniaion will be used aing he adanage of Capuo s approach ha he iniial condiions for fracional differenial equaions wih Capuo s deriaies ae on he radiional for as for ineger-order differenial equaions. We gie soe basic definiions and properies of he fracional calculus heory which were used hrough paper. Definiion.. A real funcion f ( x) x is said o be in he space C R if here exiss a real nuber ( p ) such ha f ( x) x p f( x) where f ( x ) C [ ) and i said o be in he space C iff f C N. Definiion.. The Rieann Liouille fracional inegral operaor of order of a funcion f C is defined as x J f ( x) x f ( ) d ( ) J f x ( ) f ( x). I has he following properies: For f C and :. J J f ( x) J f ( x). J J f ( x) J J f ( x). J x x. ( ) The Rieann Liouille fracional deriaie is osly used by aheaicians bu his approach is no suiable for he physical probles of he real world since i requires he definiion of fracional order iniial condiions which hae no physically eaningful explanaion ye. Capuo inroduced an alernaie definiion which has he adanage of defining ineger order iniial condiions for fracional order differenial equaions. Definiion.. The fracional deriaie of f( x ) in he capuo sense is defined as D f x J D f x x f d x ( ) * ( ) ( ) ( ) ( ) ( ) for N x f C. Lea.. If N and f C hen 58

3 IJRRAS () June Kurulay Soluions of he Space & Tie-Fracional Burgers Equaions D J f ( x) f ( x) * x J D* f ( x) f ( x) f ( ) x>.! The Capuo fracional deriaie is considered here because i allows radiional iniial and boundary condiions o be included in he forulaion of he proble. Definiion.4. For o be he salles ineger ha exceeds he Capuo ie-fracional deriaie operaor of order is defined as ux ( ) u( x ) ( ) D* u( x ) u( x ) for N ( ) d for and he space-fracional deriaie operaor of order is defined as u( ) u( x ) ( ) D* xu( x ) x u( x ) for N. x x ( x ) d for. DIFFERENTIAL TRANSFORM METHOD The DTM is applied o he soluion of elecric circui probles. The DTM is a nuerical ehod based on he Taylor series expansion which consrucs an analyical soluion in he for of a polynoial. The radiional high order Taylor series ehod requires sybolic copuaion. Howeer he DTM obains a polynoial series soluion by eans of an ieraie procedure. The ehod is well addressed in []. Consider a funcion of wo ariables u( x y ) and suppose ha i can be represened as a produc of wo singleariable funcions i.e. u( x y) f ( x) g( y). Based on he properies of generalized wo-diensional differenial ransfor [45] he funcion u( x y ) can be represened as h h u( x y) F ( )( x x ) G ( h)( y y ) U ( h)( x x ) ( y y ) h h where U ( h) F ( ) G ( h) is called he specru of u( x y ). The generalized wodiensional differenial ransfor of he funcion u( x y ) is gien by h U ( h) ( D* x ) ( D * y ) u( x y) (.) h x y where D* x D * x D * x D * x -ies. In case of and he generalized wo-diensional differenial ransfor (.) reduces o he classical wo-diensional differenial ransfor.[6]. The operaors in wo-diensional differenial ransforaion Mehod [6]: Le U ( h) V ( h) and W ( h) be he differenial ransforaions of he funcions u( x y ) ( x y ) and w( x y ) : (.) 59

4 IJRRAS () June Kurulay Soluions of he Space & Tie-Fracional Burgers Equaions (a) If u( x y) ( x y) w( x y) hen U ( h) V ( h) W ( h) (b) If u( x y) a( x y) a R hen U ( h) av ( h) (c) If u( x y) ( x y) w( x y) hen U ( h) V ( r h s) W ( r s) h r s n (d) If u( x y) ( x x) ( y y) hen U ( h) ( n) ( h ) (e) If u( x y) ( x y) w( x y) q( x y) hen (f) If r h U ( h) V ( r h s p) W ( s) Q ( r p) r u( x y) D ( x y) hen x ( ) U ( h) V ( h) (g) If u( x y) f ( x) g( y) and he funcion f ( x) x h( x) where hx ( ) has he generalized Taylor series expansion h( x) an( x x) and [6] n (i). and arbirary or arbirary and a for n... where. (ii). Then he generalized differenial ransfor (.) becoes h U ( h) D* x ( D * y ) u( x y) h x y (h) If (i) If (j) If x n u( x y) D ( x y) and ( x y) f ( x) g( y) * x hen U ( h) V ( / h). u( x y ) D ( x y ) hen ( ) U ( h ) V ( ). h ( ) u( x. y) a( x y) x y x h i j hen. U( h) ( i ) A( i j) U( i h j) The proofs of he soe properies can be found in [6]. 4. APPLICATIONS In order o illusrae he adanages and he accuracy of he DTM for soling nonlinear fracional Burgers equaion we hae applied he ehod o wo differen exaples. In he firs exaple we consider a nonlinear ie-fracional Burgers equaion while in he second exaple we consider a nonlinear space-fracional equaion. All he resuls are calculaed by using he sybolic copuaion sofware Maple. 4.. Approxiae soluion of ie-fracional Burgers equaion we consider he following ie-fracional Burgers equaion [5]: u u u u. (4.) x x 6

5 IJRRAS () June Kurulay Soluions of he Space & Tie-Fracional Burgers Equaions We consider Eq. (4.) wih and he following iniial condiion [7]: u ( )exp( ) ( x ) g ( x ) exp( ) (4.) where ( x ) and he paraeers and are arbirary consan. Taing he differenial ransfor of Eq. (4.) by using he relaed propery we hae h ( ( h ) ) U ( h ) ( i ) U ( i j) U ( i h j) ( )( ) U ( h). ( h ) i j We sar wih an iniial condiions ha was gien by Eq. (4.). The soluion for he fracional Burgers Eq. (4.) in a series for is gien by ( )exp( ) u( x ) [ g gg] exp( ) ( ) ( ) Thus we hae he soluion of (4.) in a series for for (4) [ g( g) g g 4 gg gg g ]. ( - )exp( ) exp( ) [ exp( )][exp( ) -] u( x ) exp( ) [ exp( )] [ exp( )] [ 4 exp( )][ 4exp( ) exp( )] [ exp( )] 4 ( - )exp[ ( x )] u( x ) exp[ ( x )] which are exacly sae as obained by Adoian decoposiion [5] and ariaional ieraion ehods [6]. (a) (b) Fig.. The surface shows he soluion u( x ) for Eq. (4.): (a) exac soluion; (b)approxiae soluion when and. 4.. Approxiae soluion of space-fracional Burgers equaion In his secion we consider he following space-fracional Burgers equaion [5]: 6

6 IJRRAS () June Kurulay Soluions of he Space & Tie-Fracional Burgers Equaions u u u u u x x x x (4.) subjec o he iniial condiions u( ) ux( ). (4.4) Taing he differenial ransfor of Eq. (4.) we hae ( ( ) ) U ( h) ( )( ) U ( h) ( ) h ( i ) U ( i j) U ( i h j) ( h ) U ( h ). (4.5) i j For purposes of illusraion of he differenial ransfor ehod for soling Burgers equaion wih space-fracional deriaie consider (4.) wih and subjec o he iniial condiions x u( ) ux ( ) u( x) x anh. (4.6) We subsiue he iniial condiions (4.6) ino (4.5) for he special case we obain x x x u( x ) x x which is he soluion of (4.) in series for. The exac soluion [5] for he special case is gien by x x u( x ) anh. (a) (b) Fig.. The surface shows he soluion u( x ) for Eq. (4.): (a) exac soluion; (b)approxiae soluion. The paraeers hae he following alues and. 6

7 IJRRAS () June Kurulay Soluions of he Space & Tie-Fracional Burgers Equaions 5. CONCLUSIONS The fundaenal goal of his wor has been o obain analyical soluions of he space- and ie fracional Burgers equaions. This goal has been achieed by using differenial ransfor ehod and an approxiaion series soluion can be obained o any desired nuber of ers. We exend he differenial ransfor ehod o sole nonlinear fracional parial differenial equaions. The ieand space-fracional Burgers equaion wih iniial condiions is chosen o illusrae he proposed ehod. As resuls based on sybolic copuaion syse Maple soe approxiae soluions of fracional Burgers equaion wih high accuracy are obained. The obained resuls deonsrae he reliabiliy of he algorih and is wider applicabiliy o nonlinear fracional parial differenial equaions. We hope oher radiional analyic ehods for nonlinear differenial equaions of ineger order can be exended o nonlinear fracional calculus equaions and his will be inesigaed in following wor. REFERENCES [] I. Podlubny Fracional differenial equaions. San Diego: Acadeic Press; 999. [] Wes BJ Bolognab M Grigolini P. Physics of fracal operaors. New Yor: Springer;. [] Sao SG Kilbas AA Mariche OI. Fracional inegrals and deriaies: heory and applicaions. Yerdon: Gordon and Breach; 99. [4] M. Capuo Linear odels of dissipaion whose Q is alos frequency independen. Par II J. Roy. Ausral. Soc. (967) [5] S. Moani Non-perurbaie analyical soluions of he space- and ie-fracional Burgers equaions Chaos Solions Fracals 8 (6) 9 97 [6] M Inc The approxiae and exac soluions of he space- and ie-fracional Burgers equaions wih iniial condiions by ariaional ieraion ehod J. Mah. Anal. Appl. 45 (8) [7] Qi Wang Hooopy perurbaion ehod for fracional KdV-Burgers equaion Chaos Solions & Fracals Vol. 5 No. 5 (8) pp [8] Qi Wang Nuerical soluions for fracional KdV-Burgers equaion by Adoian decoposiion ehod. Applied aheaics and copuaion. Vol.8 No. (6) pp [9] Sugioo N. Burgers equaion wih a fracional deriaie; Herediary effecs on non-linear acousic waes. J Fluid Mech 5 (99) 6 5. [] J.K.Zhou Differenial ransforaion and is applicaions for elecrical circuis Huazhong uniersiy Pres WuhanChina 986. [] O.Abdulaziz I. Hashi E.S.İsail Approxiae analyical soluion o fracional odified KdV equaions Maheaical and Co. Modelling. 49 (9) [] M. Kurulay M. Bayra Approxiae analyical soluion for he fracional odified KdV by differenial ransfor ehod Coun. Nonlinear Sci. Nuer. Siul (in press) Jan. (9). [] S. Moani Z. Odiba V. Erür Generalized differenial ransfor ehod for soling a space and ie fracional diffusion-wae equaion Phys. Le. A. Volue 7 Issues Ocober 7 Pages [4] N.Bildi A.Konuralp F. Be S. Kucuarslan Soluion of differenial ype of he parial differenial equaion by differenial ransfor ehod and Adoian s decoposiion ehod Appl. Mah.Copu. 7 (6) [5] I.H. Abdel-Hali Hassan Coparison differenial ransforaion echnique wih Adoian decoposiion ehod for linear and nonlinear iniial alue probles Chaos Solions & Fracals Volue 6 Issue April 8 Pages [6] S. Moani Z.Odiba A noel ehod for nonlinear fracional parial differenial equaions: Cobinaion of DTM and generalized Taylor s Forula Forula Journal of Copuaional and Applied Mah.. (8) [7] Ali AHA Gardner GA Gardner LRT. A collocaion soluion for Burgers equaion using B-spline finie eleens. Copu Mah. Appl. Mech. Eng. 997;:5 7. 6