LIE SYMMETRY ANALYSIS AND CONSERVATION LAWS FOR TIME FRACTIONAL COUPLED WHITHAM-BROER- KAUP EQUATIONS

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1 U.P.B. Sc. Bull., Seres A, Vol. 80, Iss. 3, 08 ISSN LIE SYMMETRY ANALYSIS AND CONSERVATION LAWS FOR TIME FRACTIONAL COUPLED WHITHAM-BROER- KAUP EQUATIONS Had Rooha GHEHSAREH, Ahmad MAJLESI, Al ZAGHIAN 3 I he curre work, me fracoal coupled Whham-Broer-Kaup equaos whch descrbes he aomalous bdrecoal propagao of log waves shallow waer s vesgaed. A Le symmery aalyss s formulaed ad used o he goverg model. The symmery reducos of he model are cosruced ad he sysem of olear me fracoal paral dffereal equaos s smlary reduced o a sysem of olear frcaoal ordary dffereal equaos Erdely-Kober dervave sese. Moreover, he resula symmery geeraors are used o calculae coserved vecors for he me fracoal problem. Two dffere kds of coservao laws of he problem have bee cosruced. Keywords: Le symmery aalyss; Tme fracoal coupled Whham-Broer- Kaup equaos; Erdely-Kober operaors; Coservao laws. Iroduco I rece decades heory of fracoal calculus [, ] has gaed oable aeo of may researchers scece ad egeerg due o s hgh ably for descrbg varous complcaed aural pheomea. May of aomalous pheomea ad comple processes aural scece have bee successfully descrbed by usg he heory of fracoal calculus ad mahemacally modeled as fracoal dffereal or egral equaos[3, 4, 5, 6]. The ma advaage of fracoal modelg s ha hey are ecelle ools o appropraely characerze heredary ad here memory properes of he aomalous pheomea. I rece years several sem-aalycal mehods ad umercal echques have bee developed ad employed for solvg ad vesgag varous praccal problems whch have bee mahemacally modeled as fracoal equaos [7, 8, 9, 0,,, 3, 4, 5, 6]. I hs sudy he followg mpora mahemacal model, he me fracoal Whham-Broer-Kaup equaos, whch descrbes he aomalous bdrecoal propagao of log waves shallow waer wll be vesgaed[7]. Prof., Dep. of Mahemacs, Malek Ashar Uversy Of Techology, Shah Shahr, , Ira, e-mal: hadrooha6@gmal.com PhD sude, Dep.of Mahemacs, Malek Ashar Uversy Of Techology, Shah Shahr, , Ira, e-mal: ahmad.majless@gmal.com 3 Prof., Dep.of Mahemacs, Malek Ashar Uversy Of Techology, Shah Shahr, , Ira, e-mal: al_zagha338@yahoo.com

2 54 H. Rooha Ghehsareh, A. Majles, A. Zhaga u v u u q u = 0, 3 v ( uv ) u v + + p q = 0, 3 () where u (, ) ad v (, ) deoe he flud velocy alog he horzoal dreco ad vercal dsplaceme of he flud from s equlbrum poso, respecvely. The cosas p ad q ( ( pq, ) (0, 0) ) are dffuso coeffces ad (.) deoes he Rema-Louvlle paral dervave operaor of order (0 < ) wh respec o me compoe,, whch s defed as follows : ( ) ( u (, )) = m ( ) u (, ) d, m < < m, m N, 0 ( u (, )), = N, where (.) s he gamma fuco. For = he aomalous bdrecoal propagao model () s reduced o he classcal Whham-Broer-Kaup problem. Several umercal ad sem-aalycal mehods have bee formulaed ad employed o vesgae he classcal Whham-Broer-Kaup problem[8, 9, 0,, ]. The classcal Whham-Broer-Kaup (WBK) equao has bee wdely used o sudy of solary wave heory shallow waer. However, he classcal Whham-Broer-Kaup (WBK) equao s o adequae o descrbe complcaed mechasm of propagao of shallow waer waves porous medum, such as suam wave propagao. I hese suaos he emporal dervave he WBK equao ad correspods o he varao he flu,, should be replaced wh fracoal me dervave,, whch properly descrbes varao he flu hrough he fracal boudary, where deoes he fracal dmesos of he porous medum [3]. Saha Ray [7] has dscussed he me-fracoal WBK equaos () ad appromaed he ravelg wave soluos of he model her geeralzed Taylor epaso forms. I [4] a sem-aalycal echque, called he resdual power seres mehod s formulaed ad used o appromae ravelg soluos of he model (). By choosg p = ad q =0 he () he so-called me fracoal modfed Boussesq sysem s cocluded. Whle f p =0 ad

3 Le symmery aalyss ad coservao laws [ ] coupled Whham-Broer-Kaup equaos 55 0 q aoher specal case of he me fracoal coupled Whham-Broer-Kaup equaos, called me fracoal coupled appromae log wave equaos s obaed. I [5] he me-space fracoal Whham-Broer-Kaup equaos has bee vesgaed. Recely, Amjad Al e.al employed a umercal mehod based o he Adoma decomposo mehod coupled wh he Laplace rasforms o calculae he appromae ravelg wave soluos of he me-space fracoal coupled Whham-Broer-Kaup equaos[6]. I rece years, Le symmery aalyss has bee developed ad wdely used o deal wh several ypes of complcaed olear dffereal problems[7, 8, 9, 30]. Buckwar ad Luchko [3] compued vara soluos of fracoal dffereal equaos by employg he scalg rasformaos. Moreover, a symmery group of scalg rasformaos for me-space fracoal paral dffereal equao s eraced [3]. Gazzov e al. roduced a prologao formula for he Rema-Louvlle fracoal dervave operaor[33]. I [34] hey ulzed he proposed Le po symmery mehod for solvg he olear me fracoal dffuso problem. Recely, Hashem e al. formulaed ad used he Le symmery approach for vesgag varous ypes of he fracoal dffereal equaos[35,36,37,38,39,40,4]. Sgla ad Gupa eraced a Le po symmery aalyss for sysems of coupled me fracoal paral dffereal equaos [4,43]. Moreover, hey developed he Le symmery approach for space-me fracoal sysems of paral dffereal equaos [44, 45]. I hs sudy, he epaded Le symmery approach proposed by Sgla ad Gupa s ulzed for symmery aalyss ad smlary reducos of he me fracoal Whham-Broer-Kaup equaos (). The dea of coservao law plays a mpora role o aalyze he fudameal properes of he physcal models[46]. Relao bewee symmeres of dffereal equaos ad coservao laws s eplaed by he Noeher s heorem [47]. The Noeher s heorem s vald for dffereal equaos havg Lagragas. The classcal Noeher s heorem have bee geeralzed ad employed o fd coservao laws for several fracoal dffereal equaos havg fracoal Lagragas [48,49,50,5]. Ibragmov [5] eraced a ew geeralzed coservao heorem based o he adjo equaos for he olear dffereal equaos o havg Lagragas. Lukashchuk has used he ew coservao heorem o fd coservao laws for me fracoal subdffuso ad dffuso-wave equaos[53]. Gazzov e al. foud coservao laws for he me-fracoal Kompaees equaos based o he geeralzao of fracoal Noeher s operaor [54]. Very recely, Sgla ad Gupa [55,56] have eraced he fracoal Noeher s operaors o calculae coserved vecors of he me ad space-me fracoal olear sysems of paral dffereal equaos, respecvely. Majles e.al. performed a Le symmery aalyss o a coupled sysem of me fracoal Jaule-Modek equaos ad cosruced he

4 56 H. Rooha Ghehsareh, A. Majles, A. Zhaga coservao laws of he problem [57]. I he curre work he geeralzao of fracoal Noeher s operaor proposed by Sgla ad Gupa [55] s employed for calculag coserved vecors for he goverg sysem of olear me fracoal paral dffereal equao ().. Le symmery aalyss ad smlary reducos of he me fracoal Whham-Broer-Kaup sysem I hs seco a vara aalyss for he me fracoal Whham-Broer- Kaup sysem () would be preseed. Moreover, he symmery reducos of he goverg me fracoal olear sysem based o he symmery groups are vesgaed. For hs purpose, frsly he ma deals of Le symmery aalyss for sysems of me fracoal PDEs are brefly descrbed.. Descrpo of he Le symmery aalyss for sysems of me fracoal PDEs Here we cosder a coupled sysem of wo me fracoal olear PDEs as follows[4]: u F (,, u, v, u, v,) = 0, () v G (,, u, v, u, v,) = 0, where { u (, ), v (, )} ad {, } are depede ad depede varables respecvely. >0 s a real umber ad (.) deoes he Rema-Louvlle paral dervave operaor ad subscrps deoe eger paral dervaves. Accordg o he Le symmery aalyss, assume he me fracoal sysem () s vara uder he followg oe parameer Le group of rasformaos : = + (,, u, v ) + O( ), u v O = + (,,, ) + ( ), u u u v O = + (,,, ) + ( ), v v u v O = + (,,, ) + ( ), u u v v, = + + O ( ),, = + + O ( ), (3)

5 Le symmery aalyss ad coservao laws [ ] coupled Whham-Broer-Kaup equaos 57 j j u u j, = + + O( ), j j j =,,3, j j v v j, = + + O( ), j j j =,,3,,,, where s a parameer, (,,, ) s he se of fesmals ad, ad j, j,,,( j =,,) are eeded fesmals of order ad j respecvely. The fesmal geeraor of he me fracoal sysem () s a vecor feld as follows: V = (,, u, v ) + (,, u, v ) + (,, u, v ) + (,, u, v ). (4) u v The vecor feld (4) geeraes a symmery of () provded ha adms he followg fesmal varace crera: Pr V ( ) = 0, =0, =0 Pr V ( ) = 0, (5) =0, =0 where Pr V s he geeralzed fracoal prologao operaor[4,43] :, m,,, m,,,, Pr V = V, ( ) ( ) u u u v v (6) v m where m, N deoe orders of paral dffereal equaos (). I he above j, j, relao he eger order eeded fesmals ad are defed as follows: j j j, j, u u = D ( D) ( D ) ( ), j j j j j, j, v v = D ( D) ( D ) ( ), j =,,, j j where D deoes he oal dervave operaor wh respec o D = + u + u + + v + v +, u u v v, Moreover he -order eeded fesmal s defed as follows: = D + D ( u ) D ( u ) + D ( D ( ) u ) D ( u ) + D ( u ),, + + where D s he oal fracoal dervave wh respec o. Usg he geeralzed Lebz rule [] ad geeralzed cha rule [5] ad afer some, smplfcaos he -order eeded fesmal s gve he followg form (see for more deals [4, 43]) :

6 58 H. Rooha Ghehsareh, A. Majles, A. Zhaga where, u u v v = + ( u D ( )) u + ( v v ) u + v + [ D ( )] D ( u) D ( v ) + = + = D ( ) D ( u ) + +, = (7) D deoes he oal dervave operaor wh respec o ad j r j r s j + r r s s ( u ) = ( ) u, j j r = j = r = s =0 j s r! ( + ) u j r j r s j + r r s s ( v ) = ( ) v. j j r = j = r = s =0 j s r! ( + ) v, Smlary, he -order eeded fesmal ca be smplfed as follows : where, v v u u = + ( v D ( )) v + ( u u ) v + u + [ D ( )] D ( v ) D ( u ) + = + = D ( ) D ( v ) + +, = (8) j r j r s j + r r s s ( u ) = ( ) u, j j r = j = r = s =0 j s r! ( + ) u j r j r s j + r r s s ( v ) = ( ) v, j j r = j = r = s =0 j s r! ( + ) v From he above relaos ca be easly cocluded ha wheever fesmals ad are lear wh respec o each depede varables u ad v, he epressos,, ad vash decally.

7 Le symmery aalyss ad coservao laws [ ] coupled Whham-Broer-Kaup equaos 59. The symmery reducos of he me fracoal Whham-Broer-Kaup sysem Here, he proposed Le symmery aalyss s formulaed ad employed o reduce he problem (). For hs purpose, by mposg he prologaos (6),,,3, Pr V ( ) ad Pr V ( ) o problem () uder group of rasformaos (3), he followg varace creros are cocluded:,,,, [ + + u + u + q ] =0, =0 = 0,, 3,,,, [ p u v v u q ] =0, =0 = (9) By subsug he eger ad order eeded fesmals o (9) ad equag he coeffces of varous powers ad paral dervaves of depede varables u ad v o zero, a se of fracoal ad classcal paral dffereal equaos wh respec o he varables,, ad s obaed. By solvg he resula sysem of dffereal equaos symbolcally he followg values for fesmals fucos are compued: = c + c, = c + c, = c u, = c v, (0) 3 where c, c ad c 3 are free cosa parameers. Noce ha he lower lm of he Rema-Louvlle paral dervave operaor s fed, so o esure ha s vara uder group of rasformaos (3), he followg al codo should be sasfed. (,, u, v ) = 0. () =0 So c =0 3 ad he followg ses of he fesmal geeraors are obaed: V = + u v, V =. u v Clearly, he above vecor felds form a closed Le algebra: [ V, V ] = [ V, V ] = 0, [ V, V ] = V = [ V, V ]. For he fesmal geeraor V, he followg characersc equaos s cocluded : d d du dv = = =. u v By solvg he above characersc equaos he followg smlary varable ad vara soluos are obaed. =, (, ) = ( ), (, ) = ( ), z u F z v G z (3) where z ad F ( z ), G ( z ) are ew depede varable ad depede varables respecvely. Now, he reduco form of he me fracoal sysem () wh respec o he above preseed symmery geeraors s gve by he followg heorem. ()

8 60 H. Rooha Ghehsareh, A. Majles, A. Zhaga Theorem.Correspodg o he fesmal geeraor V, he smlary varable z = wh he smlary rasformaos u (, ) = F ( z ) v (, ) = G ( z ), reduced he me fracoal sysem of paral dffereal equaos () o he followg sysem of fracoal ordary dffereal of equaos: 3, ' ' ' ( P F )( z ) = G ( z ) + F ( z ) F ( z ) + qf ( z ), 3, ' ' ' ad ( P G )( z ) = pf ( z ) + ( F ( z ) G ( z )) qg ( z ), (4) where he lef-had sde Erdely-Kober fracoal dffereal operaor, m, +, m + j =0 dz, ( P ) defed as[58] : d ( P H )( z ) = ( j z )( K H )( z ), z > 0, > 0, (5) where [ ] + f N, m = f N, ad ( + ), ( ) H ( z ) d f > 0 ( K H )( z ) = ( ) H( z) f = 0 Proof: Accordg o he smlary soluos (3), for < < ( N ) he -order Rema-Louvlle me fracoal dervave for u (, ) s gve as follows: u = ( s) s F ( s ) ds, ( ) (6) 0, s Now by usg he chage of varable, = s, he relao (6) s rasformed o followg form : 3 ( 3 ) u + = [ ( ) F ( z ) d], ( ) (7) by comparg wh he Erdely-Kober fracoal dffereal operaor, he above relao ca be preseed as follows:

9 Le symmery aalyss ad coservao laws [ ] coupled Whham-Broer-Kaup equaos 6 3 u, = [ ( K F )( z )]. So we have, 3 u, = ( [ ( K F )( z )]) 3 3 3, d, z K F z + K F z dz 3 3 z d, + K F z dz 3 3 d, z K F z dz = [( ) ( )( ) ( )( ) ] = [ ( )( )( )] = [ ( )( )( )]. Followg he above process for ( ) mes, we easly oba: u = [ ( K F )( z )] 3, 3 3 d, + j =0 dz = ( j z )( K F )( z ) 3 3, = ( P F )( z ). (8) (9) (0) I smlar maer, he -order Rema-Louvlle me fracoal dervave for v (, ) wh respec o he smlary rasforms (3) ca be epressed as follows: v, = ( P G )( z ). Moreover for = N we have: u u = = ( F ( z )) = [ ( F ( z ))] d = [ ( z ) F ( z )] dz () 3 3 d = ( + j z ) F ( z ) dz j =0

10 6 H. Rooha Ghehsareh, A. Majles, A. Zhaga 3 3, = ( P F )( z ), ad smlarly, for = N we have: v v, = = ( P G )( z ). Now by subsug he proposed smlary rasformaos (3) he ma problem () ad from relaos (0) ad () he reduced sysem of fracoal ordary equaos (4) s cocluded. Moreover, for he fesmal geeraor V, by solvg he assocaed characersc equaos he vara soluos of he problem () are obaed as u (, ) = f ( ) ad v (, ) = g ( ). Subsug he above obaed group of vara soluos he goverg problem (), he followg reduced sysem s cocluded: f () =0 g () =0 Iegrag boh sdes of he above sysem, he followg vara soluos are arrved: f ( ) =, g ( ) =, () ( ) ( ) where ad are arbrary cosas. 3. Coservao laws of he me fracoal Whham-Broer-Kaup sysem I hs seco a ew approach whch has bee developed ad mplemeed o he me-fracoal sysem of PDEs [55] s employed o cosruc coservao laws for he me fracoal Whham-Broer-Kaup sysem (). Accordg o he approach proposed by Sgla ad Gupa[55], he formal Lagraga for he me fracoal PDEs sysem () s cosdered as follows: u L = f (, )[ + v (, ) + u (, ) u (, ) + qu (, )] (3) v + g (, )[ + pu (, ) + u (, ) v (, ) + u (, ) v (, ) qv (, )], where f (, ) ad g (, ) are ewly roduced depede varables. The adjo equaos for he formal Lagraga operaor (3) are calculaed as follows:

11 Le symmery aalyss ad coservao laws [ ] coupled Whham-Broer-Kaup equaos 63 L u ad * = ( D ) f u (, ) f (, ) v (, ) g (, ) + qf (, ) pg (, ) = 0, L v * = ( D ) g f (, ) g (, ) u (, ) qg (, ) = 0, where he Euler-Lagrage operaor,, s defed as : * k = + ( D ) + ( ) Dd D, ( ) d D d D k ( ) where ( ) * D k = d, d,, d k (4) (5) deoes he adjo operaor for he Rema-Louvlle fracoal operaor ( D ) ad equals o he sadard rgh-had sded Capuo fracoal dffereal operaor [55]. Accordg o [55], he me fracoal sysem of equaos(.5) s sad o be olearly self-adjo wheever by roducg ew varables f = (,, u, v ) ad g = (,, u, v ), where a leas oe of hem s o-zero ad subsug hem ad her relaed paral dervaves wh respec o o adjo equaos (4) ad (5), he resula equaos are sasfed for all soluos of he goverg problem (). I meas ha for olear selfadjoess of () he followg codos should be held : L u v = ( + v + uu + qu ) + ( + pu + uv + vu qv ), u L u v = 3( + v + uu + qu ) + 4( + pu + uv + vu qv ), v where, ( =,,3,4) are ukow coeffces. The above relaos are gve as follows: * ( D ) u(, )( + u + v ) v (, )( + u + v ) u v u v + q( + u + v + u + u v + u + v + v ), u, v u, u u, v u v, v v, p[ + 3 u v + 3 v v + 3 u v + 3 u u + 3 u v,, u, u, v v, v u, v, v u, u, u, v, + 6 u v + 3 u v + 3 u + 3 v + 3 u + 3 v, u, v u, v,,, u,, v, u,, v, + u + v + 3 u + 3 v + u + v ] 3 3 u v, u, u, v, v u, u, u v, v, v u v = ( + v + uu + qu ) + ( + pu + uv + vu qv ), (6) * ad ( D ) ( + u + v ) ( + u + v ) u q( + u u v u v, u

12 64 H. Rooha Ghehsareh, A. Majles, A. Zhaga + v + u + u v + u + v + v ), v u, u u, v u v, v v, u v = 3( + v + uu + qu ) + 4( + pu + uv + vu qv ). (7) Now by balacg he coeffces of he dffere powers of depede varables u ad v ad her relaed paral dervaves boh sdes of (6) ad (7) a sysems of paral dffereal equaos should be cocluded. Solvg he resula sysem of dffereal equaos aalycally, he followg resuls s obaed: = 0, =,,3,4, (,, u, v ) = f (, ) = A, (,, u, v ) = g (, ) = B, (8) where A ad B are free cosas. Ths cofrms he olear self-adjoess of he problem (). Correspodg o each vecor feld V = + + u + v here es Le characersc fucos whch are defed as follows: u v W = u u, W = v v. So Correspodg o he vecor felds () for he me fracoal Whham-Broer- Kaup problem he followg Le characersc fucos are cocluded, respecvely: u v W = u u u, W = v v v, (9) ad u v W = u, W = v. (30) Now by usg he above cosruced Le characersc fucos, coserved vecors correspodg o he goverg problem should be calculaed. A vecor = ( C C, C ) s sad o be a coserved vecor for he problem () f he followg coservao equao s held for, D ( C ) + D ( C ) = 0. (3) () Clearly he goverg sysem of fracoal paral dffereal equaos (), all of he paral dervaves wh respec o he depede varable are of eger-orders, so he compoes of he coserved vecors are obaed based o he followg classcal forms [5]: u u u v v C = ( W L D ( W ) L D ( W ) L ) ( W L D ( W ) L ). (3) u u u v v Moreover, a eeded formula o compue he -compoe of coserved vecor for he me fracoal sysem of equaos are roduced as follows [55] : k k u k L k v k L C = ( ) [ D ( W ) D ( ) + D ( W ) D ( )] ( D u) ( D v ) k =0

13 Le symmery aalyss ad coservao laws [ ] coupled Whham-Broer-Kaup equaos 65 u L v L ( ) [ J ( W, ) + J ( W, )], (33) ( D u ) ( D v ) where = [ ] + ad J ( f, g) deoes a egral rasform whch defed as follows: p f (, s ) g (, r) J ( f, g ) = drds. 0 ( ) ( r s ) + Leg A = ad B = ad by subsug Le characersc fucos (9) ad (30) o relaos (3) ad (33), he followg coserved vecors for he me fracoal problem () s obaed. Case. 0 < <, coserved vecors are gve as follows: C = u ( u + u + u + v + v + v ) v ( u + u + u ) q(u + u + u 3v v v ) p(3 u + u + u ) (v + v + v ), C I u I v I u I v I v I u = ( ( ) ( )) ( ( ) ( )) ( ) ( ), C = ( ) ( ), u u v vu v q u + v pu = ( ) C ( ). I u I v Case. < <, coserved vecors are gve as follows: C = u ( u + u + u + v + v + v ) v ( u + u + u ) q(u + u + u 3v v v ) p(3 u + u + u ) (v + v + v ), = ( ( ) ( )) ( ( ) ( )) ( ) ( ), (34) C D u D v D u D v D v D u C = ( ) ( ), u u v vu v q u + v pu = ( ) C ( ). D u D v 4. Coclusos I hs paper, he Le group aalyss approach s used o sudy me fracoal coupled Whham-Broer-Kaup equaos wh Rema-Louvlle dervave operaors. Based o he Le symmeres aalyss, he goverg sysem of me fracoal paral dffereal equaos s smlary reduced o a sysem of olear frcaoal ordary dffereal equaos Erdely-Kober dervave (35)

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