THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A, OF THE ROMANIAN ACADEMY Volume 16, Number 1/2015, pp. 3 10

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1 THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY Series A OF THE ROMANIAN ACADEMY Volume 6 Number /5 pp. ON THE SOLUTIONS OF TIME-FRACTIONAL GENERALIZED HIROTA-SATSUMA COUPLED-KDV EQUATION WITH MODIFIED RIEMANN-LIOUVILLE DERIVATIVE BY AN ANALYTICAL TECHNIQUE Mehme MERDAN Gümüşhae Uiversiy Facuy of Egieerig Mahemaical Egieerig Gümüşhae Turkey mmerda@gumushae.edu.r I his paper he fracioal variaioal ieraio mehod (FVIM) was implemeed o solve imefracioal geeralized Hiroa-Sasuma coupled-kdv euaio wih modified Riema-Liouville derivaive. A ew applicaio of fracioal variaioal ieraio mehod (FVIM) was eeded o derive aalyical soluios i he form of a series for hese euaios. I is idicaed ha he soluios obaied by he FVIM are reliable ad effecive mehod for srogly oliear couple parial euaios wih modified Riema-Liouville derivaive. The effecs of differe fracioal derivaive for he coupled sysem are ivesigaed. Key words: fracioal variaioal ieraio mehod ime-fracioal geeralized Hiroa-Sasuma coupled-kdv euaio Riema-Liouville derivaive. INTRODUCTION I rece years sigificaly ieres i fracioal calculus used i may areas such as elecrical eworks corol heory of dyamical sysems probabiliy ad saisics elecrochemisry of corrosio chemical physics opics egieerig acousics maerial sciece ad sigal processig ca be successfully modeled by liear or oliear fracioal order differeial euaios [ ]. The variaioal ieraio mehod (VIM) which firs proposed by Ji-Hua He [ ] was successfully applied o auoomous ordiary ad parial differeial euaios ad oher fields. Ji-Hua He [] was he firs o apply he variaioal ieraio mehod o fracioal differeial euaios. I rece years a ew modified Riema-Liouville lef derivaive is suggesed by G. Jumarie [4 6]. I his paper we eed he applicaio of he VIM i order o derive aalyical approimae soluios o oliear ime-fracioal geeralized Hiroa-Sasuma coupled-kdv euaio u + uu vw wv u = () ( ) v uv + v = () ( ) w uw + w = < > R. () Subjec o he iiial codiios: β k u = + k ah ( k) (4)

2 4 Mehme Merda ( β + ) ( β + ) 4k c k 4k k ah k v = + (5) c c w = c + c ah k (6) where c k c ad β are arbirary cosas. The Hiroa-Sasuma sysem was proposed by Hiroa ad Sasuma [7] o describe he ieracio of wo log waves wih differe dispersio relaios. Recely a umerical mehod is proposed for a ime-fracio geeralized Hiroa-Sasuma coupled KDV euaio by Gaji e.al [8] hey are used he homoopy perurbaio mehod. A ew ieraive echiue [9] is employed o solve a sysem of oliear fracioal parial differeial euaios. The goal of his paper is o eed he applicaio of he variaioal ieraio mehod o solve imefracioal geeralized Hiroa-Sasuma coupled-kdv euaio wih modified Riema-Liouville derivaive.. BASIC DEFINITIONS Here some basic defiiios ad properies of he fracioal calculus heory which ca be foud i [ ] Defiiio.. The lef-sided Riema-Liouville fracioal iegral operaor of order of a fucio f C μ μ is defied as a = ( ) d for > > ad a =. Γ a J f f J f f (7) ( ) ( )d Ja f = f > > Γ a (8) The properies of he operaor J ca be foud i [ 6 ]. Defiiio.. The modified Riema-Liouville derivaive[4 6] is defied as where [ ] d D f = ( ) ( f f() ) d Γ( ) d < ad. (9) Defiiio.. Fracioal derivaive of compouded fucios [ 46] is defied as Defiiio.4. The iegral wih respec o ( d) differeial euaio defied as d f Γ + d f< <. () [4-6] is defied as he soluio of he fracioal dy f d y() = < <. () LEMMA.. Le f ( ) deoe a coiuous fucio [4 6] he he soluio of he E. () is y = f d = ( ) f d < <. ()

3 O he soluios of ime-fracioal geeralized Hiroa-Sasuma coupled-kdv euaio 5 For eample f = β i E. () oe obais ( β ) ( ) Γ ( + β + ) Γ + Γ + β β d = + < <. Firsly le s discuss VIM for he differeial euaio i follows. (). VARIATIONAL ITERATION METHOD Accordig o he variaioal ieraio mehod [] we cosider he followig he parial differeial differeial euaio: L u + N u = g( ) (4) where L is a liear operaor N is a o-liear operaor ad g( ) is he source erm. The we ca cosruc a correc fucioal as follows: { } u + ( ) = u ( ) + λ L u ( s) + N u ( s) g( s) d s (5) where λ is a geeral Lagragia muliplier [ ] which ca be ideified opimally via variaioal heory. The secod erm o he righ is called he correcio ad u is cosidered as a resriced variaio i.e. δ =. u 4. FRACTIONAL VARIATIONAL ITERATION METHOD To describe he soluio procedure of he fracioal variaioal ieraio mehod we cosider he followig fracioal differeial euaio [4 6 ]: u( ) = L[ ] u( ) v( ) w( ) u = f < > R (6) where L[ ] is he differeial operaor i f ( ) is coiuous fucio. Accordig o he VIM we ca cosruc a correc fucioal for E. (6) as follows u () u + ( ) = u( ) + I λ( ) L [ ] u( ) v( ) w( ) u () u + ( ) = u( ) + ( ) ( ) L [ ] u( ) v( ) w () d Γ λ Usig E. () we obai a ew correcio fucioal (7) u () + ( ) = () + ( ) L [ u ] ( ) v ( ) w( ) ( d ). Γ ( + ) λ u u I is obvious ha he seueial approimaios uk k ca be esablished by deermiig λ a geeral Lagrage s muliplier which ca be ideified opimally wih he variaioal heory. The iiial values are usually used for choosig he zeroh approimaio u. Wih λ deermied he several approimaios uk k follows immediaely[]. Coseuely he eac soluio may be procured by usig (8)

4 6 Mehme Merda 4 ( ) lim ( ) u = u. (9) 5. APPROXIMATE SOLUTIONS OF THE TIME-FRACTIONAL GENERALIZED HIROTA- SATSUMA COUPLED-KDV EQUATION I his secio we prese he soluio of ime-fracioal geeralized Hiroa-Sasuma coupled-kdv euaio as he applicabiliy of FVIM. Accordig o he FVIM we cosruc a correcio fucioal for he Es. ( ). We have u() u() w() + u( ) v ( ) u + ( ) = u( ) + ( ) ( d ) Γ ( + ) λ v() u() () w () v v v () u v () v () + ( ) = ( ) + ( ) ( ) ( d ) Γ ( + ) λ + w w () u w () w () + ( ) = ( ) + ( ) ( ) ( d ) Γ ( + ) λ + we have u() u() w() + u( ) v ( ) δu+ ( ) = δu( ) + δ ( ) ( d ) = Γ ( + ) λ v() u() w () λ() = δu + λδu δu = ()d Γ ( + ) v() v() v() δv+ ( ) = δv() + δ ( ) u() ( d ) = Γ ( + ) λ + = () v + v λ δ λδ δv ()d Γ + = ( ) () () () (4) δ = δ () + δ λ ( ) () + ( d) = w () w () w () w+ w u Γ ( + ) = () w + w λ δ λδ = w()d Γ ( + ) δ where λ λ ad λ are geeral Lagrage mulipliers deoes he h approimaio ad u v w deoes resriced variaios i.e. δ u = δv = δ w = ad ( ) ( ) ( ) + λ = + λ = + λ = = = = λ( ) λ( ) λ ( ) = = =. (5) (6)

5 5 O he soluios of ime-fracioal geeralized Hiroa-Sasuma coupled-kdv euaio 7 The geeralized Lagrage muliplier ca be ideified by he above euaios λ = λ = λ =. (7) Subsiuig E. (7) io Es. ( ) produces he ieraio formulaio as follows u() u() w() ( ) ( ) + u v + ( ) = ( ) ( d ) Γ ( + ) v() u() w ( ) u u (8) v v u v() v() v() + ( ) = ( ) ( ) ( d ) Γ ( + ) + (9) w w ( ) u w ( ) w ( ) Takig he iiial value β k u ( ) = u = + k ah ( k) 4k c ( β + k ) 4k ( β + k ) ah( k) v ( ) = v ( ) = + c c w = w = w = c + c ah k + ( ) = ( ) ( ) ( d ) Γ ( + ) + we ca derive ad so o i he same maer he res of he compoes of he ieraio formulae (8 ) ca be calculaed by Maple. The he approimae soluios i a series form are β k u ( ) = Lim u ( ) = + k ah ( k) + 4k ah( k) sech ( k) β + Γ + () () 4k c + k 4k + k ah k v () = Limv ( ) = + + c c c Γ + 4k ββ ( + k ) sech ( k) β β ( ) + () w ( ) = Lim w ( ) = c + cah ( k) + ck sech ( k). β Γ + + () For he special case = = = c = β is ( ) β k u = + k ah ( k( c) ) (4) ( β+ ) ( β+ k ) ( k( c) ) 4k c k 4k ah v ( ) = + (5) c c ( ) w = c+ cah k c (6)

6 8 Mehme Merda 6 which is a eac soluio o he coupled-kdv ( ) euaio. Fig. The FVIM resuls for u( ) for he approimaio show: a) eac soluio; b) u ( ) ; c) u ( ) ; d) u for = = = whe k =. β =.5 c =.5 c =.5 for he soliary wave soluios wih he iiial codiios (4) of E. () respecively. Fig. The FVIM resuls for v( ) for he approimaio show: a) eac soluio; b) v ( ) ; c) v ( ) ; d) v for = = = whe k =. β =.5 c =.5 c =.5 for he soliary wave soluios wih he iiial codiios (5) of E. () respecively. Fig. The FVIM resuls for w( ) for he approimaio show a) eac soluio; b) w ( ) ; c) w ( ) ; d) w for = = = whe k =. β =.5 c =.5 c =.5 for he soliary wave soluios wih he iiial codiios (6) of E. () respecively. Fig. 4 The surface idicaes he soluio u( ) for E.() whe k =. β =.5 c =.5 c =.5 : a) approimae soluio u ( ) for =.9 =.8 =.7 ; b) approimae soluio u ( ) for =.7 =.8 =.9 ; c) approimae soluio u ( ) =.6 =.6 =.6 ; d) approimae soluio u ( ) for =. =. =.. for

7 7 O he soluios of ime-fracioal geeralized Hiroa-Sasuma coupled-kdv euaio 9 Fig. 5 The surface idicaes he soluio v( ) for E.() whe k =. β =.5 c =.5 c =.5 : a) approimae soluio v ( ) for =.9 =.8 =.7 ; b) approimae soluio v ( ) for =.7 =.8 =.9 ; c) approimae soluio v ( ) =.6 =.6 =.6 ; d) approimae soluio =. =. =.. for v for Fig. 6 The surface idicaes he soluio ( ) ( ) w for w for E.() whe k =. β =.5 c =.5 c =.5 : a) approimae soluio = = = ; b) approimae soluio ( ) ( ) =.6 =.6 =.6 ; d) approimae soluio ( ) w for w for =.7 =.8 =.9 ; c) approimae soluio w for =. =. =.. 6. CONCLUSIONS I his paper he He's variaioal ieraio mehod was used for fidig solio soluios of a geeralized ime fracioal Hiroa-Sasuma coupled KdV euaio wih modified Riema-Liouville derivaive. I his paper we have discussed modified variaioal ieraio mehod havig iegral w.r.. ( d) used for he firs ime successfully by Jumarie. The obaied resuls idicae ha his mehod is powerful ad meaigful for solvig he oliear couple fracioal differeial euaios. The soluios obaied by usig FVIM havig iegral w.r.. ( d) are i agreeme wih hose obaied by HPM[8] which is available i he lieraure. REFERENCES. OLDHAM K.B. SPANIER J. The Fracioal Calculus Academic Press New York PODLUBNY I. Fracioal Differeial Euaios Academic Press New York KILBAS A.A. SRIVASTAVA H.M. TRUJILLO J.J. Theory ad Applicaios of Fracioal Differeial Euaios Elsevier Amserdam LAVOIE J.L. OSLER T.J. TREMBLAY R. Fracioal derivaives ad special fucios SIAM Review. 8 pp CAPUTO M. Liear models of dissipaio whose Q is almos freuecy idepede Par II J. Roy. Asr. Soc. p MILLER K.S. ROSS B. A Iroducio o he Fracioal Calculus ad Fracioal Differeial Euaios Wiley New York SAMKO S.G. KILBAS A.A. MARICHEV O.I. Fracioal Iegrals ad Derivaives: Theory ad Applicaios Gordo ad Breach Yverdo 99.

8 Mehme Merda 8 8. MERDAN M. Soluios of ime-fracioal reacio-diffusio euaio wih modified Riema-Liouville derivaive Ieraioal Joural of Physical Scieces 7 5 pp MERDAN M. MOHYUD-DIN S.T. A New Mehod for Time-fricioal Coupled-KDV Euaios wih Modified Riema- Liouville Derivaive Sudies i Noliear Scieces pp MERDAN M. GÖKDOĞAN YILDIRIM A. MOHYUD-DIN S.T. Numerical simulaio of fracioal Forberg-Whiham euaio by differeial rasformaio mehod Absrac ad Applied Aalysis Aricle ID HE J.H. Some applicaios of oliear fracioal differeial euaios ad heir approimaios Bull. Sci. Techol. 5 pp HE J.H. Variaioal ieraio mehod- a kid of o-liear aalyical echiue: Some eamples I. J. Noliear Mech. 4 pp HE J.H. WU X.H. Variaioal ieraio mehod: New developme ad applicaios Compu. Mah. Appl pp JUMARIE G. New sochasic fracioal models for Malhusia growh he Poissoia birh process ad opimal maageme of populaios Mah. Compu. Model 44 pp JUMARIE G. Laplace s rasform of fracioal order via he Miag-Leffler fucio ad modified Riema-Liouville derivaive Appl. Mah. Le. pp JUMARIE G. Table of some basic fracioal calculus formulae derived from a modified Riema-Liouvillie derivaive for odiffereiable fucios Appl. Mah. Le. pp HIROTA R. SATSUMA J. Solio soluios of a coupled Koreweg-de Vries euaio Phys. Le. A pp GANJI Z.Z. GANJI D.D. ROSTAMIYAN Y. Soliary wave soluios for a ime-fracio geeralized Hiroa-Sasuma coupled KdV euaio by a aalyical echiue Appl. Mah. Mod. pp SHATERI M. GANJI D.D. Soliary Wave Soluios for a Time-Fracio Geeralized Hiroa-Sasuma Coupled KdV Euaio by a New Aalyical Techiue I. J. Diff. E. Aricle ID LUCHKO A.Y. GRONEFLO R. The iiial value problem for some fracioal differeial euaios wih he Capuo derivaive Prepri series A8-98 Fachbreich Mahemaik ud Iformaik Freic Uiversia Berli FARAZ N. KHAN Y. JAFARI H. YILDIRIM A. MADANI M. Fracioal variaioal ieraio mehod via modified Riema-Liouville derivaive J. Kig. Saud. Uiv.(Sciece) pp Received February 6