Exact and Solitary Wave Solutions to the Generalized Fifth-order KdV Equation by Using the Modified Simple Equation Method

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1 Applied nd Compionl Mhemic ; (: -9 Pblihed online April 7, (hp:// doi:.68/j.cm.. ISSN: 8-6 (Prin; ISSN: 8-6 (Online Ec nd Soliry Wve Solion o he Generlized Fifh-order KdV Eqion by Uing he Modified Simple Eqion Mehod M. Ahrfzzmn Khn, M. Ali Akbr Deprmen of Applied Mhemic, Univeriy of Rjhhi, Rjhhi, Bngldeh Emil ddre: khnmh@yhoo.com (M. A. Khn, li_mh7@yhoo.com (M. A. Akbr To cie hi ricle: M. Ahrfzzmn Khn, M. Ali Akbr. Ec nd Soliry Wve Solion o he Generlized Fifh-order KdV Eqion by Uing he Modified Simple Eqion Mehod. Applied nd Compionl Mhemic. Vol., No.,, pp. -9. doi:.68/j.cm.. Abrc: Alhogh he modified imple eqion (MSE mehod effecively provide ec rveling wve olion o nonliner evolion eqion (NLEE in he field of engineering nd mhemicl phyic, i h ome limiion. When he blnce nmber i greer hn one, lly he mehod doe no give ny olion. In hi ricle, we hve epoed proce how o implemen he MSE mehod o olve NLEE for blnce nmber wo. In order o verify he proce, he generlized fifh-order KdV eqion h been olved. By men of hi cheme, we fond ome freh rveling wve olion o he bove menioned eqion. When he prmeer receive pecil vle, oliry wve olion re derived from he ec olion. We nlyze he oliry wve properie by he grph of he olion. Thi how he vlidiy, eflne, nd neceiy of he proce. Keyword: MSE Mehod, Nonliner Evolion Eqion, Soliry Wve Solion, Ec Solion, Generlized Fifh-Order Kdv Eqion. Inrodcion Nonliner evolion eqion occr no only from mny field of mhemic, b lo from oher brnche of cience ch phyic, meril cience, mechnic ec. Inriccy of NLEE nd chllenge in heir heoreicl dy h rced lo of enion from nmero mhemicin nd cieni who re concern wih nonliner cience. Therefore, he die of ec olion o NLEE ply very imporn role o know he inner rcre of he nonliner phenomen. B he bic problem i, i i no ey o in heir ec olion. Therefore, in order o emine ec olion, differen grop of mhemicin nd phyici re working joinly. In he recen yer, coniderble developmen hve been mde for erching ec olion o NLEE. They eblihed everl mehod, ch,he invere cering rnformion mehod [], he Hiro biliner mehod [], he Bcklnd rnformion mehod ([][], he Drbo rnformion mehod [], he Pinleve epnion mehod [6], he Adomin decompoiion mehod ([7][8], he He homoopy perrbion mehod ([9][], he Jcobi ellipic fncion mehod ([][], he Mir rnformion mehod [], he ine-coine mehod ([][], he homogeneo blnce mehod [6], he nhfncion mehod ([7][8], he eended nh-fncion mehod ([9] [], he fir inegrion mehod [], he F- epnion mehod [],he iliry eqion mehod [], he Lie grop ymmery mehod [], he vriionl ierion mehod [], he nz mehod ([6][7], he Ep-fncion mehod ([8][9], he ( G / G -epnion mehod ([]- [], he modified imple eqion mehod ([6]-[],he ep( φ( η -epnion mehod ([][], ec. The modified imple eqion mehod ([6]-[] i recenly developed riing mehod. I compion i righforwrd, yemic, nd no need he ymbolic compion ofwre o mniple he lgebric eqion. B, he mehod h ome horcoming, when he blnce nmber i greer hn one, lly he mehod doe no give ny olion. To he be of or knowledge, ill now only wo ricle re vilble in he lierre concerning higher blnce nmber (for blnce nmber wo. Slm [] ed he MSE mehod o he modified Lioville eqion (wherein he blnce nmber i wo nd wrie-down olion o hi eqion. However, nfornely he obined olion doe

2 Applied nd Compionl Mhemic ; (: -9 no ify he eqion. And in Ref. [], Zyed nd Arno olved he KP-BBM eqion by men of he MSE mehod nd fond ome olion of hi eqion. B here i no gideline in hi ricle, how one cn olve oher NLEE for he higher blnce nmber. In he preen ricle, we hve developed echniqe o h he MSE mehod cn be eploied o olve NLEE for blnce nmber wo. Inering he med olion o he correponding ordinry differenil eqion nd hen eqing he coefficien of j ( ξ,( j =,,, L, N yield n over-deermined e of lgebric nd differenil eqion. Dring deerminion he nknown fncion, here born hird order liner ordinry differenil eqion in nd ξ. If in he olion of, ξ pper polynomil, i will no be eligible o receive oliry wve olion, bece for oliry wve olion, we know h ξ ± [7]. Therefore, he coefficien of he polynomil m be zero. Thi conrin i eenil o olve NLEE for higher blnce nmber. The ricle i orgnized follow: In ecion, we mmrize he decripion of he mehod. In ecion, we employ he mehod o NLEE wih blnce nmber wo. In ecion, he phyicl eplnion of he olion re preened nd in ecion, we hve drwn or conclion.. The Mehod Le conider he nonliner evolion eqion of he form H(,,, y, z,,,... =, (. where = (, y, z, i n nidenified fncion, H i polynomil in (, y, z, nd i pril derivive, which inclde he highe order derivive nd nonliner erm of he highe order, nd he bcrip denoe pril derivive. In order o olve (. by men of he MSE mehod [6-], we hve o eece he following ep: Sep : The rveling wve vrible, (, y, z, = ( ξ, ξ = k( + y + z ± ω (. permi o rnform he Eq. (. ino he following ordinry differenil eqion (ODE: G(,,, L =, (. whereg i polynomil in ( ξ nd i derivive, wherein d ( ξ =. dξ Sep :We ppoe h he olion of (. cn be epre in he form, i N ( ξ ( ξ = i i= ( ξ, (. where i, ( i =,,, L, N re nknown conn o be deermined, ch h N, nd ( ξ i n nknown fncion o be evled. In nh-fncion mehod, ( G / G - epnion mehod, ine-coine mehod, Jcobi ellipic fncion mehod, Ep-fncion mehod ec., he olion re propoed in erm of ome fncion eblihed in dvnce, b in he MSE mehod, ( ξ i no pre-defined or no olion of ny pre-defined differenil eqion. Therefore, i i no poible o conjecre from erlier wh kind of olion one my ge hrogh hi mehod. Thi i he individliy nd diincion of hi mehod. Therefore, ome freh olion migh be fond by hi mehod. Sep : The poiive ineger N ppering in Eq. (. cn be deermined by king ino ccon he homogeneo blnce beween he highe order nonliner erm nd he derivive of highe order occrring in Eq. (.. Sep : We bie (. ino (. nd hen we ccon he fncion ( ξ. A rel of hi biion, we ge nd i derivive. In he reln polynomil, we eqe ll he coefficien of polynomil of ( ( ξ/ ( ξ ( i ( ξ, ( i =,,,..., N o zero. Thi procedre yield yem of lgebric nd differenil eqion which cn be olved for geing i ( i =,,, L, N, ( ξ nd he vle of he oher needfl prmeer. Thi complee he deerminion of he olion o he eqion (... Applicion of he Mehod In hi ecion, we will eece he MSE mehod o erc oliry wve olion o he generlized fifh-order KdV eqion which i very imporn in he field of rfce wve propgion on hllow wer rfce. Le conider he generlized fifh-order KdV eqion of he form + α + + γ + = (. where α,, γ nd re he rel conn. To conrc oliry wve olion of he generlized fifh-order KdV eqion by pplying he MSE mehod, we e he wve vrible (, = U( ξ, ξ = k( ω. (. The rveling wve rnformion (. redce Eq. (. o he following ODE in he form: ( v ωu + αuu + U U + γ k U + k U = (. where prime denoe he derivive wih repec o ξ. Now, inegring he Eq. (. wih repec o ξ, we ge new ODE in he form U U ωu α γ k U k U ( iv = (. Blncing he highe order derivive erm U nd he nonliner erm of he highe order U occrring in (., we (iv

3 M. Ahrfzzmn Khn nd M. Ali Akbr:Ec nd Soliry Wve Solion o he Generlized Fifh-order KdV Eqion by Uing he Modified Simple Eqion Mehod ge N =. Th, he olion (. ke he form U( ξ = + +. (. where, nd re conn o be deermined, ch h, nd (ξ i n nknown fncion o be deermined. Now, i i ey o ccomplih ( ( U = + +. (.6 ( U = U U = + { 6( ( +. (.7 ( [ { ( + 7 ( + ( ( ( iv 7 + } 6 ( { ( ( + ( ( }] iv ( iv = ( 6 ( + ( ( ( ( iv ( ( v 6 } + [ 6( 68 ( 6 ( { ( ( ( iv 6 + } { ( ( ( iv + + ( ( v + + }]. (.8. (.9 (iv Sbiing he vle of U, U, U, U nd U from (.-(.9 ino Eq. (. nd hen eqing he coefficien of,,,, o zero, we repecively obin 6 ( 6 + α + = ω. (. {( ( v ω + α + + k γ + k }. (. = { v + k + ( + k } { k + ( + k ( iv } ( ( ( iv α + + k γ k + ( ( ( k γ + ( ω + α + ( =. (. + k ( + [{ k γ + ( α + }( { ( + }] k γ ( { ( + + 9( ( iv k } ( α + ( 6 k ( + k { } ( 7( ( 6k γ + ( { ( k + 6k }( = =. (. =. (.. (. ( 6k + ( 6 = From Eq. (. nd (.6, we obin. (.6 i6 k = nd = ±, ince. Therefore, he following ce rie depending on he vle of. 6i k Ce : When =, hen from Eq. (.- (., we obin nd k γ i = ± γ ω = mi / + i α k α / γ ± i ξ, + α, k ( e / ξ = + c k c γ + i α γ + i α where c nd c re inegring conn. Now, biing he vle of,, nd (ξ ino Eq. (., we obin.

4 Applied nd Compionl Mhemic ; (: -9 U 6k ( γ + i α ( ξ = ± c γ + i α mi / / k c / c e ce ± i ξ ± i ξ γ + i k / γ + i k α α /. (.7 Simplifying he reqired olion (.7, we obin he following cloe-form olion o he generlized fifh-order KdV eqion (.: (, = m ( + / 6 / { k c ( co( θ ± iin ( θ ± ic γ + i α ( co( θ miin ( θ } / k γ i α c c. (.8 Where ( / / / ( γ + i α + 8 γ + i α γ + α θ = Since c nd c re rbirry conn, one my rndomly pick heir vle. If we chooe / c = γ + i α nd c = k hen from (.8, we obin he following oliry wve olion: ( γ + iα γ α (, = θ ( α i γ in. (.9 wrien he following oliry wve olion in he form: ( ( i γ + α, =. (. θ + co 6i k Ce : When =, hen from Eq. (.-(., we ge Agin, if we chooe c = γ + α nd / i c = k hen from (.8, we obin ( γ + iα γ α (, = θ ( α i γ + in.(. And = ± 8 γ ω = 6k i γ i α / γ α, + α On he oher hnd, if c = γ + α nd / i c = ±i k, from olion (.8, we derive he oliry wve olion in he form: ( ( i γ + α, =. (. θ + co i / k ( e / ξ = ± + c k c γ i α mi ξ γ i α.w here c nd c re inegring conn. Now, biing he vle of, nd ( ξ, ino Eq. (., we obin he olion in he form: Alo when c = γ + α nd / i c = mi k hen from olion (.8 cn be

5 6 M. Ahrfzzmn Khn nd M. Ali Akbr:Ec nd Soliry Wve Solion o he Generlized Fifh-order KdV Eqion by Uing he Modified Simple Eqion Mehod U ( ξ ( / γ i α ± i ξ / k / 6k γ i α cce = ± γ i α mi ξ / / k ± i k ce + c γ i α. (. Simplifying he reqired eponenil olion (. ino rigonomeric fncion, we derive he olion of he Eq. (. in he following: where Th, we ge he ec olion (. of he generlized fifh-order KdV eqion (.. B c nd c re rbirry conn, o, one my rbirrily pick heir vle. Therefore, if we chooe c = γ α nd / i c = k, hen he oliry wve olion (. become ( γ iα γ α (, = ϑ ( i γ + α in. (. Agin, if we chooe c = γ α nd / i c = k, hen from (., we obin he following oliry wve olion: 6 / 6k ( γ α (, i = m ϑ = ξ ( γ iα γ α (, = ϑ ( i γ + α + in (.6 On he oher hnd, if we ke c = γ α / i nd c = ±i k, hen from (., we ge he olion in he form: i ( ( γ iα, = 7. (.7 ϑ + co Alo if c = γ α nd / { k ( ( ( ( ( ( } / c co ϑ miin ϑ mic γ i α co ϑ ± iin ϑ γ i α = / k i + c = mi k, hen he oliry wve olion (. cn be wrien in he form: / c c ( 8 γ iα γ + α / / / ( γ i α ( ( i γ + α, =.. (. 8. (.8 ϑ + co B, ince c nd c re rbirry conn for oher choice of cnd c, we migh obin mch new nd more generl ec olion o Eq. (. by he MSE mehod wiho ny id of ymbolic compion ofwre. The mjor dvnge of he MSE mehod i h he clclion re no ophiiced nd ey o conrol. I i no reqired ny comper lgebr yem o fcilie he clclion i ke o he Ep-fncion mehod, he ( G / G -epnion, he nh-fncion mehod, he homoopy nlyi mehod ec. B he olion obined by he MSE mehod re eqivlen o hoe olion obined by he bove menioned mehod. Remrk: Solion (.9-(. nd (.-(.8 hve been verified by ping hem bck ino he originl eqion nd fond correc.. Eplnion nd Phyicl Inerpreion of he Solion In hi ecion, we will depic he grph nd ignify he obined olion o he generlized fifh-order KdV eqion. The olion (.9 o (. re repreen he periodic olion. Thee olion re rveling wve olion h re ll periodic bell hpe b differen, ch h Fig. how he bell hpe of he olion (.9 o (. nd Fig. drwn he bell hpe olion (. o (. re repecively for α = γ = = k = nd = wihin,. Solion re oliry wve wih reilien cering propery. On he oher hnd, Fig. nd Fig. re ploed he periodic olion (.-(.6 nd (.7-(.8 repecively for α = = nd γ = = k =

6 Applied nd Compionl Mhemic ; (: -9 7 wihin,. Here, he Figre o nd he figre o re kech of me ype b ll re differen Fig.. Bell hpe mli-olion of olion (.9 nd (. o he generlized fifh-order KdV eqion ( Fig.. Drk olion of olion (. nd (.6 o he generlized fifhorder KdV eqion ( Fig.. Bell hpe mli-olion of olion (. nd (. o he generlized fifh-order KdV eqion (.. Fig.. Drk olion of olion (.7 nd (.8 o he generlized fifhorder KdV eqion.

7 8 M. Ahrfzzmn Khn nd M. Ali Akbr:Ec nd Soliry Wve Solion o he Generlized Fifh-order KdV Eqion by Uing he Modified Simple Eqion Mehod. Conclion If he blnce nmber i greer hn one, in generl he MSE mehod doe no provide ny olion. For hi ce, we hve eblihed procedre in order o implemen he MSE mehod o olve NLEE for blnce nmber wo. If he olion of ( ξ coni of polynomil of he wve vrible ξ, i will no be he oliry wve olion, ince i doe no mee he condiion ξ ± for oliry wve olion. In hi ce, ech coefficien of he polynomil m be zero. Thi conrin i crcil o olve NLEE for higher blnce nmber. By ing hi chieved proce, we hve olved he generlized fifh-order KdV eqion nd fond ome new rveling wve olion. When he prmeer receive pecil vle, oliry wve olion re derived from he ec olion. We nlyze he oliry wve properie of he olion vi he grph. Reference [] M.J. Ablowiz nd P.A. Clrkon, Solion, nonliner evolion eqion nd invere cering, Cmbridge Univeriy Pre, New York, 99. [] R. Hiro, The direc mehod in olion heory, Cmbridge Univeriy Pre, Cmbridge,. [] C. Roger nd W.F. Shdwick, Bcklnd rnformion nd heir pplicion, Vol. 6 of Mhemic in Science nd Engineering, Acdemic Pre, New York, USA, 98. [] L. Jinming, D. Jie nd Y. Wenjn, Bcklnd rnformion nd new ec olion of he Shrm-To-Olver eqion, Abrc Appl. Anlyi, ( Aricle ID 97, 8 pge. [] V.B. Mveev nd M.A. Slle, Drbo rnformion nd olion, Springer, Berlin, 99. [6] J. Wei, M. Tbor nd G. Crnevle, The Pinlevé propery for pril differenil eqion, J. Mh. Phy., (98,pp. -6. [7] A.M. Wzwz, Pril Differenil eqion: Mehod nd Applicion, Tylor nd Frnci,. [8] M.A. Hell nd M.S. Mehn, A comprion beween wo differen mehod for olving Boineq-Brger eqion, Cho, SolionFrc., 8 (6,pp. -6. [9] D.D. Gnji, The pplicion of He homoopy perrbion mehod o nonliner eqion riing in he rnfer, Phy. Le. A, (6, pp. 7-. [] D.D. Gnji, G.A. Afrozi nd R.A. Tlrpohi, Applicion of vriionl ierion mehod nd homoopy perrbion mehod for nonliner he diffion nd he rnfer eqion, Phy. Le. A, 68 (7, pp. -7. [] G. X, An ellipic eqion mehod nd i pplicion in nonliner evolion eqion, Cho, SolionFrc., 9 (6, pp [] E. Yfogl nd A. Bekir, Ec olion of copled nonliner evolion eqion, Cho, olionfrc., 7 (8, pp [] T.L. Bock nd M.D. Krkl, A wo-prmeer Mir rnformion of he Benjmin-Ono eqion, Phy. Le. A, 7 (979,pp [] A.M. Wzwz, A ine-coine mehod for hndle nonliner wve eqion, Appl. Mh. Comp. Modeling, ( [] E. Yfogl, nd A. Bekir, Solion nd periodic olion of copled nonliner evolion eqion by ing ine-coine mehod, In. J. Comp. Mh., 8 ( (6, pp [6] M. Wng, Soliry wve olion for vrin Boineq eqion, Phy. Le. A, 99 (99,pp [7] W. Mlflie nd W. Heremn, The nh mehod II: Perrbion echniqe for conervive yem, Phy. Scr., (996, pp [8] H.A. Nr, M.A. Abdel-Rzek nd A.K. Seddeek, Epnding he nh-fncion mehod for olving nonliner eqion, Appl. Mh., (,pp [9] A.J.M. Jwd, M.D. Pekovic, P. Lke nd A. Biw, Dynmic of hllow wer wve wih Boineq eqion, ScieniIrnic, Trn. B: Mech. Engr., ( (, pp [] M.A. Abdo, The eended nh mehod nd i pplicion for olving nonliner phyicl model, Appl. Mh. Comp., 9 ( (7,pp [] N. Tghizdeh nd M. Mirzzdeh, The fir inegrl mehod o ome comple nonliner pril differenil eqion, J. Comp. Appl. Mh., (,pp [] M.L. Wng nd X.Z. Li, Eended F-epnion mehod nd periodic wve olion for he generlized Zkhrov eqion, Phy. Le. A, (, pp. 8-. [] Sirendoreji, Ailiry eqion mehod nd new olion of Klein-Gordon eqion, Cho, SoliionFrc., (7, pp [] A.L. Go nd J. Lin, Ec olion of (+-dimenionl HNLS eqion, Commn. Theor. Phy., (, pp [] S.T. Mohyd-Din,, M.A. Noor nd K.I. Noor, Modified vriionl ierion mehod for olving ine-gordon eqion, World Appl. Sci. J., 6 (7 (9, pp [6] H. Triki, A. Chowdhry nd A. Biw, Soliry wve nd hock wve olion of he vrin of Boineq eqion, U.P.B. Sci. Bll., Serie A, 7( (,pp. 9-. [7] H. Triki, A.H. Kr nd A. Biw, Domin wll o Boineq ype eqion in (+-dimenion, Indin J. Phy., 88(7 (,pp [8] J.H. He nd X.H. W, Ep-fncion mehod for nonliner wve eqion, Cho, SolionFrc., (6,pp [9] H. Nher, A.F. Abdllh nd M.A. Akbr, New rveling wve olion of he higher dimenionl nonliner pril differenil eqion by he Ep-fncion mehod, J. Appl. Mh., ( Aricle ID 787, pge. [] M. Wng, X. Li nd J. Zhng, The ( G / G -epnion mehod nd rveling wve olion of nonliner evolion eqion in mhemicl phyic, Phy. Le. A, 7 (8, pp. 7-.

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