Solitary Wave Solutions for the Boussinesq and Fisher Equations by the Modified Simple Equation Method

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1 Mhemics Leers ; (): -8 hp:// doi:.48/j.ml.. Soliry Wve Solions for he Bossinesq nd Fisher Eqions by he Modified Simple Eqion Mehod Md. Ashrfzzmn Khn, M. Ali Akbr, Fehi Bin Mhmmd Belgcem Deprmen of Applied Mhemics, Universiy of Rjshhi, Rjshhi, Bngldesh Deprmen of Mhemics, Fcly of Bsic Edcion, PAAET, Al-Ardhyi, Kwi Emil ddress: khnmh@yhoo.com (M. A. Khn), li_mh74@yhoo.com (M. A. Akbr), fbmbelgcem@gmil.com (F. B. M. Belgcem) To cie his ricle: Md. Ashrfzzmn Khn, M. Ali Akbr, Fehi Bin Mhmmd Belgcem. Soliry Wve Solions for he Bossinesq nd Fisher Eqions by he Modified Simple Eqion Mehod. Mhemics Leers. Vol., No.,, pp. -8. doi:.48/j.ml.. Received: November 4, ; Acceped: Mrch 3, ; Pblished: Jne 3, Absrc: Alhogh he modified simple eqion mehod effecively provides ec rveling wve solions o nonliner evolion eqions in he field of engineering nd mhemicl physics, i hs some drwbcks. Priclrly, if he blnce nmber is greer hn, he mehod cnno be epeced o yield ny solion. In his ricle, we presen process o implemen he modified simple eqion mehod o solve nonliner evolion eqions for blnce nmber greer hn, nmely wih blnce nmber eql o. To vlide or heory hrogh pplicions, wo eqions hve been chosen o ndergo he proposed process, he Bossinesq nd he Fisher eqions, o which rveling wve re fond nd nlyzed. For specil prmeers vles, soliry wve solions re origined from he ec solions. We nlyze he soliry wve properies by he grphs of he solions. This shows he vlidiy, seflness, nd necessiy of he process. Keywords: Bossinesq Eqion, Fisher Eqion, Modified Simple Eqion Mehod, Nonliner Evolion Eqions, Soliry Wve Solions. Inrodcion The mhemicl modeling of comple phenomen h chnge over ime depends closely on he sdy of vriey of sysems of ordinry nd pril differenil eqions. Similr models re developed in diverse fields of sdy, rnging from he nrl nd physicl sciences, poplion ecology o economics, infecios disese epidemiology, nerl neworks, biology, mechnics ec. In spie of he eclecic nre of he fields wherein hese models re formled, differen grops of hem conribe deqe common ribes h mke i possible o emine hem wihin nified heoreicl srcre. Sch sdies mke for lrge re of fncionl nlysis, slly clled he heory of evolion eqions (EEs) which my be liner or nonliner. The ler re slly more chllenging hn heir liner conerprs, nd richer in erms of deqely modeling nd describing comple phenomen. Therefore, he invesigion of solions o nonliner evolion eqions (NLEEs) plys very imporn role o ncover he obscriy of mny phenomen nd processes hrogho he nrl sciences. However, one of he essenil problems is o obin heir ec solions. Therefore, in order o find o ec solions o NLEEs differen grops of mhemicins, physicis, nd engineers hve been working irelessly. Accordingly, in he recen yers, hey esblish severl mehods o serch ec solions, for insnce, he inverse scering mehod [], he Hiro s biliner rnsformion mehod [], he Bcklnd rnsformion mehod [3], [4], he Drbo rnsformion mehod [], he Pinleve epnsion mehod [], he Adomin decomposiion mehod [7], [8], he He s homoopy perrbion mehod [9], [], he Jcobi s ellipic fncion mehod [], [], he Mir rnsformion mehod [3], he sine-cosine mehod [4], [], he homogeneos blnce mehod [], he nh-fncion mehod [7], [8], he eended nh-fncion mehod [9], [], he firs inegrion mehod [], he F-epnsion mehod [], he iliry eqion mehod [3], he Lie grop symmery mehod [4], he vriionl ierion mehod [], he nsz mehod [], [7], he Ep-fncion mehod [8], [9], he ( G / G) -epnsion mehod [3], [3], [3], [33], [34], [3], he ep( φ( η)) -epnsion mehod [4], [4], nd he vrios versions nd improvemens of he ( G / G) -epnsion mehod [4], [4], [47], [48], [49], nd []. The modified simple eqion mehod, [3], [37], [38],

2 Md. Ashrfzzmn Khn e l.: Soliry Wve Solions for he Bossinesq nd Fisher Eqions by he Modified Simple Eqion Mehod [39], [4], being recenly developed, is rising in se. Is compion is srighforwrd, sysemic, nd needs no he symbolic compion sofwre o mniple he lgebric eqions. However, he mehod hs some shorcomings. The min problem is h when he blnce nmber is greer hn one, he mehod slly does no give ny solion. To he bes of or knowledge, ill now only wo ricles re vilble in he lierre concerning higher blnce nmber (for blnce nmber wo). In [43], Slm sed he MSE mehod o he modified Lioville eqion (wherein he blnce nmber is wo) nd wrien down solion o his eqion. However, nfornely he obined solion does no sisfy he eqion. Also, in Ref. [44], Zyed nd Arnos solved he KP-BBM eqion by mens of he MSE mehod nd fond some solions of his eqion. Unfornely, here is no gideline in his ricle, how one cn solve oher NLEEs for he higher blnce nmber. In he presen ricle, we hve considered wo eqions; he blnce nmber for ech of hese eqions is wo. If he blnce nmber is greer hn one, slly here rise difficlies in solving he NLEEs by mens of he MSE mehod. One cnno se he MSE mehod in srigh wy. In his cse, we need o ke in some sregy. Insering he ssmed solion o he corresponding ordinry differenil eqion nd hen j eqing he coefficiens of s( ξ), ( j,,,, N) yields n over-deermined se of lgebric nd differenil eqions. Dring deerminion of he nknown fncion, here born hird order liner ordinry differenil eqion in snd ξ.a polynomil ppering in he solion of s, ξ will mke i ineligible for soliry wve solion, becse in his cse, we hve s ξ ± [7]. Therefore, he coefficiens of he polynomil ms be zero. This consrin is essenil o solve NLEEs for higher blnce nmber. The ricle is orgnized s follows: In secion, we smmrize he descripion of he mehod. In secion 3, we employ he mehod o NLEEs wih blnce nmber is, nd in secion 4, we give conclsions.. The Modified Simple Eqion (MSE) Mehod To elbore on he MSE mehod, le s consider he nonliner evolion eqion of he form, H(,,, y, z,,,...), () where (, ) is n nidenified fncion, H is polynomil in (, ) nd is pril derivives, which inclde he highes order derivives nd nonliner erms of he highes order, nd he sbscrips denoe pril derivives. In order o solve Eqion () by mens of he MES mehod [3], [37], [38], [39], nd [4], we hve o eece he following seps: Sep : The rveling wve vrible, (, y, z, ) ( ξ),wih, ξ k( + y + z ± ω), () Permis for he chnge of Eq. (), ino he following ordinry differenil eqion (ODE): G (,,, ), (3) Where, G is polynomil in ( ξ ) nd is derivives, d wherein ( ξ). dξ Sep : We sppose h Eq. (3) hs he solion in he form, N s ( ξ) ( ξ) i i s( ξ), (4) where i, ( i,,,, N) re nknown consns o be deermined, sch h N, nd s( ξ ) is n nknown fncion o be evled. In sine-cosine mehod, nhfncion mehod, ( G / G) -epnsion mehod, Jcobi ellipic fncion mehod, Ep-fncion mehod ec., he solions re proposed in erms of some fncions esblished in dvnce, b in he MSE mehod, s( ξ) is no pre-defined or no solion of ny pre-defined differenil eqion. Therefore, i is no possible o conjecre from erlier wh kind of solions one my ge hrogh his mehod. This is he individliy nd disincion of his mehod. Therefore, some fresh solions migh be fond by his mehod. Sep 3: The posiive ineger N ppering in Eq. (4) cn be deermined by king ino ccon he homogeneos blnce beween he highes order nonliner erms nd he derivives of highes order occrring in Eq. (3). If he degree of ( ξ) is deg[ ( ξ )] N, herefore, he degree of he oher epressions will be s follows: nd, m d ( ξ) deg[ ] N + m m, d ξ l m d( ξ) p deg[ ( ) ] mn + p( N + l) l. d ξ Sep 4: We sbsie Eq. (4) ino Eq. (3) nd hen we ccon he fncion s( ξ ). As resl of his sbsiion, we ge polynomil of ( s ( ξ)/ s( ξ) ) i nd is derivives. In he resln polynomil, we eqe ll he coefficiens of ( ) i s( ξ), ( i,,,..., N) o zero. This procedre yields sysem of lgebric nd differenil eqions which cn be solved for geing he vles of i ( i,,,, N), s( ξ ) nd he vle of he oher needfl prmeers. 3. Applicions of he MSE Mehod In his secion, we will eece he MSE mehod o erc

3 Mhemics Leers ; (): -8 3 soliry wve solions o he Bossinesq eqion nd he Fisher eqions which re very imporn in he fields of srfce wve propgion in cosl regions, he nd mss rnsfer, biology, ecology, gene propgion, physiology, crysllizion, plsm physics, nd recion-diffsion sysems. 3.. The Bossinesq Eqion In his sb-secion, we will implemen he MSE mehod o find he ec solions nd hen he soliry wve solions o he Bossinesq eqion in he form: + +. () α( ) β whereα nd β re rel consns. To consrc soliry wve solions of he Bossinesq eqion by pplying he MSE mehod, we se he wve vrible (, ) U( ξ), ξ k( ω). () The rveling wve rnsformion (3.) redces Eq. (3.) o he following ODE: ω + α + β, (7) ( iv) ( ) U ( U ) ku where prime denoes he ordinry derivives wih respec o ξ. Now, inegring Eq. (7) wice wih respec o ξ nd seing he consn of inegrion o zero, we obin new ODE in he form: ω + α + β. (8) ( ) U U ku Blncing he highes order derivive erm U nd he nonliner erm of he highes order U, we obin N. Therefore, he solion (4) kes he form, s s U( ξ) + + s s, (9) where, nd re consns sch h nd s( ξ ) is n nknown fncion o be deermined. Now, i is esy o mke o, 3 s s s s s U + +. () 3 s s s s 3 4 s s 3ss s s s s s s U () s s s s s s s Sbsiing he vles of U, U nd U from (9)-() ino Eq. (8) nd hen eqing he coefficiens of 3 4 s, s, s, s, s o zero, we respecively obin ( + ω + α), () α ( s ) 3 k β s s + ( + ω + α)( s ) + β + β k ( s ) k s s (4) ( s ) ( ( k β + α ) s k β s ), () ( k β + α )( s ). () 4 From Eq. () nd Eq. (), we obin ω k β, nd, since. α α Therefore, we obin he following wo cses rises for he vles of. Cse : When, hen From Eq. (3)-(), we ge ± ξ ω k β ( ω ) ± k βc k β Ands( ξ) e + c, ω wherec ndc re inegring consn. Now, sing he vles of,, nd s( ξ) ino Eq. (9), we obin he solion U( ξ) k β α ( + ω ) cc ξ ω ± k β + c αk βce ω ( ) e ξ ω ± k β. (7) Simplifying he reqired solion (7), we derive he following close-form solion of he Bossinesq eqion: ( + ω ) k β cc (, ) ( ω) ω ( ω) ω k βc cosh ± sinh β β α ( ω) ω ( ω) ω ( + ω ) c cosh sinh β β. (8) Since c nd c re rbirry consns, one my rbirrily pick heir vles. If we choose c + ω nd c k β hen from solion (8), we obin ( ω) ω 3( + ω ) cosech β (, ). (9) α Agin if we choosec + ω nd c k β hen from solion (8), we hve he following soliry wve solion: ( + ω + α) s + k βs, (3)

4 4 Md. Ashrfzzmn Khn e l.: Soliry Wve Solions for he Bossinesq nd Fisher Eqions by he Modified Simple Eqion Mehod ( ω) ω 3( + ω ) sech β (, ). () α On he oher hnd, if c nd c, from solion (8), we derive he soliry wve solions in he form: (, ) k β ( + ω ) 3 ( ω) ω ( + k β + ω ) cosh β α ( ω) ω ± ( + k β + ω ) sinh β. () Also whenc nd c, hen from solion (8) cn be wrien s he following soliry wve solions in he form: ( ω ) k β + 4(, ). () ( ω) ω ( + k β + ω ) cosh β α ( ω) ω ± ( + k β ω ) sinh β The solions (9)-() re ploed nd shown in Figre nd solions ()-() re ploed nd shown in Figre nd 3 respecively forα β k, ω. Cse : When obin, ω hen from Eqs.(3)-(), we α k β ( + ω ) ±, while α ξ + ω ± k β k βc s( ξ) e + c, ω U( ξ) Where, c nd c re inegring consn. Now, sing he vles of,, nd s( ξ ) in Eq. (9), we obin he solion in he form: ξ + ω ξ + ω 4 k ce k cc e c ± ± k β k β ( + ω ) β 4 β ( + ω ) + ( + ω ) α k βce ω ξ + ω ± k β + + c ( ) Swiching he eponenil solion (3) ino rigonomeric fncion, we derive he solion of he Eq. (): (, ) ( ). (3) 4 ( ω) ω ( ω) ω k β c cosh ± sinh β β ω 4k βcc ( + ω ) ( ω) ω ( ω) ω + ( + ω ) c cosh sinh β β. (4) ( ω) ω ( ω) ω k βc cosh ± sinh β β α ( ω) ω ( ω) ω + ( + ω ) c cosh sinh β β Ths, we ge he ec solion (4) o he Bossinesq eqion (). B, since c nd c re rbirry consns, one my rndomly pick heir vles. So, if we ke c ω + nd c k β hen he soliry wve solion (4) becomes,

5 Mhemics Leers ; (): -8 ( ) ( ) ( ) ω ω ω ω ω (, ) + cosh sech. () α β β Agin, if we choose form: c ω + nd c k β hen from (4), we obin he following soliry wve solions in he ( ) ( ) ( ) ω ω ω ω ω (, ) + cosh cosech. () α β β c On he oher hnd, if we ke c ω + nd 4 ± hen from (4), we ge he solion in he form: 3 k β (, ) + isinh α isinh ( ω ) ( ω) ( ω) ω β. (7) ω β 4 Also, ifc + ω nd c k β hen we derive he soliry wve solion (4) cn be wrien in he form: (, ) 4 isinh α + isinh ( ω ) ( ω) ( ω) ω β. (8) ω β The solions ()-() re ploed nd shown in Figre 4 nd solions (7)-(8) re ploed nd shown in Figre, for α β, ω. The mjor dvnge of he MSE mehod is h he clclions re no sophisiced nd esy o conrol. I is no reqired ny comper lgebr sysem o fcilie he clclions s i ke o he Ep-fncion mehod, he ( G / G) -epnsion, he nh-fncion mehod, he homoopy nlysis mehod ec. B he solions obined by he MSE mehod re eqivlen o hose solions obined by he bove menioned mehod. Since c nd c re rbirry consns for oher choices of c nd c, we migh obin mch new nd more generl ec solions of Eq. () by he MSE mehod wiho ny id of symbolic compion sofwre. Remrk : Solions (9)-() nd ()-(8) hve been verified by ping hem bck ino he originl eqion nd fond correc. 3.. Physicl Inerpreions of he Bossinesq Eqion Solions In his sb-secion, we will depic he grph nd signify he obined solions o he Bossinesq eqion. The solions (9) nd () represen he singlr periodic solions. Periodic solions re rveling wve solions h re periodic, sch h Fig. shows he shpe of he solions (9) nd () for α β, ω wihin,. Solions re soliry wves wih resilien scering propery. Solions () nd () re comple solions, herefore, he modls nd rgmens of hese solions hve been ploed. The grph of modls of he solions () nd () hve been shown in Fig., nd heir rgmens hve been shown in Figs. 3 nd 4 respecively for α β k, ω wihin,. On he oher hnd, Fig. shows h he solion () is he bell shpe solion nd solion () is singlr bell shpe solion. Also, he solions (7) nd (8) re comple solions, herefore, he modls nd rgmens of hese solions hve been ploed. The grph of modls nd rgmens of h solions hve been shown in Fig. nd 7 respecively for α β, ω, sch h,, Fig.. Periodic solions nd in (9) nd () o he Bossinesq eqion ().

6 Md. Ashrfzzmn Khn e l.: Soliry Wve Solions for he Bossinesq nd Fisher Eqions by he Modified Simple Eqion Mehod Fig.. Plo of he modls of he solions nd in () nd () o he Bossinesq eqion () Fig. 3. Plo of he rgmens of he solion 3 in () o he Bossinesq eqion () Fig. 4. Plo of he rgmens of he solion 4 in () o he Bossinesq eqion () Fig.. Bell shpe solion in () nd singlr bell shpe solion in () o he Bossinesq eqion.

7 Mhemics Leers ; (): Fig.. Plo of he modls nd rgmens of he solions 3 in (7) o he Bossinesq eqion () Fig. 7. Plo of he modls nd rgmens of he solions 4 in (8) o he Bossinesq eqion () The Fisher Eqion In his sb-secion, we will mke se of he MSE mehod o obin new nd more generl ec solions nd hen he soliry wve solions o he Fisher eqion in he form: ( ). (9) The rveling wve rnsformion in Eq. () helps redce Eq. (9) o he following ODE, ω ku + k U + U( U), (3) where prime denoes he derivives wih respec o ξ. Blncing he highes order derivive erm U nd highes order nonliner erm U, we hve N. Therefore, he form of he solion of Eq. (3) is similr o he form of he solion in Eq. (9). Sbsiing he vles of U, U nd U from (9)-() ino Eq. (3) nd hen eqing he coefficiens of 3 4 s, s, s, s, s o zero, we respecively obin he following eqions, ( + ). (3) { s k ( ωs ks )} ( ) + +. (3) ( ω ) ( )( s ) k ( s ) ks ( ωs ks ). (33) ( s) k s s + ks { } { ( ω )} +. (34) ( s ) ( k ) s k s ks ( k )( s ). (3) 4 From Eq. (3), (34) nd (3), we obin,; k, since 3k c ξ ( kω ) 3k s ( ξ ) e + c kω, nd, where c nd c re consns of inegrion. Hence he following wo cses rise for he vles of. Cse : When, hen from Eq. (3)-(33), we obin, ω ±, { ± ik, ω }, i ± i k, ω.,

8 8 Md. Ashrfzzmn Khn e l.: Soliry Wve Solions for he Bossinesq nd Fisher Eqions by he Modified Simple Eqion Mehod Therefore, hree cses rises depending on he vles for. Cse (): When vles of,, nd ( ) U ( ) ξ nd ω ±, hen sing he s ξ in Eq. (3.), we ge ξ k 3 k c e kc e ξ k c. (3) Simplifying he solion (3) of he Fisher eqion (9), we obin (, ) k c cosh sinh ± kc cosh sinh ± + c cosh sinh +. (37) Hence we ge he more generl ec solion (37) of he Eq. (9). Since, c nd c re rbirry consns, we migh rndomly op heir vles. Therefore, if we ke c ± k nd c, we derive he soliry wve solion of he Fisher eqion (9) from Eq. (37) s follows: (, ) cosech cosh ± sinh.(38) 4 Agin, if choose c k nd c hen from he solion Eq. (37),we obin he following soliry wve solion: (, ) sech cosh ± sinh. (39) 4 Solion (38)is ploed nd shown in Figre 8 nd solion (39)is ploed nd shown in Figre 9. Cse (b): When ± ik nd ω hen sbsiing he vles of,, nd s( ξ ) in Eq. (9), we ge U ikcc e ( ξ ) iξ k iξ k kc e ic. (4) Now, simplifying he eponenil solion (4), we obin ikcc (, ) kc cos isin + c i cos + sin. (4) wve is no more rveling wve. Therefore, we re no ineresed o discss his cse. Hence i is rejeced. i Cse (c): When ± i k nd ω hen sbsied he vles of,, nd s( ξ) in Eq. (9), we ge U ( ξ ) iξ iξ ± ± k k kce 3kce ± i c. (4) iξ ± k kce ± ic Now, simplifying he solion (4), we obin he following rigonomeric fncion solion: (, ) kc cosh ± i sinh + ± i 3k c cosh ± i + sinh ± i ± i c cosh ± i sinh ± i. (43) kc cosh ± i sinh + ± i ± ic cosh ± i sinh ± i Ths, we ge he more generl ec solion (43) of he Fisher eqion. Since c nd c re consns of inegrion, we my iniively choose heir vles. Therefore, if we choose c k, nd, c ± i, hen he solion(43)is simplified o yield, (, ) 3 + nh ± i nh ± i. (44) 4 Agin if c k nd c i hen from he ec solion (43) we derived he following soliry wve solion, (, ) 3 + coh ± i coh ± i. (4) 4 Anoher wy is h we choose c ± i nd c 3k hen from solion (43), we ge, 3 (, ) 4 + cosh ± i sinh + ± i. (4) 4 + cosh ± i + 3sinh ± i Also, if we ke c i nd c 3k hen he solion (43) becomes Since, in his cse ω, i.e. he wve speed is zero, he

9 Mhemics Leers ; (): (, ) 4 + cosh ± i sinh + ± i.(47) 4 + cosh ± i 3sinh i + ± These solions (44)-(47) re ploed s shown in Figre o, respecively. Cse : When hen from Eq. (3)-(33), we obin i, ω ±, ± k, ω,,. { ± k ω } Agin for vles of we cn discss he following hree cses. i Cse (): When,, nd, ω ±, hen sing he vles of sing he vles of,, nd s( ξ ) in Eq. (9), we obin eponenil solion: U ( ) ξ i k 3 k c e ξ + ± i kc e + ξ i k c. (48) Afer simplificion, from solion (48) we obin he following rigonomeric solion of he Fisher eqion (9): c cosh ± i sinh + ± i ± i kc cosh ± i sinh ± i + c cosh ± i + sinh ± i. (49) (, ) ± i kc cosh ± i sinh ± i + c cosh ± i + sinh ± i The solion (49)is more generl ec solion of he Fisher eqion(9). If we pickc ± i k nd c, from solion(49), we obin he following soliry wve solion: (, ) 3 + nh ± i nh ± i. () 4 Agin, if we p, c i k, nd, c hen he solion(49) of he Fisher eqion becomes, (, ) 3 + coh ± i coh ± i. () 4 On he oher hnd, if c ± i k nd c hen we derive he following ec solion of Eq. (49): 3 (, ) 4 + cosh ± i sinh + ± i. () 4 + cosh ± i + 3sinh ± i Also, if we choose c i k nd c hen from Eq. (49), we obin he following soliry wve solion: 4 (, ) 4 + cosh ± i sinh + ± i. (3) 4 + cosh ± i + 3sinh ± i On he oher hnd, if we se c nd c, from Eq. (3.4), we ge k i cosh ± i sinh + ± i (, ) k ± i ( k ) cosh ± i i ( k ) sinh i + ±. (4) Finlly if we ke c nd c hen he ec solion (49) cn be wrien s: k ± i cosh ± i sinh + ± i (, ) k i ( k ) cosh ± i i ( k ) sinh i ± + ±. () The solions ()-() re drwn in Figre -4, respecively. Cse (b): When ± k nd ω hen sbsiing he vles of,, nd s( ξ) in Eq. (3.), we ge U ( ) ξ ξ k 3 c e kc c e ξ k. () Now, simplifying he solion () we obin he following close-form of he Fisher eqion: (, ) c cosh ± sinh ± ± kc cosh ± sinh + ± + c cosh ± sinh ± ±. (7) Since c nd c re rbirry consns, so we my ke c ± k nd c, hen he generl solion(7) cn be

10 Md. Ashrfzzmn Khn e l.: Soliry Wve Solions for he Bossinesq nd Fisher Eqions by he Modified Simple Eqion Mehod wrien s he following form: (, ) cosech ± cosh ± + sinh ± 4. (8) Agin, if we sec k nd c hen from Eq. (7), we obin he following soliry wve solion, (, ) sech ± cosh ± + sinh ±.(9) 4 Solions (8) - (9) re ploed s shown in Figre -, respecively. Cse (c): When ± knd ω, hen sbsiing he vles of,, nd s( ξ ) ino Eq. (3.), we derive he eponenil form generl solion: U ( ξ ) ξ ξ k k ± 4 + ξ k kc ic e k c kcc e c e. () Simplifying he eponenil solion rnsformed o he rigonomeric fncion in he following close-form solion of he Eq. (9): (, ) { cosh( ) sinh( ) } ± 4kcc + c { cosh( ) ± sinh( ) }. () k c kc cosh sinh c cosh ± sinh This is he more generl ec solion ()o he Fisher eqion(9). B, his is no he rveling wve solion, since he wve velociy is nll. So, we re no ineresed o discss he generl solion () o he Fisher eqion (9). Imperive is i now o poin o h solions derived by he MSE mehod re eqipoenil o hose solions obined by he previosly menioned mehod. Since, c nd c re rbirry consns, we my obin new nd/or more generl ec solions o Eq. (9) by he MSE mehod wiho ny id of symbolic compion sofwre. Remrk : Solions (38)-(39), (4),(44)-(47), ()-(), (8)-(9) nd () hve been confirmed by seing hem ino he originl eqion Physicl Inerpreions of he Fisher Eqion Solions In his sb-secion, we discss he physicl inerpreion of he solions o he Fisher eqion. The solion (38) represens he bell shpe solion nd he solion (39) represens he kink. The bell shpe solion is loclized srfce wve envelope h cses emporry increse in wve mplide nd he kink wves re rveling wves which rise from one sympoic se o noher. The kink solions re pproch o consn infiniy. Figs. 8 nd 9 show he shpe of he solions (38) nd (39) wihin,.solions (44)-(47) nd ()-() re comple solions, herefore, he modls nd rgmens of hese solions hve been ploed. The grph of modls of he solions (44)-(47) nd ()-() hve been shown in Figs., 3,, 9 nd respecively. On he oher hnd, he grph of rgmens of he solions (44)-(47) nd ()- () hve been shown in Figs.,, 4,, 7, 8,,, 3 nd 4 respecively. These solions re ploed for k.wihin,. Solions (8) represen he bell shpe solions nd solion (9) represens he kink. The shpes of hese solions re ploed in Figs. nd wihin, Fig. 8. Bell shpe soliry wve solion given in (38) o he Fisher eqion (9).

11 Mhemics Leers ; (): Fig. 9. Kink solion obined from (39) o he Fisher eqion (9) Fig.. Plo of he modls of he soliry wve solions nd given in 44) nd (4) o he Fisher eqion (9) Fig.. Plo of he rgmens of he soliry wve solion in (44) o he Fisher eqion (9).

12 Md. Ashrfzzmn Khn e l.: Soliry Wve Solions for he Bossinesq nd Fisher Eqions by he Modified Simple Eqion Mehod Fig.. Plo of he rgmens of he solion (4) o he Fisher eqion (9) Fig. 3. Grph of he modls of he soliry wve solions 3 nd 4o he Fisher eqion Fig. 4. Plo of he rgmens of he soliry wve solion 3 in (4) o he Fisher eqion (9).

13 Mhemics Leers ; (): Fig.. Figre of he rgmens of he solion 4 in (47) o he Fisher eqion Fig.. Figre of he modls of he solions in () nd in () o he Fisher eqion (9) Fig. 7. Skech of he rgmens of he soliry wve solion in () o he Fisher eqion.

14 4 Md. Ashrfzzmn Khn e l.: Soliry Wve Solions for he Bossinesq nd Fisher Eqions by he Modified Simple Eqion Mehod Fig. 8. Skech of he rgmens of he solion in () o he Fisher eqion Fig. 9. Plo of he modls of he solions () nd (3) o he Fisher eqion Fig.. To he figre of he rgmens of he solion 3 in () o he Fisher eqion.

15 Mhemics Leers ; (): Fig.. Skech of he rgmens of he solion 4 in (3) o he Fisher eqion Fig.. Plo of he modls of he solion in (4) nd in () o he Fisher eqion (9) Fig. 3. Plo he rgmens of he solion in (4) o he Fisher eqion.

16 Md. Ashrfzzmn Khn e l.: Soliry Wve Solions for he Bossinesq nd Fisher Eqions by he Modified Simple Eqion Mehod Fig. 4. Plo of he rgmens of he soliry wve solion in () o he Fisher eqion (9) Fig.. Skech of he Bell shpe soliry wve solion (8) o he Fisher eqion (9) Fig.. Kink solions obined from he soliry wve solion in (9) o he Fisher eqion (9). 4. Conclsions In his ricle, we considered he Bossinesq eqion nd he Fisher eqions for boh of hem he blnce nmber is wo. If he blnce nmber is greer hn one, in generl he MSE mehod does no provide ny solion. For his cse, we hve esblished he procedre in order o implemen he MSE mehod o solve NLEEs for blnce nmber wo. If he

17 Mhemics Leers ; (): -8 7 solion of s( ξ ) consiss of polynomil of he wve vrible ξ, i will no be he soliry wve solion, since i does no mee he condiion s ξ ± for soliry wve solion. In his cse, ech coefficien of he polynomil ms be zero. This consrin is crcil o solve NLEEs for higher blnce nmber. By sing his chieved process, we solved he bove menioned NLEEs nd fond some new rveling wve solions. When he prmeers receive specil vles, soliry wve solions re derived from he ec solions. Alhogh he mehod hs been pplied in wo eqions, i cn clerly be pplied o mny oher nonliner evolion eqions whose blnce nmber is eql o. Acknowledgmens Fehi Bin Mhmmd Belgcem wishes o cknowledge he conined sppor of he Pblic Ahoriy for Applied Edcion nd Trining Reserch Deprmen, (PAAET RD), Kwi. The Ahors wish o hnk he ML Chief Edior, Professor Onr Ilhn, nd he ML Referees, whose professionl sppor nd deiled commens helped improve he originl mnscrip. References [] M. J. Ablowiz nd P.A. Clrkson, Solion, nonliner evolion eqions nd inverse scering, Cmbridge Universiy Press, New York, 99. [] R. Hiro, The direc mehod in solion heory, Cmbridge Univ. Press, Cmbridge, 4. [3] C. Rogers nd W.F. Shdwick, Bcklnd rnsformions nd heir pplicions, Vol. of Mhemics in Science nd Engineering, Acdemic Press, New York, USA, 98. [4] L. Jinming, D. Jie nd Y. Wenjn, Bcklnd rnsformion nd new ec solions of he Shrm-Tsso-Olver eqion, Absrc Appl. Anlysis, () ID 937, 8 pges. [] V. B. Mveev, M.A. Slle, Drbo rnsformion nd solions, Springer, Berlin, 99. [] J. Weiss, M. Tbor nd G. Crnevle, The Pinlevé propery for pril differenil eqions, J. Mh. Phys., 4 (98) -. [7] A. M. Wzwz, Pril Differenil eqions: Mehod nd Applicions, Tylor nd Frncis,. [8] M. A. Hell nd M.S. Mehn, A comprison beween wo differen mehods for solving Bossinesq-Brgers eqion, Chos, Solions Frc., 8 () 3-3. [9] D. D. Gnji, The pplicion of He s homoopy perrbion mehod o nonliner eqions rising in he rnsfer, Phys. Le. A, 3 () [] D. D. Gnji, G.A. Afrozi nd R.A. Tlrposhi, Applicion of vriionl ierion mehod nd homoopy perrbion mehod for nonliner he diffsion nd he rnsfer eqions, Phys. Le. A, 38 (7) [] G. X, An ellipic eqion mehod nd is pplicions in nonliner evolion eqions, Chos, Solions Frc., 9 () [] E. Ysfogl nd A. Bekir, Ec solion of copled nonliner evolion eqions, Chos, Solions Frcls, 37, (8) [3] T. L. Bock nd M.D. Krskl, A wo-prmeer Mir rnsformion of he Benjmin-One eqion, Phys. Le. A, 74 (979) [4] A. M. Wzwz, A sine-cosine mehod for hndle nonliner wve eqions, Appl. Mh. Comp. Modeling, 4 (4) [] E. Ysfogl, nd A. Bekir, Solions nd periodic solions of copled nonliner evolion eqions by sing Sine-Cosine mehod, In. J. Comp. Mh., 83 () () [] M. Wng, Soliry wve solions for vrin Bossinesq eqions, Phy. Le. A, 99 (99) 9-7. [7] W. Mlflie nd W. Heremn, The nh mehod II: Perrbion echniqe for conservive sysems, Phys. Scr., 4 (99) 3-9. [8] H. A. Nssr, M.A. Abdel-Rzek nd A.K. Seddeek, Epnding he nh-fncion mehod for solving nonliner eqions, Appl. Mh., () 9-4. [9] A.J.M. Jwd, M.D. Pekovic, P. Lke nd A. Bisws, Dynmics of shllow wer wves wih Bossinesq eqion, Scieni Irnic, Trns. B: Mech. Engr., () (3) [] M. A. Abdo, The eended nh mehod nd is pplicions for solving nonliner physicl models, Appl. Mh. Comp., 9 () (7) [] N. Tghizdeh nd M. Mirzzdeh, The firs inegrl mehod o some comple nonliner pril differenil eqions, J. Comp. Appl. Mh., 3 () [] M. L. Wng nd X.Z. Li, Eended F-epnsion mehod nd periodic wve solions for he generlized Zkhrov eqions, Phys. Le. A, 343 () [3] Sirendoreji, Ailiry eqion mehod nd new solions of Klein-Gordon eqions, Chos, Soliions Frc., 3 (7) [4] A. L. Go nd J. Lin, Ec solions of (+)-dimensionl HNLS eqion, Commn. Theor. Phys., 4 () 4-4. [] S. T. Mohyd-Din,, M.A. Noor nd K.I. Noor, Modified Vriionl Ierion Mehod for Solving Sine-Gordon Eqions, World Appl. Sci. J., (7) (9) [] H. Triki, A. Chowdhry nd A. Bisws,Soliry wve nd shock wve solions of he vrins of Bossinesq eqion, U.P.B. Sci. Bll., Series A, 7(4) (3) 39-. [7] H. Triki, A.H. Kr nd A. Bisws, Domin wlls o Bossinesq ype eqions in (+)-dimensions, Indin J. Phys., 88(7) (4) 7-7. [8] J. H. He nd X.H. W, Ep-fncion mehod for nonliner wve eqions, Chos, Solions Frc., 3 () [9] H. Nher, A.F. Abdllh nd M.A. Akbr, New rveling wve solions of he higher dimensionl nonliner pril differenil eqion by he Ep-fncion mehod, J. Appl. Mh., () Aricle ID 7387, 4 pges.

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