A new approach to Kudryashov s method for solving some nonlinear physical models

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1 Ieriol Jourl of Physicl Scieces Vol. 7() pp My 0 Avilble olie hp:// DOI: 0.897/IJPS.07 ISS Acdeic Jourls Full Legh Reserch Pper A ew pproch o Kudryshov s ehod for solvig soe olier physicl odels Yusuf Pdir Yusuf Gurefe * d Eie Misirli Depre of Mheics Fculy of Sciece d Ars Bozo Uiversiy 6600 Yozg Turey. Depre of Mheics Fculy of Sciece Ege Uiversiy 00 Borov-Izir Turey. Acceped 0 My 0 I his pper we give ew versio of he Kudryshov's ehod for solvig o-iegrble pril differeil equios i heicl physics. Soe ec soluios icludig -solio d sigulr solio soluios of he K ( ) equio wih geerlized evoluio d ie depede dpig d dispersio re obied by usig his ew pproch. Key words: K ( ) equio wih geerlized evoluio Kudryshov's ehod syeric Fibocci fucios solio soluio. ITRODUCTIO The sudy of olier evoluio equios hs becoe very ipor i he rece yers. There re lo of olier evoluio equios h re solved usig differe heicl ehods. For hese physicl probles solio soluios copcos coidl wves sigulr solios d he oher soluios hve bee foud. These ypes of soluios pper i vrious res of pplied scieces d egieerig. I his pper we cosider he K ( ) equio wih geerlized evoluio log wih ie-depede dpig d iedepede dispersio s follows (Bisws 00): ( l q ) ( q dq q ( q ) = 0. () Here ( d b ( re rel-vlued fucios while l d re posiive iegers. I his pper we ssue l = d Equio ges ( q ) ( q dq q ( q ) = 0. () There re y ehods h re used o obi he iegrio of olier pril differeil equios. Soe of he re he ep-fucio ehod (He d Wu 006; Misirli d Gurefe 00 b 0; Ebid 0; Gurefe d Misirli 0) he ril equio ehod (Gurefe e l. 0 0) he (G /G)-epsio ehod (Gurefe d Misirli 0 b) he Hiro's ehod (Sls e l. 0; Gurefe e l. 0) he uiliry equio ehod (Zhg e l. 009) d y ore. I his reserch we odify Kudryshov's (0) ehod o rise he effeciveess of his ehod. Our ey ide is h rdiiol bse e of he epoeil fucio is replced by rbirry bse. So ew ec soluios of olier evoluio equios y be obied by his siple odificio. THE MODIFIED KUDRYASHOV METHOD We cosider he followig olier pril differeil equio for fucio q of wo rel vribles spce d ie : P q q q q q q...) = 0. () ( The i seps of he odified Kudryshov ehod is surize s follows: Sep *Correspodig uhor. E-il: ygurefe@gil.co. Our firs sep is o obied he rvellig wve soluio of

2 Pdir e l. 86 Equio of he for q( = u( ) = w( d (4) is free cos. Equıo ws reduced o olier ordiry differeil equio of he for: ( u u u...) = 0 () he prie deoes differeiio wih respec o. Suppose h he highes order olier ers i l ( s) Equio re u ( ) u ( ) d Sep ( p) (u ) Suppose h he ec soluios of Equio c be obied i he followig for: u( ) = y( ) = (6) equio =0 =. Where he fucio is soluio of = l( ). (7) Sep Accordig o he proposed ehod we ssue h he soluio of Equio c be epressed i he for u ( ) = (8) To clcule he vlue i Equio 8 h is he pole order for he geerl soluio of Equio we proceed logously s i he clssicl Kudryshov ehod o blcig he highes order olier ers i Equio (). More precisely by srighforwrd clculios we hve u( ) = (9) u( ) = ( ) (0) ( s) s u ( ) = ( ) ( s ) (). l ( s) ( l) s u u ( ) = ( ) ( s ) () ( p) ( ) ( s ) ( p) ( u ) ( ) = () d re cos coefficies. Blcig he highes order olier ers of Equios d () we hve ( l ) s = ( p) so (4) s p =. l () Sep 4 Subsiuig Equio 6 io Equio yields polyoil R () of. Seig he coefficies of R () o zero we ge syse of lgebric equios. Solvig his syse we shll deerie w ( d he vrible coefficies of (... ( ). Thus we obi he 0( ec soluios o Equio. APPLICATIO OF KUDRYASHOV METHOD TO THE K ( ) EUATİO I his sudy he odified Kudryshov ehod is pplied o hdle he K ( ) equio Equio 4 is reduce o he ordiry differeil equio by subsiuig i io Equio which c be wrie s ( u ) d d w( ( u u u ( ( u ) = 0. (6) Upo iegrio d if ( = 0 he Equio 6 becoes ( u ) d d u w( ( ( u ) = C (7) C is he iegrio cos. For sipliciy we e C = 0. Usig he rsforio u ( ) = V ( ) (8)

3 86 I. J. Phys. Sci. Equio 7 becoes d ( ( ) ( V w( V ( V) VV = 0. (9) d ( ) y = l () 4 y = l (6) Fro he firs er i Equio 9 we e d d = 0 (0) As resul of his we hve he syse of lgebric equios h c be solved wih Mheic. Solvig he syses we obi he coefficies ( ) d ( ) s follows: 0( so = cos (. Also we e V ( ) = y( ) = () =0 =. We oe h he fucio is soluio of equio = l( ). () Usig he blce forul (Equio ) for he olier ers VV d V i Equio 9 we copue =. () Therefore we hve V ( ) = y( ) = = (4) =0 d we subsiue derivives of he fucio y ( ) wih respec o. The required derivives i Equio 9 re obied 0 (l ) ( )( ( ) ( 4 ) ) 0( = 0 ( = ) (7) (l ) ( ) ) ( 4 ) ) ( = ) (l ) ) ( ) ) w ( ) = ( ) (8) b ( is rbirry fucio is free cos d re posiive iegers. Subsiuig Equios 7 d 8 io 4 we hve (l ) ( )( ( ) ( 4 ) ) V ) = ) ( (9) (l ) ) ( ) ) = d ( ). Subsiuig Equio 9 io 8 we c wrie he soluios ( ) ( ) = b u (l ) ( )( ( ) ( 4 ) ) ) (l ) ) ( ) ) d ( ) (l ) ) ( ) ) d ( ) (0)

4 Pdir e l. 86 ( ) ( ) = b u (l ) ( )( ( ) ( 4 ) ) ) (l ) ) ( ) ) d ( ) (l ) ) ( ) ) d ( ). () Figure. Soluio of ( u is show = d = = = b = (. Figure. Soluio of ( u is show = d = = = ( = si( b. Applyig severl siple rsforios o hese soluios we obi ew ec soluios o Equio respecively: u ( = u ( = cfs sfs A ( [ B v( ] A ( [ B v( ] () () ( ) A ( = ( =.) v = l ( )( ( ) ( 4 ) ) ) B = d (l ) (( ) ( ) ) d ( ) (. Here A A ( ) represe he pliude of he ( solios while B is he iverse widh of he solios d v = v( represe he velociy of he solios. Also Equio represes sigulr solio soluio for Equio. Figures o 7. show he soluios

5 864 I. J. Phys. Sci. Figure. Soluio of ( u is show = d = = = = Weiersrss ellipic fucio [ /]. Figure 4. Soluio of ( u is show = d = = = = Jcobi ellipic fuciosc [ /]. Figure. Soluio of ( u is show = d = = = ( = si( b. u ( u( for he vlues = d = = = he fucio b ( d es respecively Golde Me e d 0. If we e = e i Equio he we c fid he soluio obied by usig he Asz ehod i (Bisws 008). REMARKS AD COCLUSIOS Our i i his secio is o show h geerl Ep - fucio wih Kudryshov ehod could be used i he soluios i he for of syericl hyperbolic Fibocci

6 Pdir e l. 86 Figure 6. Soluio of ( u is show = d = = = = Weiersrss ellipic fucio [ /]. Figure 7. Soluio of ( u is show = d = = = = Jcobi ellipic fuciosc [ /]. d Lucs fucios. We highligh briefly he defiiios of syericl hyperbolic Fibocci d Lucs fucios. Also Shov d Rozi (00) defied ll deils of syericl hyperbolic Fibocci d Lucs fucios. We oly give severl foruls wih respec o hese fucios here. Syericl Fibocci si cosie sec d cosec fucios re respecively defied s ( sfs ) = cfs ( ) = scfs( ) = csfs( ) =. (4) Alogously syericl Lucs sie d cosie fucios re respecively defied s sls( ) = cls( ) = () = which is ow i lierure s Golde Me (Ahd d Ezz 00). Fro his sudy i is herefore possible o fid ore geerl (or ore lrger clsses of) soluios i pplyig he odified Kudryshov ehod wih Syericl Fibocci fucios. However If = e he he oher soluios c be obied. REFERECES Ahd TA Ezz RH (00). Geerl Ep -fucio ehod for olier evoluio equios. Appl. Mh. Copu. 7(): Bisws A (008). -solio soluio of he K() equio wih geerlized evoluio. Phys. Le. A 7(): Bisws A (00). -solio soluio of he K() equio wih geerlized evoluio d ie-depede dpig d dispersio. Copu. Mh. Appl. 9: Ebid A (0). A iprovee o he Ep-fucio ehod whe blcig he highes order lier d olier ers. J. Mh. Al. Appl. 9(): -. Gurefe Y Soezoglu A Misirli E (0). Applicio of he ril equio ehod for solvig soe olier evoluio equios risig i heicl physics. Pr-J. Phys. 77(6): 0-

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