Approximate Analytic Solution of (2+1) - Dimensional Zakharov-Kuznetsov(Zk) Equations Using Homotopy

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1 Arcle Inernaonal Journal of Modern Mahemacal Scences, 4, (): - Inernaonal Journal of Modern Mahemacal Scences Journal homepage: ISSN: 66-86X Florda, USA Approxmae Analyc Soluon of (+) - Dmensonal Zakharov-Kuznesov(Zk) Equaons Usng Homoopy Perurbaon Mehod wh Hyperbolc & Perodc Inal Condons E.S. Fahmy Basc Scence Deparmen, Faculy of Engneerng, Ocober 6 Unversy E-Mal: esfahmy@yahoo.com Arcle hsory: Receved 4 December, Receved n revsed form 8 March 4, Acceped 4 March 4, Publshed Aprl 4. Absrac: In hs paper, we consruc an approxmae analyc soluon of (+) dmensonal Zakhorov-Kuznesov equaons usng Homoopy eraon mehod wh hyperbolc & perodc nal condons and he resul s compared wh he exac soluons obaned by sne-cosne mehod. Keywords: Homoopy eraon mehod; Zakhorov-Kuznesov equaons; sne-cosne mehod.. Inroducon The kdv equaon s consdered a spaally one-dmensonal model. The bes known wodmensonal generalzaons of kdv equaon are Kadomosov-Pevshvll (KP) equaons. Wazwaz [] exended he coupled Zakhorov-Kuznesov(ZK) equaons. The coupled ZK equaons descrbe he nonlnear developmen of on-acousc waves n magnezed plasma under he resrcons of small wave amplude, weak dsperson, and srong magnec felds. The physcal Copyrgh 4 by Modern Scenfc Press Company, Florda, USA

2 In. J. Modern Mah. Sc. 4, (): - phenomena for hs equaon were nvesgaed n [-4]. In hs paper we nroduce (+) - dmensonal ZK-equaons whch s gven as u ( uv ) ( vw ) ( u u ), x x xxx yyx v ( uw ) ( v v ), x xxx yyx w ( uv ) ( w w ). x xxx yyx () where,, and are arbary consans. In [5-] many soluons was obaned and one of hs soluons s he exac soluons obaned by sne-cosne mehod [5], egh cases of exac soluons was nroduced. In hs work we consder wo cases of hese soluons o be he nal condons for Homoopy perurbaon mehod [-]. The Homoopy mehod was nroduced by He [-6] n whose mehod he soluon was consdered as he summaon of an nfne seres whch usually converges readly o he exac soluon. Ths paper s organzed n he followng way: n secon we descrbe he Homoopy perurbaon mehod, n secon, we apply he Homoopy mehod o oban an analyc approxmae soluon for ZK- equaons () under wo knds of nal condons and we dscuss our numercal resuls usng fguers, show ha he approxmae soluons s comparson wh he exac soluons.. He Homoopy Perurbaon Mehod Consder he followng nonlnear dfferenal equaon A( u) f ( r), r () Wh he boundary condons: u B ( u, ), n where A r s he general dfferenal operaor, B funcon and s he boundary of he donan. Equaon () can be wren as () s he boundary operaor and f ( r ), s analyc L( u) N ( u) f ( r), (4) where Lu ( ) s he lnear par and N ( u ) s he nonlnear par. By usng Homoopy echnque, we can defne H ( v, p) ( p)[ L( u) L ( u)] p[ A( v ) f ( r)], r (5) Copyrgh 4 by Modern Scenfc Press Company, Florda, USA

3 In. J. Modern Mah. Sc. 4, (): - Or H ( v, p) L( u) L ( u) pl ( u) p( N ( v ) f ( r)), r (6) where p [,] s an embeddng condon, hen from equaons (5,6) we ge H ( v,) L( u ) L ( u ), H ( v,) A( u ) f ( r). (7) Accordng o he Homoopy echnque, we can frs use he parameer p as a small parameer and wre equaon () as a power seres n p, we have v v pv p v (8) If p n he above equaon, we ge approxmae soluon of (4) n he followng form: u lmv v pv p v (9) p The combnaon of he perurbaon mehod and he Homoopy mehod s called Homoopy perurbaon mehod whch elmnaons of he radonal mehod.. Mehod of Soluon By rewren equaon () as u ( uv vu ) ( vw wv ) ( u u ), x x x x xxx yyx v ( uw wu ) ( v v ), x x xxx yyx w ( uv vu ) ( w w ). x x xxx yyx () Applyng he Homoopy mehod, we can pu,, () u p u v p v w p w Subsung no equaon (), we ge Copyrgh 4 by Modern Scenfc Press Company, Florda, USA

4 In. J. Modern Mah. Sc. 4, (): - 4 x x ( p u ) u ( x, y,) [ p ( (( p u )( p v ) ( p v )( p u ) ) d p v v x ( ) ( x x p ( (( p w )( p v ) ( p v )( p w ) ) d xxx yyx p (( p u ) ( p u ) ) d xxx yyx x x x x, y,) [ p ( (( p u )( p w ) ( p w )( p u ) ) d p (( p v ) ( p v ) ) d ( p w ) w ( x, y,) [ p ( (( p u )( p v ) ( p v )( p u ) ) d p (( p v ) ( p v ) ) d xxx yyx () Comarng he coeffcen of he same powers of p, we ge he followng se of equaons u ( x, y, ) u ( x, y,), u ( x, y, ) [ ( u v v u ) ( v w w v ) ( u u )] d, x x x x xxx yyx u ( x, y, ) [ ( u v v u u v v u ) x x x x + ( v w w v v w w v ) ( u u )] d x x x x xxx yyx For v ( x, y, ) : v ( x, y, ) v ( x, y,), v ( x, y, ) [ ( u w w u ) ( v v )] d, x x xxx yyx v ( x, y, ) [ ( u w w u u w w u ) ( v v )] d x x x x xxx yyx And for w ( x, y, ) : w ( x, y, ) w ( x, y,), w ( x, y, ) [ ( u v v u ) ( w w )] d, x x xxx yyx w ( x, y, ) [ ( u v v u u v v u ) ( w w )] d x x x x xxx yyx ().. Case One Copyrgh 4 by Modern Scenfc Press Company, Florda, USA

5 In. J. Modern Mah. Sc. 4, (): - 5 form: From [5], we can oban he hyperbolc nal condon of ZK equaons n he followng u x y k x y (,,) sech ( ( )), v x y k x y (,,) sech ( ( )), w x y k x y (,,) sech ( ( )). (4) Where k c k c /, / 4 ( 4, k c c / 4 ( 4, and / 8. (5) Then equaon () gves us he frs few approxmaons of u( x, y, ) : u x y k x y (,,) sech ( ( )), u ( x, y,) 4 ( k k k k k k cosh ( ( x y )) 4 4 sech ( ( x y ))anh ( ( x y )) u ( x, y,) ( 46 k 96 k k 4k k 4k k 4 (8 k 7 k ( k k ) ( k k + k ( k k ) k ( k k )cosh( ( x y )) ( cosh( ( x y )) k cosh(6 ( x y ))sech (6 ( x y )). 4 8 (6) The frs few approxmaons of v ( x, y, ) : v x y k x y (,,) sech ( ( )), v ( x, y,) 4 ( k k k k cosh( ( x y )) 4 sech ( ( x y ))anh( ( x y )) v ( x, y,) ( 46 k 96 k k 4k k k 4k k 4k k 4 (8 k 7 k k k k ( k k k k )) cosh( ( x y )) ( k k k )cosh(4 ( x y )) k cosh(6 ( x y ))sech ( ( x y )). 4 8 (7) The frs few approxmaons of w ( x, y, ) : Copyrgh 4 by Modern Scenfc Press Company, Florda, USA

6 In. J. Modern Mah. Sc. 4, (): - 6 w x y k x y (,,) sech ( ( )), w ( x, y,) 4 ( k k k k cosh( ( x y )) 4 sech ( ( x y ))anh( ( x y )) w ( x, y,) ( 46 k 96 k k 4k k k 4k k 4 k k 4 (8 k 7 k k ( k k k k k k )) cosh( ( x y )) ( k k k )cosh(4 ( x y )) k cosh(6 ( x y ))sech ( ( x y )). 4 8 (8) Thus, we oban he hyperbolc approxmae soluons of () as followng u ( x, y, ) k sech ( ( x y )) 4 ( k k k k k k cosh ( ( x y ))sech ( ( x y ))anh ( ( x y )) 4 4 ( 46 k 96 k k 4k k 4k k 4 (8 k 7 k ( k k ) ( k k + k ( k k ) k ( k k )cosh( ( x y )) ( cosh( ( x y )) k cosh(6 ( x y ))sech (6 ( x y )). 4 8 v ( x, y, ) k sech ( ( x y )) 4 ( k k k k cosh( ( x y )) sech ( ( x y ))anh( ( x y )) ( 46 k 96 k k 4 4 4k k k 4k k 4 k k (8 k 7 k k ( k k k k ))cosh( ( x y )) ( k k k ) cosh(4 ( x y )) k cosh(6 ( x y ))sech ( ( x y )). 4 8 k k w ( x, y, ) k sech ( ( x y )) 4 ( k k k k 4 cosh( ( x y ))sech ( ( x y ))anh( ( x y )) ( 46 k 96 k k 4k k k 4 4k k 4 k k (8 k 7 k k ( k k k k k k ))cosh( ( x y )) ( k k k )cosh(4 ( x y )) k cosh(6 ( x y ))sech ( ( x y )). 4 8 (9) Copyrgh 4 by Modern Scenfc Press Company, Florda, USA

7 In. J. Modern Mah. Sc. 4, (): [a]: The approxmae soluon u( x, y, ), s comparson wh he exac soluon [b]: The approxmae soluon v ( x, y, ), s comparson wh he exac soluon [c]: The approxmae soluon w ( x, y, ), s comparson wh he exac soluon Fgure : The approxmae soluons s comparson wh he exac soluons for hyperbolc nal condons a.,. and y. Copyrgh 4 by Modern Scenfc Press Company, Florda, USA

8 In. J. Modern Mah. Sc. 4, (): Case Two From [5], we can oban he perodc nal condon of ZK equaons n he followng form: u x y k x y (,,) sec ( ( )), v x y k x y (,,) sec ( ( )), w x y k x y (,,) sec ( ( )). () Where k c k c /, / 4 ( 4 ), k c c / 4 ( 4, and / 8. Then equaon () gves us he frs few approxmaons of u( x, y, ) : u x y k x y (,,) sec ( ( )), u ( x, y,) 4 ( k k k k k k cos ( ( x y )) 4 4 sec ( ( x y ))an ( ( x y )) u ( x, y,) ( 8 k 7 k ( k k ) ( k k 4 k ( k k ) k ( k k )cos( ( x y )) ((64 k k k 74 k ( k k ) k ( k k ) k ( k k ) ( k k k k k )cos(4 ( x y )) k cos(6 ( x y ))sec ( ( x y )). 4 8 () The frs few approxmaons of v ( x, y, ) : v x y k x y (,,) sec ( ( )), v ( x, y,) 4 ( k k k k k k cos( ( x y )) 4 4 sec ( ( x y ))an ( ( x y )) v ( x, y,) ( 8 k 7 k k k ( k k k 4 k ))cos( ( x y )) ((64 k 74 k k k ( k k k k )) ( k k k ) cos(4 ( x y )) k cos(6 ( x y ))sec ( ( x y )). 4 8 () The frs few approxmaons of w ( x, y, ) : Copyrgh 4 by Modern Scenfc Press Company, Florda, USA

9 In. J. Modern Mah. Sc. 4, (): - 9 w x y k x y (,,) sec ( ( )), w ( x, y,) 4 ( k k k k cos( ( x y )) 4 4 sec ( ( x y ))an ( ( x y )) w ( x, y,) ( 8 k 7 k k ( k k k k 4 kk ))cos( ( x y )) ((64 k 74 k k k ( k k k k )) ( k k k ) cos(4 ( x y )) k cos(6 ( x y ))sec ( ( x y )). 4 8 () Thus, we oban he hyperbolc approxmae soluons of () as followng u ( x, y, ) k sec ( ( x y )) 4 ( k k k k k k cos ( ( x y ))sec ( ( x y ))an ( ( x y )) 4 4 ( 8 k 7 k ( k k ) ( k k k ( k 4 k ) k ( k k )cos( ( x y )) ((64 k k k 74 k ( k k ) k ( k k ) k ( k k ) ( k k k 8 k k )cos(4 ( x y )) k cos(6 ( x y ))sec ( ( x y )). 4 v ( x, y, ) k sec ( ( x y )) 4 ( k k k k k k cos( ( x y ))sec ( ( x y ))an ( ( x y )) 4 4 ( 8 k 7 k k k 4 ( k k k k ))cos( ( x y )) ((64 k 74 k k k ( k k k k )) ( k k k )cos(4 ( x y )) k 4 8 cos(6 ( x y ))sec ( ( x y )). w ( x, y, ) k sec ( ( x y )) 4 ( k k k k cos( ( x y ))sec ( ( x y ))an ( ( x y )) 4 4 ( 8 k 7 k k ( k k k k kk 4 ))cos( ( x y )) ((64 k 74 k k k ( k k k k )) ( k k k ) cos(4 ( x y )) k cos(6 ( x y ))sec ( ( x y )). 4 8 (4) Copyrgh 4 by Modern Scenfc Press Company, Florda, USA

10 In. J. Modern Mah. Sc. 4, (): [a]: The approxmae soluon 5 5 u( x, y, ), s comparson wh he exac soluon [b]: The approxmae soluon v ( x, y, ), s comparson wh he exac soluon [c]: The approxmae soluon w ( x, y, ), s comparson wh he exac soluon Fgure : The approxmae soluons s comparson wh he exac soluons for hyperbolc nal condons a.,. and y. 4. Conclusons Copyrgh 4 by Modern Scenfc Press Company, Florda, USA

11 In. J. Modern Mah. Sc. 4, (): - In hs paper we consruced an analyc approxmae soluons for (+) dmensonal Zakhorov-Kuznesov(zk)-equaons usng Homoopy perurbaon mehod, under wo knds of nal condons, we have obaned resuls wh excellen accuracy as shown n fgures, where he approxmae soluons are successfully confrm o he exac soluons. References [] A. M. Wazwaz, Compleely negerable coupled kdv and coupled kp-sysem, Commun Nonlnear Sc. do:.6 lj. cnsns. 9. [] J. WU, New explc ravellng wave soluons for hree nonlnear evaluaon equaon. Appl. mah. compu., 7(): [] Z Y. Qn, A fne - dmensonal negerable sysem relaed o a new coupled kdv equaon. Phys. Le. A, 55(6): [4] A. M. Wazwaz, The exended anh mehod for abundan solary wave soluons of nonlnear wave equaons, Appl. Mah. Compu., 87(7): -4. [5] Y. Xe, S. Tang, Sne-Cosne mehod for new Coupled Zk-sysem, App. Mah. Scence. 5()(): [6] S. Monro, E. J. Parkes, The dervaon of a modfed ZakharovK- zunesov equaon and he sably of s soluons, Journal of Plasma Physcs, 6()(999): 5-7. [7] S. Munro, EJ. Parkes. Sably of solary-wave soluons o a modfed ZakharovCK - uznesov equaon, J Plasma Phys., 64(): [8] C. M. Khalgue, Exac Explc Soluons and Conservaon Lows for a coupled Zakhorov- Kuzneov sysem, problems n Engneerng, (), ID. 467, 5 pages. [9] M. Inc, Exac soluons wh solary paerns for he Zakharov-Kuznesov equaons wh fully nonlnear dsperson, Chaos Solons Fracals, (5)(7): [] J. Wu. New explc ravellng wave soluons for hree nonlnear evoluon equaons, Appl Mah Compu., 7(): , [] Kanglgl, F and F. A yaz, Solary wave Soluon for kdv and M kdv equaons by dfferenonal ransform mehod, Chaos solons and fracals, do :6/j. Chaos..9. [] J. H. He, Homoopy perurbaon mehod for nonlnear oscllaors wh dsconnues, Appled Mahemacs and Compuaon, 5()( 4): [] J. H. He, Comparson of Homoopy perurbaon mehod and Homoopy analyss mehod, Appled Mahemacs and Compuaon, 56()( 4): [4] J.H.He, Asympoology by Homoopy perurbaon mehod, Appled Mahemacs and Compuaon, 56()(4): Copyrgh 4 by Modern Scenfc Press Company, Florda, USA

12 In. J. Modern Mah. Sc. 4, (): - [5] J.H.He, Homoopy perurbaon mehod for solvng boundary value problems, Physcs Leers. A, 5(-)(6): [6] J.H.He, Applcaon of Homoopy perurbaon mehod o nonlnear wave equaons, Chaos, Solons and Fracals, 6()(5): [7] Targ. M. Elzak and J. Bazar, Homoopy Perurbaon Mehod and Elzak Transform for Solvng Sysem of Nonlnear Paral Dfferenal Equaons, World Appl. Sc. J.,4(7)(): [8] Sharma, P.R. and Grraj Meh,. Applcaons of Homoopy Perurbaon mehod o Paral dfferenal equaons, Asan Journal of Mahemacs and Sascs, 4()(): 4-5. [9] J. Bazar, F. Badpemaa, F. Azm, Applcaon of he homoopy perurbaon mehod o Zakharov-Kuznesov equaons, Compuers and Mahemacs wh Applcaons, 58(9): [] D.D.Ganj, H. Babazadeh, F. Noor, M.MProuze, M.Janpour, An Applcaon of Homoopy perurbaon mehod for Non-lnear Blasus equaon o Boundary Layer Flow Over a Fla, plae, In. J. of nonlnear Sc, (7)(4)(9): Copyrgh 4 by Modern Scenfc Press Company, Florda, USA