The modified Exp-function method and its applications to the generalized K(n,n) and BBM equations with variable coefficients

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1 IJST () A3 (Special isse-mahemaics): Iraia Joral of Sciece & Techology hp:// The modified Ep-fcio mehod ad is applicaios o he geeralized K() ad BBM eqaios wih varile coefficies E. M. E. Zayed* ad M. A. M. Abdelaziz Mahemaics Deparme Facly of Sciece Zagazig Uiversiy Zagazig Egyp e.m.e.zayed@homail.com & mamdelaziz@homail.com Absrac I his aricle he modified ep-fcio mehod is sed o cosrc may eac solios o he oliear geeralized K() ad BBM eqaios wih varile coefficies. Uder differe parameer codiios eplici formlas for some ew eac solios are sccessflly obaied. The proposed solios are fod o be impora for he eplaaio of some pracical physical problems. Keywords: Geeralized K() eqaio wih varile coefficies; geeralized BBM eqaio wih varile coefficies; eac ravelig wave solios; Ep-fcio mehod. Irodcio I he oliear sciece may impora pheomea i varios fields ca be described by he oliear evolio eqaios (NLEEs). Searchig for eac solio solios of NLEEs plays a impora ad sigifica role i he sdy o he dyamics of hose pheomea. Wih he developme of solio heory may powerfl mehods for obaiig he eac solios of NLEEs have bee preseed sch as he eeded ahfcio mehod [-5] he ah-sech mehod [6-8] he sie-cosie mehod [9-] he homogeeos balace mehod [ 3] he ep-fcio mehod [4-7] he Jacobi ellipic fcio mehod [8-] he F-epasio mehod [] he homoopy perrbaio mehod [3 4] he variaioal ieraio mehod [5] he iverse scaerig rasformaio mehod [6] he Bäckld rasformaio mehod [7] he Hiroa biliear mehod [8 9] ad so o. To or kowledge mos of he aforemeioed mehods are relaed o cosa coefficies models. Recely mch aeio has bee paid o he varile-coefficie oliear eqaios which ca describe may oliear pheomea more realisically ha heir cosacoefficie oes. The objecive of his aricle is o apply he modified ep-fcio mehod sig a geeralized wave rasformaio o fid he eac solios of he followig wo oliear dispersive eqaios wih varile coefficies: *Correspodig ahor Received: 8 December / Acceped: 8 Febrary. The geeralized K() eqaio wih varile coefficies [3] a( ) b( )( ) k ( )( ) () where a() b() ad k() are ozero fcios of. As a model ha characerizes log waves i oliear dispersive media Eq. () was formally derived o describe he propagaio of srface waer waves i a iform chael. Now i has bee eslished ha he eqaio provides a model for o oly he srface waves of log wavelegh i liqids b also hydromageic waves i cold plasma acosic waves i aharmoic crysals ad acosic graviy waves i compressible flids. Eq. () has bee discssed by Wazwaz [3] sig siecosie mehod ad ah-mehod whe he fcios a() b() ad k() are ozero cosas.. The geeralized Bejami-Boa-Mahoy (BBM) eqaio wih varile coefficies [3] a( ) b( )( ) k ( )( ) () where a() b() ad k() all are ozero fcios of. The case = wih cosa coefficies whe a()=b()=k()= correspods o he BBM eqaio which was firs proposed by Bejami e al. [3]. Eq. () is a aleraive o Eq. () ad also describes he idirecioal propagaio of smallamplide log waves o he srface of waer i a chael. This eqaio is o oly coveie for shallow waer waves b also for hydromageic

2 IJST () A3 (Special isse-mahemaics): - 36 ad acosic waves ad herefore i has some advaages compared wih he KdV eqaio. Whe = Eq. () is called he modified BBM eqaio. Lv e al [3] have discssed Eqs. () ad () sig he ailiary differeial eqaio mehod ad obaied varios eac ravelig wave solios.. Descripio of he modified ep-fcio mehod To illsrae he basic idea of ep-fcio mehod he followig oliear evolio eqaios are cosidered: (...). (3) Where =() is a kow fcio is a polyomial i ad is parial derivaives i which he highes order derivaives ad oliear erms are ivolved. I he followig we give he mai seps of his mehod: Sep. Usig he geeralized wave rasformaio ( ) ( ) h ( d ) (4) where h is a cosa while τ() is a iegrle fcio of o be deermied. The Eq. (3) is redced o he followig ODE: p h h h ( )...) ' ' '' ( ( ) ( ) '' '' (5) d where ' ad P is a polyomial i ad is oal d derivaives. Sep. The ep-fcio mehod is based o he assmpio ha ravelig wave solios for Eq. (5) ca be epressed i he followig form d a ep c ( ) q b m epm m p (6) where c d p ad q are posiive iegers which are kow o be deermied laer a ad b m are kow cosas. Eq. (6) ca be wrie i he form ac ep( c)... ad ep( d) ( ) a ep( p)... a ep( d) p q (7) Sep 3. Deermie he vales of c ad p by balacig he liear erm of highes order i Eq. (5) wih he highes order oliear erm. Similarly deermie he vales of d ad q by balacig he liear erm of lowes order i Eq. (5) wih he lowes order oliear erm. Sep 4. Sbsie (7) io Eq. (5) ad collec all erms wih he same order of e ogeher he lefhad side of Eq. (5) is covered io a polyomial i e. The se each coefficie of his polyomial o zero o derive a se of algebraic eqaios for a b. i j Sep 5. Solve he algebraic sysem obaied i Sep4 by he se of Maple or Mahemaica. The we ca obaie he eac solios of Eq. (3). 3. Applicaio I his secio he modified ep-fcio mehod is applied o cosrc he eac solios of wo oliear evolio eqaios wih varile coefficies via he oliear geeralized K() eqaio wih varile coefficies () ad he oliear geeralized Bejami-Boa-Mahoy (BBM) eqaio wih varile coefficies (). 3.. Eample. The geeralized K() eqaio wih varile coefficies I order o obai he eac solios of Eq. () i is assmed ha he solio of his eqaio ca be wrie i he form ( ) ( ) (8) Sbsiig (8) io () we have a 4 4 ( ) ( ) 6 5 k ( ) ( ) 3 ( ) k ( ) ( ) ( ) ( ) b ( ) ( ) ( ) 3 ( )(3 ) k ( ) ( ) ( ) ( ). (9) Usig he geeralized wave rasformaio (4) Eq. (9) covers o he oliear ODE: ( ) ( ) ha ( ) 4 ' 3 3 ' (6 5 ) h k ( ) 3 ' '' 3 ( )(3 ) h k( ) ( ) ( ) ( ) h k 3 ''' ( ) ( ) ( ) ( ) ' ( ) hb ( ) ( ) ( ). () Usig he asaz (7) sice here is o liear erm i Eq. () i order o deermie he vales of c d p ad q we balace he oliear erm of highes

3 36 IJST () A3 (Special isse-mahemaics): - ''' order ( ) ( ) wih he oliear erm 4 ' ( ) ( ) as follows: ''' c ep[(3c7 p) ( ) ( ) c ep[ p 4 4 ' c3 ep[(5c 5 p) ( ) ( ) c ep[ p () () where c i are coefficies for simpliciy. By balacig he highes order of ep-fcio i Eqs. () ad () we have 3c7p 5c 5 p which leads o p c. (3) Similarly from he asaz (7) we have '''... d ep[ (3d 7 q) ( ) ( )... d ep[ q 4 '... d3 ep[ (5d 5 q) ( ) ( )... d ep[ q 4 (4) (5) where di are coefficies for simpliciy. By balacig he lowes order of ep-fcio i Eqs. (4) ad (5) we have 3d 7q 5d 5 q which leads o d q. (6) Choosig p = c = ad q = d = Eq. (7) becomes a ep( ) a a ep( ) ( ). bep( ) b b ep( ) (7) Sbsiig Eq. (7) io Eq. () ad by he help of Maple we ge e 9e 8e 7e 6e 5e A 4 3 e e e e (8) where A e be b b e (9) ad are coefficies of e. Eqaig all he coefficies of e o zero we obai a sysem of algebraic eqaios which ca be solved by he Maple o obai he followig cases of solios Case. a a a a b b b b ( ) h a ( ) k ( ) 4 h a ( ) a ( ) b ( ) b ( ) () I his case he eac solio of Eq. () has he ae ( ) b Case. () a a a a b b b b ( ) h a ( ) k ( ) 4 h a ( ) a ( ) b ( ) b ( ) () I his case he eac solio of Eq. () has he a ( ) be Case 3. (3) a a a a b b b b b ( ) ( ) h a ( ) k ( ) 4 h a ( ) a ( ) b ( ) b ( ) (4) I his case he eac solio of Eq. () has he a e ( ) b (5) Case 4. a a a a b b b b ( ) h a ( ) k ( ) 6h a ( ) a ( ) b ( ) b ( ) (6) I his case he eac solio of Eq. () has he a ( ) be (7)

4 IJST () A3 (Special isse-mahemaics): - 36 Case 5. a a a a b b b b ( ) h a ( ) k ( ) 6h a ( ) a ( ) b ( ) b ( ) (8) I his case he eac solio of Eq. () has he ae ( ) b Case 6. (9) a a a a b b b b ( ) h a ( ) k ( ) 4 h a ( ) a ( ) b ( ) b ( ) (3) I his case he eac solio of Eq. () has he ae ( ) b Case 7. a (3) a a a a a b b b b ( ) h a ( ) k ( ) 4 h a ( ) a ( ) b ( ) b ( ) (3) I his case he eac solio of Eq. () has he a a a e ( ) b a ae Case 8. a a a a a b b b b (33) a ( ) h a ( ) k ( ) 4 h a ( ) a ( ) b ( ) b ( ) (34) I his case he eac solio of Eq. () has he a a ae ( ) b a ae where ξ i he ove cases -8 has he form (35) h a() d. (36) Case 9. a a a a b b b b b h b b ( ) b ( ) a a ( ) ( ) a b( ) k () a () a () b () b () (37) 4h I his case he eac solio of Eq. () has he a ( ) be be where (38) bb ( ) b() aa() h d. a (39) 3.. Eample. The geeralized BBM eqaio wih varile coefficies I order o obai he eac solios of Eq. () we assme ha he solio of his eqaio has he same from (8). Sbsiig (8) io () we have a 4 4 ( ) ( ) 6 5 k ( ) ( ) ( ) ( )(3 ) k( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) k b ( ) 3 k ( ) ( ) ( ) ( ). (4) Usig he geeralized wave rasformaio (4) Eq. (4) covers o he oliear ODE: ( ) ( ) ha ( ) 4 ' 3 ' (6 5 ) h k ( ) ( ) ' '' 3 ( )(3 ) h k( ) ( ) ( ) ( ) ( ) ''' ( ) h k ( ) ( ) ( ) ( ) ' ( ) hb ( ) ( ) ( ). (4) Sbsiig Eq. (7) io Eq. (4) ad by he help of Maple we ge

5 363 IJST () A3 (Special isse-mahemaics): e 9e 8e 7e 6e 5e B 4 3 e e e e (4) where B e be b b e (43) ad are coefficies of e. Eqaig all he coefficies of e o zero we obai a sysem of algebraic eqaios which ca be solved by he Maple ad ge he followig cases of solios Case. a a a a b b b b ( ) h a ( ) k ( ) 4 h a ( ) a ( ) a ( ) b ( ) b ( ) (44) I his case he eac solio of Eq. () has he ae ( ) b Case. (45) a a a a b b b b ( ) h a ( ) k ( ) 4 h a ( ) a ( ) a ( ) b ( ) b ( ) (46) I his case he eac solio of Eq. () has he a ( ) be Case 3. (47) a a a a b b b b b ( ) ( ) h a ( ) k ( ) 4 h a ( ) a ( ) a ( ) b ( ) b ( ) (48) I his case he eac solio of Eq. () has he ae ( ) b (49) Case 4. a a a a b b b b ( ) h a ( ) k ( ) 4 h a ( ) a ( ) a ( ) b ( ) b ( ) (5) I his case he eac solio of Eq. () has he a e ( ) b Case 5. (5) a a a a b b b b b ( ) ( ) h a ( ) k ( ) 4 h a ( ) a ( ) a ( ) b ( ) b ( ) (5) I his case he eac solio of Eq. () has he e a ( ) b Case 6. a a a a a b b b b (53) a b ( ) ( ) h a ( ) k ( ) 6 h a ( ) a ( ) a ( ) b ( ) b ( ) (54) I his case he eac solio of Eq. () has he a a ae ( ) b a ae Case 7. a (55) a a a a a b b b b b ( ) ( ) h a ( ) k ( ) 4 h a ( ) a ( ) a ( ) b ( ) b ( ) (56) I his case he eac solio of Eq. () has he

6 IJST () A3 (Special isse-mahemaics): a a a e ( ) b a ae (57) where ξ i ove cases -7 has he same form (36). Case 8. a a a a b b b b b ha a( ) a 8h bb k( ) () a ( ) a ( ) b ( ) b ( ) a 8h bb k( ) b( ) k (). 4 haa ( ) (58) I his case he eac solio of Eq. () has he a ( ) be be where ξ i he ove cases -8 has he form h (59) a a( ) d. a 8h (6) bb k( ) Remark: All solios of his aricle have bee checked wih Maple by pig hem back io he origial eqaios () ad (). 4. Coclsios I his aricle he modified ep-fcio mehod was applied i order o fid he eac solios of he geeralized K() ad BBM eqaios wih varile coefficies. The ep-fcio mehod is a very powerfl ad effecive echiqe i fidig he eac solios for a wide rage of problems. The solios so obaied have also bee verified o saisfy he origial eqaio. This mehod is impora becase he solios obaied ca be applied o a wide rage of problems i sciece ad egieerig. Refereces [] El-Wakil S. A. & Abdo M. A. (7). New eac ravelig wave solios sig modified eeded ahfcio mehod. Chaos Solios ad Fracals [] Fa E. (). Eeded ah-fcio mehod ad is applicaios o oliear eqaios. Phys. Le. A [3] Wazwaz A. M. (7). The ah-mehod for ravelig wave solios of oliear wave eqaios. Appl. Mah. Comp [4] Zayed E. M. E. & AbdelRahma H. M. (). The eeded ah-mehod for fidig ravelig wave solios of oliear PDEs. Noliear Sci. Le. A () 93-. [5] Zayed E. M. E. & Abdelaziz M. A. M. (). The ah-fcio mehod sig a geeralized wave rasformaio for oliear eqaios. I. J. Noliear Sci. Nmer. Simla [6] Malflie W. (99). Soliary wave solios of oliear wave eqaios. Am. J. Phys [7] Malflie W. & Herema W. (996). The ahmehod par I. Eac solios of oliear evolio ad wave eqaios. Phys. Scrip [8] Wazwaz A. M. (4). The ah-mehod for ravelig wave solios of oliear eqaios. Appl. Mah. Comp [9] Yaghobi Moghaddam M. Asgari A. & Yazdai H. (9). Eac ravellig wave solios for he geeralized oliear Schrödiger (GNLS) eqaio wih a sorce by Eeded ah-coh sie-cosie ad Ep-Fcio mehods. Appl. Mah. Comp [] Wazwaz A. M. (4). A sie-cosie mehod for hadlig oliear wave eqaios. Mah. Comp. Model [] Yag Y. Tao Z. L. & Asi R. F. (). Solios of he geeralized KdV eqaio wih imedepede dampig ad dispersio. Appl. Mah. Comp [] Fa E. & Zhag H. (998). A oe o he homogeeos balace mehod. Phys. Le. A [3] Wag M. L. (996). Eac solios of a compod Kdv-Brgers eqaios. Phys. Le. A [4] He J. H. & W X. H. (6). Ep-fcio mehod for oliear wave eqaios. Chaos Solios ad Fracals [5] He J. H. & Abdo M. A. (7). New periodic solios for oliear evolio eqaios sig Epfcio mehod. Chaos Solios ad Fracals [6] Zhag S. (8). Applicaio of ep-fcio mehod o high dimesioal evolio eqaio. Chaos Solios ad Fracals [7] Zh S. D. (7). Ep-fcio mehod for he discree mkdv laice. I. J. Noliear Sci. Nmer. Simla [8] Dai C. Q. & Zhag J. F. (6). Jacobia ellipic fcio mehod for oliear differeial-differece eqaios. Chaos Solios ad Fracals [9] Fa E. & Zhag J. (). Applicaios of he Jacobi ellipic fcio mehod o special-ype oliear eqaios. Phys. Le. A [] Li S. F Z. Li S. & Zhao Q. (). Jacobi ellipic fcio mehod ad periodic wave solios of oliear wave eqaios. Phys. Le. A [] Zhao X. Q. Zhi H. Y. & Zhag H. Q. (6). Improved Jacobi-fcio mehod wih symbolic compaio o cosrc ew doble-periodic

7 365 IJST () A3 (Special isse-mahemaics): - solios for he geeralized Io sysem. Chaos Solios ad Fracals 8-6. [] Zhag J. L. Wag M. L. Wag Y. M. & Fag Z. D. (6). The improved F-epasio mehod ad is applicaios. Phys Le. A [3] He J. H. (6). New ierpreaio of homoopy perrbaio mehod. I. J. Mod. Phys. B [4] He J. H. (5). Asympology by homoopy perrbaio mehod. I. J. Noliear Sci. Nmer. Siml [5] He J. H. & W X. H. (6). Cosrcio of soliary solios ad compaco like solio by variaioal ieraio mehod. Chaos Solios ad Fracals [6] Ablowiz M. J. & Segr H. (98). Solios ad he iverse scaerig rasform. Philadelphia SIAM. [7] Hiroa R. (98). Direc mehod of fidig eac solios of oliear evolio eqaios. I: Bllogh R Cadrey P ediors. Bäckld rasformaio Berli Spriger. [8] Hiroa R. (4). The direc mehod i solio heory. Cambridge Cambridge Uiversiy Press. [9] Wazwaz A. M. (). O mliple solio solios for copled KdV-mKdV eqaio. Noliear Sci. Le. A (3) [3] Lv X. Lai S. & W Y. (). The physical srcres of solios for geeralized K() ad BBM eqaios wih varile coefficies. Mahemaical ad Comper Modellig [3] Wazwaz A. M. (6). Two relile mehods for solvig varias of he KdV eqaio wih compac ad ocompac srcres. Chaos Solios ad Fracals [3] Bejami T. B. Boa J. L. & Mahoy J. J. (97). Model eqaios for log waves i oliear dispersive sysems. Philos Tras R Soc Lodo Ser A