Solving 2D-BKDV Equation by a Sub-ODE Method

Transcription

1 Internatonal Conference on Coputer Technology and Scence (ICCTS ) IPCSIT vol 47 () () IACSIT Press Sngapore DOI: 7763/IPCSITV4756 Solvng D-BKDV Equaton by a Sub-ODE Method Bn Zheng + School of Scence Shandong Unversty of Technology Zhangzhou Road Zbo Shandong Chna 5549 Abstract In ths paper we rve exact travelng wave solutons of D-BKDV equaton and D-BKDV equaton by a proposed Bernoull sub-ode ethod The ethod appears to be effcent n seekng exact solutons of nonlnear equatons Keywords: Bernoull sub-ode ethod travelng wave solutons exact soluton evoluton equaton BKDV equaton Introducton Durng the past four cas or so searchng for explct solutons of nonlnear evoluton equatons by usng varous dfferent ethods have been the an goal for any researchers and any powerful ethods for constructng exact solutons of nonlnear evoluton equatons have been establshed and veloped such as the nverse scatterng transfor the Darboux transfor the tanh-functon expanson and ts varous extenson the Jacob ellptc functon expanson the hoogeneous balance ethod the sne-cosne ethod the rank analyss ethod the exp-functon expanson ethod and so on [-3] In ths paper we proposed a Bernoull sub-ode ethod to construct exact travelng wave solutons for NLEES The rest of the paper s organzed as follows In Secton we scrbe the Bernoull sub-ode ethod for fndng travelng wave solutons of nonlnear evoluton equatons and gve the an steps of the ethod In the subsequent sectons we wll apply the ethod to fnd exact travelng wave solutons of the D-BKDV equaton and the D-BKDV equaton In the last Secton soe conclusons are presented Descrpton of the Bernoull Sub-ODE ethod In ths secton we present the solutons of the followng ODE: G' G G where G = G( ξ ) + = μ () When μ Eq () s the type of Bernoull equaton and we can obtan the soluton as G = μ + ξ () where d s an arbtrary constant Suppose that a nonlnear equaton say n two or three npennt varables x y and t s gven by Puuu ( u u u u u u ) = (3) t x y tt xt yt xx yy where u = u(x y t) s an unknown functon P s a polynoal n u = u(x y t) and ts varous partal rvatves + Correspondng author Tel: E-al address: zhengbn6@6co 3

2 n whch the hghest orr rvatves and nonlnear ters are nvolved By usng the solutons of Eq () we can construct a serals of exact solutons of nonlnear equatons: Step We suppose that uxyt ( ) = u( ξ ) ξ = ξ( xyt ) (4) the travelng wave varable (4) perts us reducng Eq (3) to an ODE for u = u( ξ ) Puu ( ' u '') = (5) Step Suppose that the soluton of (5) can be expressed by a polynoal n G as follows: u = G + G + (6) ( ξ) α α where G = G( ξ ) satsfes Eq () and α α are constants to be terned later α The postve nteger can be terned by consrng the hoogeneous balance between the hghest orr rvatves and nonlnear ters appearng n (5) Step 3 Substtutng (6) nto (5) and usng () collectng all ters wth the sae orr of G together the left-hand s of Eq (5) s converted nto another polynoal n G Equatng each coeffcent of ths polynoal to zero yelds a set of algebrac equatons for α α μ Step 4 Solvng the algebrac equatons syste n Step 3 and by usng the solutons of Eq () we can construct the travelng wave solutons of the nonlnear evoluton equaton (5) In the subsequent sectons we wll llustrate the proposed ethod n tal by applyng t to D-BKDV equaton and D-BKDV equaton 3 Applcaton Of the Bernoull Sub-ODE Method For D-BKDV Equaton In ths secton we wll consr the followng D-BKDV equaton: u + αuu + βu + γu = (3) t x xx xxx Suppose that uxt ( ) = u( ξ ) ξ = kx ( ct) (3) where ck are constants that to be terned later By (3) (3) s converted nto an ODE cu ' + αuu ' + βku '' + γk u ''' = (33) Integratng the ODE (33) once we obtan α cu + u + ku ' + k u '' = g (34) β γ where g s the ntegraton constant that can be terned later Suppose that the soluton of (34) can be expressed by a polynoal n G as follows: u( ξ ) = ag (35) = where a are constants and G = G( ξ ) satsfes Eq() Balancng the orr of u and u '' n Eq(34) we have = + = So Eq(35) can be rewrtten as u( ξ ) = a G + ag+ a a (36) where a a a are constants to be terned later 3

3 Substtutng (36) nto (34) and collectng all the ters wth the sae power of G together equatng each coeffcent to zero yelds a set of sultaneous algebrac equatons as follows: G : ca g+ αa = G : ca + αaa + βkaμ+ γk aμ = G : ca + βkaμ+ αaa βka+ αa + 4γk aμ 3γkaμ = G : βka γk a μ+ αaa + γk a = 3 4 G : αa + 6γk a = Solvng the algebrac equatons above yelds: Case : a β 5 a a a k β 6β + 5 α ( 5 ) = = = = a γ a β + α c g a γ = = (37) αγμ 5 γμ 5 γ 5 γ where a s an arbtrary constants Substtutng (37) nto (36) we get that β u( ξ ) = G + a β 6β + 5αa γ ξ = ( x t) (38) 5 αγμ 5 γ where kc are fned as n (37) Cobnng wth Eq () we can obtan the travelng wave solutons of (3) as follows: β ( ξ ) = ( ) + a 5 αγμ μ ξ u + where a d are arbtrary constants and Case : β 4 β β a = a = a = a k = 5 αγμ 5 αγμ 5 γμ where a s an arbtrary constants Substtutng (3) nto (36) we get that 6β + 5αa γ ( ) β ξ = x t 5 γ (39) 6β + 5 α a γ ( 5 ) c g a β + α a γ = = (3) 5 γ 5 γ β 4 β u( ξ ) = G G a 5 αγμ + 5 αγμ + β 6β + 5αa γ ξ = ( x t) (3) 5 γ where kc are fned as n (37) Cobnng wth Eq () we can obtan the travelng wave solutons of (3) as follows: β 4 β u( ξ) = ( ) + ( ) + a 5 αγμ μ ξ 5 αγμ μ ξ where a d β 6β + 5αa γ are arbtrary constants and ξ = ( x t) 5 γ Reark : Our results (39) and (3) are new fales of exact travelng wave solutons for Eq (3) 4 Applcaton Of the Bernoull Sub-ODE Method For D-BKDV Equaton In ths secton we wll consr the followng D-BKDV equaton: (3) ( u + αuu + βu + su ) + γu = (4) t x xx xxx x yy 3

4 Suppose that uxyt ( ) = u( ξ ) ξ = kx ( + y ct) (4) where ck are constants that to be terned later By (4) (4) s converted nto an ODE ( cu ' αuu ' βku '' sk u ''') ' γu '' = (43) Integratng the ODE (43) once we obtan cu ' + αuu ' + βku '' + sk u ''' + γu ' = g (44) where g s the ntegraton constant that can be terned later Suppose that the soluton of (44) can be expressed by a polynoal n G as follows: u( ξ ) = ag (45) = where a are constants and G = G( ξ ) satsfes Eq () Balancng the orr of uu ' and have + + = + 3 = So Eq(45) can be rewrtten as u ''' n Eq(44) we u( ξ ) = a G + ag+ a a (46) where a a a are constants to be terned later Substtutng (46) nto (44) and collectng all the ters wth the sae power of G together equatng each coeffcent to zero yelds a set of sultaneous algebrac equatons as follows: G : g = G : αaa + βka sk a γa ca = 3 G :ca caμ αa 3βkaμ αaa+ αaaμ = G : 3αaa μ ca μ+ 38sk a μ + βkaμ + αa a μ + γa μ sk μ a βka μ+ αa μ = 3 G : 54sak μ αa + 6sk μa+ 3αaaμ+ 6βkμa = 4 3 G :4sa k μ + αa μ = 5 3 Solvng the algebrac equatons above yelds: Case : a = β μ 4 5 s a β μ α = 5 sα a = a 5αas + 6β + 5γs β c = g = k = 5 s 5 s where a s an arbtrary constants Substtutng (47) nto (46) we get that u β μ 4 ( ) 5 G β μ ξ = sα + 5 sα G+ a β 5αas + 6β + 5γs ξ = ( x + y t) (48) 5 s 5 s where kc are fned as n (47) Cobnng wth Eq () we can obtan the travelng wave solutons of (4) as follows: β μ 4 β μ u( ξ ) = ( ) + ( ) + a 5 sα μ ξ 5 sα μ ξ where a d β 5αas + 6β + 5γs are arbtrary constants and ξ = ( x+ y t) 5s 5 s Case : β μ 5αas 6β + 5γs β a = a = a = a c= g = k = (4) 5 sα 5 s 5 s (47) (49) 33

5 where a s an arbtrary constants Substtutng (4) nto (46) we get that u β μ 4 ( ) 5 G β μ ξ = sα + 5 sα G+ a β 5αas 6β + 5γs ξ = ( x + y t) (4) 5 s 5 s where kc are fned as n (47) Cobnng wth Eq () we can obtan the travelng wave solutons of (4) as follows: βμ 4 βμ u( ξ) = ( ) + ( ) + a 5 sα μ ξ 5 sα μ ξ where a d are arbtrary constants and β 5αas 6β + 5γs ξ = ( x+ y t) 5s 5 s Reark : Our results (49) and (4) are new fales of exact travelng wave solutons for Eq (4) 5 Conclusons (4) We have seen that soe new travelng wave solutons of D and D-BKDV equaton are successfully found by usng the Bernoull sub-ode ethod The an ponts of the ethod are that assung the soluton of the ODE reduced by usng the travelng wave varable as well as ntegratng can be expressed by an -th gree polynoal n G where G = G( ξ ) s the general solutons of a Bernoull sub-ode equaton The postve nteger can be terned by the general hoogeneous balance ethod and the coeffcents of the polynoal can be obtaned by solvng a set of sultaneous algebrac equatons Also ths ethod can be used to any other nonlnear probles 6 References [] M Wang Soltary wave solutons for varant Boussnesq equatons Phys Lett A 99 (995) 69-7 [] EME Zayed HA Zedan KA Gepreel On the soltary wave solutons for nonlnear Hrota-Satsua coupled KdV equatons Chaos Soltons and Fractals (4) [3] L Yang J Lu K Yang Exact solutons of nonlnear PDE nonlnear transforatons and reducton of nonlnear PDE to a quadrature Phys Lett A 78 () 67-7 [4] EME Zayed HA Zedan KA Gepreel Group analyss and odfed tanh-functon to fnd the nvarant solutons and solton soluton for nonlnear Euler equatons Int J Nonlnear Sc Nuer Sul 5 (4) -34 [5] M Inc DJ Evans On travelng wave solutons of soe nonlnear evoluton equatons Int J Coput Math 8 (4) 9- [6] MA Abdou The extend tanh-ethod and ts applcatons for solvng nonlnear physcal ols Appl Math Coput 9 (7) [7] M Wang X L J Zhang The (G /G )-expanson ethod and travellng wave solutons of nonlnear evoluton equatons n atheatcal physcs Physcs Letters A 37 (8) [8] MJ Ablowtz PA Clarkson Soltons Nonlnear Evoluton Equatons and Inverse Scatterng Transfor Cabrdge Unversty Press Cabrdge 99 [9] MR Mura Backlund Transforaton Sprnger-Verlag Berln 978 [] C Rogers WF Shadwck Backlund Transforatons Acac Press New York 98 [] R Hrota Exact envelope solton solutons of a nonlnear wave equaton J Math Phys 4 (973) 85-8 [] R Hrota J Satsua Solton soluton of a coupled KdV equaton Phys Lett A 85 (98) [3] ZY Yan HQ Zhang New explct soltary wave solutons and perodc wave solutons for Whtha-Broer-Kaup equaton n shallow water Phys Lett A 85 ()