Exact Solutions for Nonlinear D-S Equation by Two Known Sub-ODE Methods

Transcription

1 Internatonal Conference on Compter Technology and Scence (ICCTS ) IPCSIT vol. 47 () () IACSIT Press, Sngapore DOI:.7763/IPCSIT..V47.64 Exact Soltons for Nonlnear D-S Eqaton by Two Known Sb-ODE Methods Qngha Feng School of Scence, Shandong Unversty of Technology, Zhangzho Road, Zbo, Shandong, Chna, 5549 Abstract. In ths paper, we derve exact travelng wave soltons of nonlnear D-S eqaton by a proposed Bernoll sb-ode method and the known ( /) expanson method. Keywords: Bernoll sb-ode method, ( /) expanson method, travelng wave soltons, exact solton, evolton eqaton, nonlnear D-S eqaton. Introdcton In scentfc research, seekng the exact soltons of nonlnear eqatons s a hot topc. Many approaches have been presented so far [-7]. In ths paper, we proposed a Bernoll sb-ode method to constrct exact travelng wave soltons for NLEES. The rest of the paper s organzed as follows. In Secton, we descrbe the Bernoll sb-ode method for fndng travelng wave soltons of nonlnear evolton eqatons, and gve the man steps of the method. In the sbseqent sectons, we wll apply the Bernoll Sb-ODE method and the known ( /) expanson method to fnd exact travelng wave soltons of the nonlnear D-S eqaton. In the last Secton, some conclsons are presented.. Descrpton of the Bernoll Sb-ODE method In ths secton we present the soltons of the followng ODE: ' =, (.) where, = ( ξ ) When, Eq. (.) s the type of Bernoll eqaton, and we can obtan the solton as = de ξ, (.) where d s an arbtrary constant. Sppose that a nonlnear eqaton, say n two or three ndependent varables x, y and t, s gven by Correspondng athor. Tel.: E-mal address: fqha@sna.com 34

2 P (,,,,,,,...) (.3) t, x y tt xt yt xx yy where = (x, y, t) s an nknown fncton, P s a polynomal n = (x, y, t) and ts varos partal dervatves, n whch the hghest order dervatves and nonlnear terms are nvolved. By sng the soltons of Eq. (.), we can constrct a serals of exact soltons of nonlnear eqatons:. Step.We sppose that xyt (,, ) = ( ξ ), ξ = ξ( xyt,, ) (.4) the travelng wave varable (.4) permts s redcng Eq. (.3) to an ODE for = ( ξ ) P (, ', '',...) (.5) Step. Sppose that the solton of (.5) can be expressed by a polynomal n as follows: = (.6) m m ( ξ) αm αm... where = ( ξ ) satsfes Eq. (.), and αm, αm... are constants to be determned later, αm. The postve nteger m can be determned by consderng the homogeneos balance between the hghest order dervatves and nonlnear terms appearng n (.5). Step 3. Sbstttng (.6) nto (.5) and sng (.), collectng all terms wth the same order of together, the left-hand sde of Eq. (.5) s converted nto another polynomal n. Eqatng each coeffcent of ths polynomal to zero, yelds a set of algebrac eqatons for αm, αm,...,. Step 4. Solvng the algebrac eqatons system n Step 3, and by sng the soltons of Eq. (.), we can constrct the travelng wave soltons of the nonlnear evolton eqaton (.5). In the sbseqent sectons we wll llstrate the proposed method n detal by applyng t to nonlnear D-S eqaton. 3. Applcaton Of the Bernoll Sb-ODE Method For nonlnear D-S Eqaton In ths secton, we wll consder the followng nonlnear D-S eqatons: t = ( v ) x (3.) v v 3v 3v (3.) t xxx x x Spposng that ξ = kx ωt (3.3) By (3.3), (3.) and (3.) are converted nto ODEs ω' k( v )' (3.4) 3 ωv ' k v ''' 3 kv ' 3 kv ' (3.5) Integratng (3.4) and (3.5) once, we have ω kv = g (3.6) 343

3 = (3.7) 3 ωv k v'' 3kv g Sppose that the solton of (3.6) and (3.7) can be exp-ressed by a polynomal n as follows: m ( ξ ) = a (3.8) n v( ξ ) = b (3.9) where a, b are constants, = ( ξ ) satsfes Eq. (.). Balancng the order of and v n Eq. (3.6), the order of v '' and v n Eq. (3.7), we can obtan m= n, m = m n m=, n=. So Eq.(3.8) and (3.9) can be rewrtten as ( ξ ) = a a a, a (3.) v( ξ ) = b b, b (3.) where a, a, a, b, b are constants to be determned later. Sbstttng (3.) and (3. ) nto (3.6) and (3.7) and collectng all the terms wth the same power of together and eqatng each coeffcent to zero, yelds a set of smltaneos algebrac eqatons. Solvng the algebrac eqatons above, yelds: b a =, a = k, a = k, b 3 b b,,,, k 3 3 = b b k k g ω = = = =, k 4 b ( 3 b k ) g = (3.) 3 4 4k where b Sbstttng (3.) nto (3.) and (3.), yelds: b ( ξ) = k k (3.3) 3 3 k v( ξ ) b = b (3.4) where 3 b ξ = kx t., and b k Sbstttng the general soltons of (.) nto (3.3) and (3.4), we obtan the travelng wave soltons of nonlnear D-S eqatons as follows: b ( ξ) = k ( ) k ( ) 3 ξ de 3 ξ de k b v( ξ ) = b ( ) de ξ (3.5) (3.6) where b, k,, are arbtrary constants. 4. Applcaton Of ( /) expanson Method For nonlnear D-S Eqaton 344

4 In ths secton, we apply the ( /) expanson method to obtan the travelng wave soltons of nonlnear D-S eqatons (3.)-(3.). ' Sppose that the solton of (3.6) and (3.7) can be exp-ressed by a polynomal n ( ) as follows: m ' ( ξ ) a ( ) = (4.) v n ' ( ξ ) b ( ) = (4.) where a, b are constants, = ( ξ ) satsfes the second order LODE n the form: '' ' (4.3) where and are constants. Balancng the order of and v n Eq.(4.6), the order of v '' and v n Eq.(4.7), we can obtan m= n, m = m n m=, n=. So Eq.(4.) and (4.) can be rewrtten as ( ξ ) = a ( ) a ( ) a, a (4.4) ' ' v( ξ ) = b ( ) b, b (4.5) ' where a, a, a, b, b are constants to be determned later. ' Sbstttng (4.4) and (4.5 ) nto (3.6) and (3.7) and co-llectng all the terms wth the same power of ( ) together and eqatng each coeffcent to zero, yelds a set of smltaneos algebrac eqatons as follows: For Eq.(4.6): ' ( ) : ωa g kb ' ( ) : ωa kbb ' ( ) : kb ωa For Eq.(4.7): ' 3 ( ) : ωb g k b 3kab ' 3 3 ( ) : kb 3kab ωb kb 3kba ' 3 ( ) :3kab 3k b 3kba ' 3 3 ( ) : kb 3kba Solvng the algebrac eqatons above, yelds: b 4k a = k, a = k, a =, b = b, b = b, 3 b b ( 3b 4 k k ) k = k, ω =, g =, g 3 (4.6) k k 4k where b Sbstttng (4.6) nto (4.4) and (4.5), yelds: 345

5 4 ' ' 3b 4k ( ξ) = k ( ) k ( ) k (4.7) ' v( ξ) = b( ) b (4.8) where 3b ξ = kx t. k Sbstttng the general soltons of (4.3) nto (4.7) and (4.8), we have: When 4 > k k 6 6 ( ) ( 4 ). ξ = b 4 ( ξ) = b. v Csnh 4ξ Ccosh 4ξ ( ) Ccosh 4ξ Csnh 4ξ Csnh 4ξ Ccosh 4ξ ( ) Ccosh 4ξ Csnh 4ξ 4 3b 4k 6 k b where 3b ξ = kx t, b k When 4 < v k k 6 6 ( ξ ) = (4 ). b 4 ( ξ) = b. Csnh 4 ξ Ccosh 4 ξ ( ) Ccosh 4 ξ Csnh 4 ξ Csnh 4 ξ Ccosh 4 ξ ( ) Ccosh 4 ξ Csnh 4 ξ 4 3b 4k 6 k b where 3b ξ = kx t, b k When 4 k kc 3b 4k 4 3( ξ ) = 6 3( C Cξ ) 6 k b( C C C ξ) v ( ξ ) = b 3 ( C Cξ ) where 3b ξ = kx t, b k Remark: As one can see from Secton III and Secton IV, the travelng wave soltons obtaned by the Bernoll Sb-ODE method are dfferent from those by the known ( /) expanson method 5. Conclsons We have seen that some new travelng wave soltons of nonlnear D-S eqaton are sccessflly fond by sng the Bernoll sb-ode method. The man ponts of the method are that assmng the solton of the ODE redced by sng the travelng wave varable as well as ntegratng can be expressed by an m -th degree 346

6 polynomal n, where = ( ξ ) s the general soltons of a Bernoll sb-ode eqaton. The postve nteger m can be determned by the general homogeneos balance method, and the coeffcents of the polynomal can be obtaned by solvng a set of smltaneos algebrac eqatons. Also we make a comparson between the proposed method and the known ( /) expanson method. The Bernoll Sb-ODE method method can be appled to many other nonlnear problems. 6. References [] M. Wang, Soltary wave soltons for varant Bossnesq eqatons, Phys. Lett. A 99 (995) [] E.M.E. Zayed, H.A. Zedan, K.A. epreel, On the soltary wave soltons for nonlnear Hrota-Satsma copled KdV eqatons, Chaos, Soltons and Fractals (4) [3] L. Yang, J. L, K. Yang, Exact soltons of nonlnear PDE nonlnear transformatons and redcton of nonlnear PDE to a qadratre, Phys. Lett. A 78 () [4] E.M.E. Zayed, H.A. Zedan, K.A. epreel, rop analyss. and modfed tanh-fncton to fnd the nvarant soltons and solton solton for nonlnear Eler eqatons, Int. J. Nonlnear Sc. Nmer. Sml. 5 (4) -34 [5] M. Inc, D.J. Evans, On travelng wave soltons of some nonlnear evolton eqatons, Int. J. Compt. Math. 8 (4) 9- [6] M.A. Abdo, The extended tanh-method and ts applcatons for solvng nonlnear physcal models, Appl. Math. Compt. 9 (7) [7] E.. Fan, Extended tanh-fncton method and ts applcatons to nonlnear eqatons, Phys. Lett. A 77 () -8. [8] W. Malflet, Soltary wave soltons of nonlnear wave eqatons, Am. J. Phys. 6 (99) [9] J.L. H, A new method of exact travelng wave solton for copled nonlnear dfferental eqatons, Phys. Lett. A 3 (4) -6. [] M.J. Ablowtz, P.A. Clarkson, Soltons, Nonlnear Evolton Eqatons and Inverse Scatterng Transform, Cambrdge Unversty Press, Cambrdge, 99. [] M.R. Mra, Backlnd Transformaton, Sprnger-Verlag, Berln, 978. [] C. Rogers, W.F. Shadwck, Backlnd Transformatons, Academc Press, New York, 98. [3] R. Hrota, Exact envelope solton soltons of a nonlnear wave eqaton, J. Math. Phys. 4 (973) [4] R. Hrota, J. Satsma, Solton solton of a copled KdV eqaton, Phys. Lett. A 85 (98) [5] Z.Y. Yan, H.Q. Zhang, New explct soltary wave soltons and perodc wave soltons for Whtham-Broer-Kap eqaton n shallow water, Phys. Lett. A 85 () [6] A.V. Porbov, Perodcal solton to the nonlnear dsspatve eqaton for srface waves n a convectng lqd layer, Phys. Lett. A (996) [7] E.. Fan, Extended tanh-fncton method and ts applcatons to nonlnear eqatons, Phys. Lett. A 77 ()