Appled Mathematcal Scences, Vol. 11, 2017, no. 52, 2579-2586 HIKARI Ltd, www.m-hkar.com https://do.org/10.12988/ams.2017.79280 A Soluton of the Harry-Dym Equaton Usng Lattce-Boltzmannn and a Soltary Wave Methods F. Fonseca Unversdad Naconal de Colomba Departamento de Físca Bogotá-Colomba Copyrght c 2017 F. Fonseca. Ths artcle s dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. Abstract We solve the nonlnear Harry-Dym equaton usng lattce Boltzmann and a d1q3 lattce velocty scheme. Also, we suppose a sech b soluton, resultng several famles of soltary wave solutons. We present results for two colldng solutons, usng lattce Boltzmann and showng that those exhbts solton-lke behavor. Keywords: Harry-Dym Equaton, lattce-boltzmann, sech b 1 Introducton From the early work of Harry-Dym n the seventes and Krushkal [1], Harry Dym equaton (HDEq) has shown a lot of mathematcal propertes. Among them, we can fnd ntegrablty, [3], behavour lke-solton, because t can be transformed nto the Korteweg-de Vres [4]. Also, HDEq has emerged n hydrodynamcs problems,[5]. In addton, n the last 30 years we have wtnessed the development of a group of analytcal methods desgned to provde solutons to nonlnear partal dfferental equatons, such as HDEq, called soltary wave behavor,[6]. On the other hand, lattce-boltzmann (LB) equaton gves the spato-temporal dstrbuton functon of an statstcal system, showng that s sutable to gve answer n a bg number of physcal problems, [7]-[11]. Ths work s prepared as follows. Secton (2), shows the lattce-boltzmann model. In secton (3), we defne the moments of the dstrbutons. Secton
2580 F. Fonseca (4), we get the HDEq. In Secton (5), we present the equlbrum dstrbuton functon. In addton, secton (6) postulates a sech b analytcal soluton. At last, n secton (7), we present results and conclusons. 2 The lattce Boltzmann model The lattce Boltzmann equaton s gven by [7]-[10]: f (x + e ɛ, t + ɛ) f (x, t) = 1 τ (f (x, t) f eq (x, t)) (1) Here f (x, t) s the partcle dstrbuton and f eq (x, t) s equlbrum dstrbuton functons, respectvely. Also, the poston s x, tme t, wth velocty e, and t s the tme step. τ s a nondmensonal relaxaton tme [11]. Expandng n a Taylor seres, the dstrbuton functon, up to order fourth, we have: ( f (x + e ɛ, t + ɛ) f (x, t) = ɛ + ɛ2 2 ( t + e x ) 2 f + ɛ3 6 t + e x ( ) 3 t + e f + O(ɛ 4 ) x ) f (2) Dong a perturbatve expanson of the dervatves n tme n powers of ɛ, we get: And assumng: f = f (0) + ɛf (1) + ɛ 2 f (2) + ɛ 3 f (3) (3) f (0) Where the temporal scales are defned as: = f (eq) (4) t 0 = t t 1 = ɛt t 2 = ɛt 2 t 3 = ɛt 3 (5) And the perturbatve expanson n parameter ɛ of the temporal dervatve operator t = + ɛ 1 + ɛ 2 + ɛ 3 (6) t 0 t 1 t 2 t 3 Replacng eqs. (3) and (6) n eq. (2), we get at frst, second and thrd order n ɛ, respectvely, the next set of equatons: f 0 t 0 + e f 0 x = 1 τ f 1 (7)
Soluton of Harry-Dym equaton 2581 f 0 τ(1 1 ( ) 2 t 1 2τ ) + e f = 1 t 0 x τ f 2 (8) f 0 t 1 (τ 2 τ + 1 6 ) ( f 0 + (1 2τ) + e t 2 t 0 x ( ) 3 + e f 0 t 0 x = 1 τ f 3 3 The moments of the dstrbuton The moments of the dstrbuton functon are defned as: f (0) = 1 2φ = 2 l Where δ j s Kronecker s delta. e l, e l,j f (0) l 4 The Harry-Dym Equaton Then, summng on j n eq. (7), we obtan: ) + (9) f (eq) (10) e f (0) = 0 (11) = λ φ x δ j (12) f (n) = 0; n 1 (13) j f j 0 + e j fj 0 = 1 fj 1 (14) t 0 x τ Takng nto account eqs. (10)-(11) and (13), we get: j j j f j 0 = ( 1 = 0 (15) t 0 t 0 2φ 2 ) Summng on j n eq. (8) and multplyng by ɛ ɛ j f 0 j t 1 ɛτ(1 1 2τ ) j ( ) 2 + v j f j = ɛ 1 t 0 x τ j f (2) j (16)
2582 F. Fonseca Fgure 1: The lattce velocty scheme d1q3. And usng the equatons (10)-(13) and (15), we have: ɛ ( 1 2φ 2 ) t 1 ɛτ(1 1 2τ )λ 3 φ x 3 = 0 (17) Then, summng eq. (15) and (17) and usng eq. (6) at second order, we get: If we chose D as: ( 1 2φ 2 ) t = ɛλ(τ 1 2 ) 3 φ x 3 (18) Then, eq. (24) s the Harry-Dym equaton [1]: D = ɛλ(τ 1 2 ) (19) ( 1 2φ 2 ) t = D 3 φ x 3 (20) φ t = Dφ3 3 φ x 3 (21) 5 The equlbrum dstrbuton functon We use a d1q3, see fgure (1), one-dmensonal velocty scheme wth e α = {0, c, c} [7]-[8]. Then, the one partcle equlbrum dstrbuton functon s defned as: f (eq) = λ c 2 φ 1 x 2 φ 2 = 0 λ φ = 1 2c 2 x λ φ = 2 2c 2 x (22)
Soluton of Harry-Dym equaton 2583 6 Analytcal soluton We choose n eq. (21), D = 1, then: Usng φ t = 3 φ φ3 (23) x 3 ξ = x ct + ξ 0 (24) The dervatves change lke: t = c ξ ; x = ξ ; 3 x 3 = 3 ξ 3 (25) Then, replacng n (23) d 3 φ dξ 3 (φ)3 + c dφ dξ = 0 (26) We suppose a soluton φ = Asech b (ξ) (27) d 3 (Asech b (ξ)) (Asech b (ξ)) 3 + c d(asechb (ξ)) dξ 3 dξ = 0 (28) b 1 = 2, b 2 = 1 (29) Usng b 1 = 2, and defnng 3A 3 b + 2A 3 + c = 0 (30) The solutons are: l 1 = 4 243c + 9 3 8c + 243c 2 (31)
2584 F. Fonseca Fgure 2: The spatotemporal, LB, evoluton of φ(x, t) usng a d1q3 lattce velocty, for two sech 1 eq. (27) ntal profles. A 1 = 1 9 ( 1 22/3 (l 1 ) (l 1) 1/3 1/3 2 2/3 ) (32) Usng b 1 = 1, and defnng A 2 = 1 9 + 1 + ( ) 3 1 3 92 1/3 (l 1 ) + (l1 ) 1/3 (33) 1/3 182 2/3 A 3 = 1 9 + 1 ( ) 3 1 + 3 92 1/3 (l 1 ) + (l1 ) 1/3 (34) 1/3 182 2/3 l 2 = 16 243c + 9 3 32c + 243c 2 (35) A 4 = 1 9 ( 2 421/3 (l 2 ) (l 2) 1/3 1/3 2 1/3 ) (36) A 5 = 2 ( 9 + 221/3 1 + 3 ) ( ) 1 3 (l2 ) 1/3 + 9 (l 2 ) 1/3 182 1/3 (37) A 6 = 2 ( 9 + 221/3 1 3 ) ( ) 1 + 3 (l2 ) 1/3 + 9 (l 2 ) 1/3 182 1/3 (38)
Soluton of Harry-Dym equaton 2585 7 Conclusons We solve the general nonlnear Harry-Dym equaton usng lattce-boltzmann and a sech b methods. In addton, we get twelve famles of solutons usng sech b ansatz. Also, those solutons, eq. (23), preserve the superposton prncple, fg. (2). As a future work, the model could be extended to two and three dmensons. φ(x, t) = Asech 1 (x ct) (39) φ(x, t) = Asech 2 (x ct) (40) Acknowledgements. Ths research was supported by Unversdad Naconal de Colomba n Hermes project (32501). References [1] J. Moser, (ed), Dynamcal Systems, Theory and Applcatons, Lec. Notes Phys., Sprnger, Berln, 1975. https://do.org/10.1007/3-540-07171-7 [2] W. Hereman, P. P. Banerjee and M. R. Chatterjee, Dervaton and mplct soluton of the Harry Dym equaton and ts connectons wth the Korteweg-de Vres equaton, Journal of Physcs A, 22 (1989), no. 3, 241-255. https://do.org/10.1088/0305-4470/22/3/009 [3] N. H. Ibragmov, Transformaton Group Appled to Mathematcal Physcs, Redel, 1985. [4] Z. J. Qao, A completely ntegrable system assocated wth the Harry Dym herarchy, Journal of Nonlnear Mathematcal Physcs, 1 (1994), no. 1, 65-74. https://do.org/10.2991/jnmp.1994.1.1.5 [5] G. L. Vasconcelos and L. P. Kadanoff, Statonary solutons for the Saffman-Taylor problem wth surface tenson, Phys. Rev. A, 44 (1991), 6490-6495. https://do.org/10.1103/physreva.44.6490 [6] W. Malflet and W. Hereman, The Tanh Method: I. Exact solutons of Nonlnear Evoluton and Wave Equatons, Physca Scrpta, 54 (1996), 563-568. https://do.org/10.1088/0031-8949/54/6/003 [7] D. A. Wolf-Gladrow, Lattce-Gas Cellular Automata and Lattce Boltzmann Models: An Introducton, Sprnger, Berln, 2000. https://do.org/10.1007/b72010
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