A Solution of the Harry-Dym Equation Using Lattice-Boltzmannn and a Solitary Wave Methods

Transcription

1 Appled Mathematcal Scences, Vol. 11, 2017, no. 52, HIKARI Ltd, 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

2 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)

3 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 ) ( 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)

4 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)

5 Soluton of Harry-Dym equaton 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 c + 243c 2 (31)

6 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 = ( ) /3 (l 1 ) + (l1 ) 1/3 (33) 1/ /3 A 3 = ( ) /3 (l 1 ) + (l1 ) 1/3 (34) 1/ /3 l 2 = c c + 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 ( / ) ( ) 1 3 (l2 ) 1/3 + 9 (l 2 ) 1/ /3 (37) A 6 = 2 ( /3 1 3 ) ( ) (l2 ) 1/3 + 9 (l 2 ) 1/ /3 (38)

7 Soluton of Harry-Dym equaton 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, [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, [3] N. H. Ibragmov, Transformaton Group Appled to Mathematcal Physcs, Redel, [4] Z. J. Qao, A completely ntegrable system assocated wth the Harry Dym herarchy, Journal of Nonlnear Mathematcal Physcs, 1 (1994), no. 1, [5] G. L. Vasconcelos and L. P. Kadanoff, Statonary solutons for the Saffman-Taylor problem wth surface tenson, Phys. Rev. A, 44 (1991), [6] W. Malflet and W. Hereman, The Tanh Method: I. Exact solutons of Nonlnear Evoluton and Wave Equatons, Physca Scrpta, 54 (1996), [7] D. A. Wolf-Gladrow, Lattce-Gas Cellular Automata and Lattce Boltzmann Models: An Introducton, Sprnger, Berln,

8 2586 F. Fonseca [8] Sauro Succ, The Lattce Boltzmann Equaton for Flud Dynamcs and Beyond, (Numercal Mathematcs and Scentfc Computaton), Oxford Unversty Press, [9] G. Yan, J. Zhang, A hgher-order moment method of the lattce Boltzmann model for the Korteweg-de Vres equaton, Mathematcs and Computers n Smulaton, 79 (2009), [10] Zhenhua Cha, Baochang Sh, A novel lattce Boltzmann model for the Posson equaton, Appled Mathematcal Modellng, 32 (2008), [11] P. L. Bathnagar, E. P. Gross and M. Krook, A model for collson processes n gases. I. Small ampltude processes n charged and neutral onecomponent systems, Phys. Rev., 94 (1954), Receved: September 23, 2017; Publshed: October 3, 2017