A Solution of Porous Media Equation

Transcription

1 Internatonal Mathematcal Forum, Vol. 11, 016, no. 15, HIKARI Ltd, A Soluton of Porous Meda Equaton F. Fonseca Unversdad Naconal de Colomba Grupo de Cenca de Materales y Superfces epartamento de Físca Bogotá-Colomba Copyrght c 016 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 porous meda equaton usng lattce Boltzmann for a d1q3 lattce velocty method. We fnd the equlbrum dstrbuton accordng to the porous meda equaton. Also, we use the tanh method n order to fnd several famles of soltary wave solutons. We present results for two colldng solutons, usng LB method, whch s n agreement wth the tanh soluton. Mathematcs Subject Classfcaton: 35-XX, 34-XX, 76xx Keywords: Porous Meda Equaton, Lattce-Boltzmann, Tanh Functon 1 Introducton ffuson s a common n many research felds n physcs, chemstry, bology, fnance and ngeneerng use ther methods and results [1]. A lot of nvestgaton has been made n the theory of nonlnear dffuson equatons such as classcal methods [], nonclasscal methods [3], generalzed condtonal symmetry [4] nonlocal symmetry method [5], and the truncated Panleve approach [6], etc. For a further and deep research on the mathematcal propertes of porous meda equaton should be consulted reference [7], and references theren. Also, n the search for solutons to these complex nonlnear equatons, we can fnd the so called soltary wave solutons. Among them, one of the most knwon and appled s the Tanh method [8], used n ths work.

2 7 F. Fonseca On the other hand, lattce-boltzmann LB) technque s the dscretzed verson of the Boltzmann equaton, who responses for the spato-temporal evoluton of the statstcal dstrbuton functon, whch s a qute complex ntegrodfferental equaton. LB over the years has proven ts effectveness and adaptablty to fnd solutons to a large number of problems n physcs [9]. The lattce Boltzmann model The lattce Boltzmann equaton s gven by: f x + e t, t + t) f x, t) = Ω x, t) The term Ω x, t)) represents the B.G.K. approxmaton, [10], s: Ω x, t)) = 1 τ f x, t) f eq x, t)) ) Where f x, t) s the dstrbuton functon for partcles wth velocty e at poston x and tme t, and t s the tme step. τ s a nondmensonal relaxaton tme, weghtng the approacng rate to the statstcal equlbrum, and f eq x, t) s equlbrum dstrbuton functon. Usng a unt tme gven by ɛ = δt/e, where e s the velocty dscretzed on a mesh. Then, the lattce-boltzmann equaton s: f x + e ɛ, t + ɛ) f x, t) = Ω x, t) 3) Expandng n a Taylor seres, the dstrbuton functon, up to order fourth, we have: f x + e ɛ, t + ɛ) f x, t) = ɛ + ɛ t + e x ) f + ɛ3 6 t + e x ) 3 t + e f + Oɛ 4 ) x ) f 4) ong a perturbatve expanson of the dervatves n tme n powers of ɛ, we get: And assumng: f = f 0) + ɛf + ɛ f ) + ɛ 3 f 3) 5) f 0) Where the temporal scales are defned as: = f eq) 6)

3 A soluton of porous meda equaton 73 t 0 = t t 1 = ɛt t = ɛt t 3 = ɛt 3 7) And the perturbatve expanson n parameter ɛ of the temporal dervatve operator t = + ɛ 1 + ɛ + ɛ 3 8) t 0 t 1 t t 3 Replacng eqs. 5) and 8) n eq. 4), 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 9) f 0 τ1 1 ) t 1 τ ) + e f = 1 t 0 x τ f 10) f 0 t 1 τ τ ) f τ) + e t t 0 x ) 3 + e f 0 t 0 x = 1 τ f 3 3 The moments of the dstrbuton Where φ, u are the macroscopc quanttes defned as: l f 0) = φ = ) + 1 f eq) 1) e f 0) = 0 13) e l, e l,j f 0) l = λφ m δ j 14) Where δ j s Kronecker s delta. ong some algebra, we get: ) ɛ φ ɛτ1 1 t 1 τ ) f x e e j = 1 f ) 15) τ Assumng f ) = βτφ n 16)

4 74 F. Fonseca Fgure 1: The lattce velocty scheme 1Q3. Then If we chose as: φ t = ɛλτ 1 ) φ m x + βφ n 17) = ɛλτ 1 ) 18) Then eq. 17) s the generalzed porous meda equaton [7]: 4 The dstrbuton functon φ t = φ m x + βφ n 19) We use a d1q3, see fgure ), one-dmensonal velocty scheme wth e α = {0, c, c}. Then, the one partcle equlbrum dstrbuton functon s defned as: f eq) = 5 Soltary wave soluton λ φ m = 0 c φ λ φ m = 1 c φ λ φ m = c We choose n eq. 19) wth m =, and n = 4, then: 0) Usng φ t = φ x + βφ4

5 A soluton of porous meda equaton 75 The dervatves change lke: ξ = x at + ξ 0 ) t = a ξ ; x = ξ ; x = ξ 3) Now, we apply the tanh method [8]. varable: Then, we ntroduce an ndependent The dervatves of ξ n terms of Y, are: Y x, t) = tanh ξ) 4) d dξ = 1 Y ) d dy, d dξ = Y 1 Y ) d dy + 1 Y ) d 5) dy The solutons are postulated [8] as: φξ) = m a Y 6) =1 Now, usng eqs. 3) and eq. 5) n eq., we get: a1 Y ) dφ dy = Y 1 Y ) dφ dy + 1 Y ) d φ dy + βφ4 7) The parameter m, n eq. 6), s obtaned by the balance of the hghest-order lnear term wth the nonlnear terms n the transformed equatons. So, we have: Then, eq. 6) s: Y 4 d φ dy φ4 4 + m = 4m m = 1 8) φξ) = a 0 + a 1 Y φξ) = a 0 + a 1 a 0 Y + a 1Y 9)

6 76 F. Fonseca Replacng a1 Y )a 1 = 4Y a 1 a 0 + a 1Y a 1 a 0 + a 1Y )Y ) 30) +a 11 Y + Y 4 ) + βa Y a 3 0a 1 + 6Y a 0a 1 +4Y 3 a 0 a Y 4 a 4 Groupng the coeffcents, n eq 30), n powers of Y and equatng ther coeefcents to zero, we get a set of nonlnear equatons. ong some algebra the solutons are: a 01, = ± β, a 1 1, = ± β, a 6 1 3,4 = ± β 3 a 15 = aβ β a β 8, a 16 = aβ + β a β 8 4β 4β 3) a 17 = a, a 0 3 = a + β ) 1/4 1/4, a 0 4 = a + β ) 1/4 1/4 33) a 05 = a + β ) 1/4 1/4, a 06 = a + β ) 1/4 1/4 34) a 07 = a β ) 1/4 1/4, a 0 8 = a β ) 1/4 1/4 35) a 09 = a β ) 1/4 1/4, a 010 = a β ) 1/4 1/4 36) a 011 = a + 1 ) 1/4 6 b b b 1/4 37)

7 A soluton of porous meda equaton 77 a + 1 ) 1/4 6 b b a 01 = 38) b 1/4 a 013 = a 014 = a + 1 ) 1/4 6 b b 39) b 1/4 a + 1 ) 1/4 6 b b b 1/4 40) a 015 = a ) 1/4 6 b b b 1/4 4 a ) 1/4 6 b b a 016 = 4) b 1/4 Usng a 11,,3,4 : a 017 = a 018 = a ) 1/4 6 b b 43) b 1/4 a ) 1/4 6 b b b 1/4 44) ong the next defntons: a 019,...,6 = ± a + 8a 1 6βa 1 45) l 1 = 187a 4 b a b b ) a 8 b a 6 b a 4 b 8 6

8 78 F. Fonseca Fgure : Two colldng functons, eq. 5), usng LB method. l = 7a b b 4 47) l 3 = 16 7b + 4 l ) 7b 3 l 1 ) + l 1/3 48) 1/3 54b 3 l 4 = 3 7b 4 l ) 7b 3 l 1 ) l 1/3 a 1/3 54b 3 3b 49) l 3 We get: l 5 = 3 7b 4 l ) 7b 3 l 1 ) + l 1/3 a 1/3 54b 3 3b 50) l 3 a 031 = 3b + 1 l3 1 l4 5

9 A soluton of porous meda equaton 79 a 131 = a 4 + a 3b 4 l 3 3 l 3 9a b l3 3 l 1 ) 1/3 4 b l 3 ) 3/ + 18b4 l 3 + l 1/3 l 3 9a b l b4 3 l 1 ) 1/3 l 1 ) 1/3 3 l 1 ) 1/3 l4 5) + l 1/3 l 4 + b l 3 l b l 4 ) 3/ )/a) a 03 = 3b + 1 l3 1 l4 53) a 13 = a 4 + a 3b 4 l 3 3 l 3 9a b l3 3 l 1 ) 1/3 4 b l 3 ) 3/ + 18b4 l 3 + l 1/3 l 3 9a b l b4 3 l 1 ) 1/3 l 1 ) 1/3 3 l 1 ) 1/3 l4 54) + l 1/3 l 4 + b l 3 l b l 4 ) 3/ )/a) a 033 = 3b + 1 l3 + 1 l4 55) a 133 = a 4 + a 3b 4 l 3 3 l 9a b l3 3 l 1 ) 1/3 4 b l 4 ) 3/ 56) + l 1/3 l 3 l 1/3 l 4 l4 3 4 b l 3 ) 3/ + 9a b l 4 18b4 l 1 ) 1/3 3 l 1 ) 1/3 b l 3 l4 + 18b4 l 3 3 l 1 ) 1/3 )/a) a 034 = 3b + 1 l3 + 1 l4 57) a 134 = 4 3 l 3 9a b l b4 l3 58) l 1 ) 1/3 3 l 1 ) 1/3 3 4 b l 4 ) 3/ 3 4 b l 3 ) 3/ + 9a b l 4 18b4 l 1 ) 1/3 3 l 1 ) 1/3 a 4 + a 3b l 1/3 l4 l 3 l4 b l 3 l4 + l 1/3 l 3 )/a)

10 730 F. Fonseca a 035 = 3b 1 l3 1 l5 59) a 135 = l 3 + 9a b l 3 18b4 l3 60) l 1 ) 1/3 3 l 1 ) 1/3 l 1/3 l b l 3 ) 3/ 9a b l 5 l 1 ) 1/3 + 18b4 l 5 + l 1/3 l 4 b l 3 l 1 ) 1/3 3 l b l 5 ) 3/ a 4 a 3b l 3 )/a) a 036 = 3b 1 l3 1 l5 6 a 136 = a 3b + 4 l 3 3 l 3 + 9a b l3 18b4 l3 6) l 1 ) 1/3 3 l 1 ) 1/3 a 4 l 1/3 l b l 3 ) 3/ 9a b l b4 l 1 ) 1/3 3 l 1 ) 1/3 l5 + l 1/3 l 5 b l 3 l b l 5 ) 3/ )/a) a 037 = 3b 1 l3 + 1 l5 63) a 137 = a 4 a 3b + 4 l 3 3 l 3 + 9a b l3 64) l 1 ) 1/3 l 1/3 l 3 3 l 1 ) 1/3 18b4 l b l 3 ) 3/ + 9a b l 5 l 1 ) 1/3 18b4 l 5 l 1/3 l 5 + b l 3 l 1 ) 1/3 3 l5 3 4 b l 5 ) 3/ )/a)

11 A soluton of porous meda equaton 731 a 038 = 3b 1 l3 + 1 l5 65) a 138 = a 3b + 4 l 3 3 l 3 + 9a b l3 18b4 l3 66) l 1 ) 1/3 3 l 1 ) 1/3 l 1/3 l 3 l b l 3 ) 3/ + 9a b l 5 18b4 l 1 ) 1/3 3 l 1 ) 1/3 a 4 l 1/3 l5 + b l 3 l5 3 4 b l 5 ) 3/ )/a) We fnd thrty eght famles of solutons usng Tanh method. famles are: So, the φ 1 a 01, a 15 ), φ a 01, a 16 ), φ 3 a 0, a 15 ) 67) φ 4 a 0, a 16 ), φ 5 a 01, a 17 ), φ 6 a 0, a 17 ) 68) φ 7 a 03, a 11 ), φ 8 a 04, a 11 ), φ 9 a 05, a 11 ) 69) φ 10 a 06, a 11 ), φ 11 a 07, a 1 ), φ 1 a 08, a 1 ) 70) φ 13 a 09, a 1 ), φ 14 a 010, a 1 ), φ 15 a 011, a 13 ) 7 φ 16 a 01, a 13 ), φ 17 a 013, a 13 ), φ 18 a 014, a 13 ) 7) φ 19 a 015, a 14 ), φ 0 a 016, a 14 ), φ 1 a 017, a 14 ) 73) φ a 018, a 14 ), φ 3 a 019, a 11 ), φ 4 a 00, a 11 ) 74)

12 73 F. Fonseca φ 5 a 01, a 1 ), φ 6 a 0, a 1 ), φ 7 a 03, a 13 ) 75) φ 8 a 04, a 13 ), φ 9 a 05, a 14 ), φ 30 a 05, a 14 ) 76) φ 31 a 031, a 131 ), φ 3 a 03, a 13 ), φ 33 a 033, a 133 ) 77) φ 34 a 034, a 134 ), φ 35 a 035, a 135 ), φ 36 a 036, a 136 ) 78) 6 Conclusons φ 37 a 037, a 137 ), φ 38 a 038, a 138 ) 79) We solve the general nonlnear porous meda equaton usng lattce-boltzmann and the Tanh method. Also, we get thrty eght famles of solutons usng Tanh method. As far as we know, ths procedure s totally orgnal and stablshes a contrbuton to the soluton to the porous meda equaton. As a future work, the model could be extended to two and three dmensons. Acknowledgements. Ths research was supported by Unversdad Naconal de Colomba n Hermes project 350. References [1] J. Janssen, O. Manca and R. Manca, Appled ffuson Processes from Engneerng to Fnance, John Wley & Sons, [] V.A. orodntsyn, On nvarant solutons of the equatons on non-lnear heat conducton wth a source, USSR Computatonal Mathematcs and Mathematcal Physcs, 198), [3] W.I. Fushchch, W.M. Shtelen, N.I. Serov, Symmetry Analyss and Exact Solutons of Equatons of Nonlnear Mathematcal Physcs, Kluwer Academc, ordrecht,

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