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Travelling wave solutions for some time-delayed equations through factorizations
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Chaos, Solitons and Fractals 38 (2008) 1209–1216 www.elsevier.com/locate/chaos
Travelling wave solutions for some time-delayed equations through factorizations E.S. Fahmy King Saud University, Women Students Medical Studies & Sciences Sections, Mathematics Department, P.O. Box 22452, Riyadh 11495, Saudi Arabia Accepted 19 February 2007
Abstract In this work, we use factorization method to find explicit particular travelling wave solutions for the following important nonlinear second-order partial differential equations: The generalized time-delayed Burgers–Huxley, timedelayed convective Fishers, and the generalized time-delayed Burgers–Fisher. Using the particular solutions for these equations we find the general solutions, two-parameter solution, as special cases. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction A number of nonlinear phenomena in many branches of sciences such as physical [1], chemical, economical [2] and biological processes [3,4] are described by reaction and diffusion or by the interaction between convection and diffusion. One of the well known partial differential equations which governs a wide variety of them is Burgers equation which provides the simplest nonlinear model of turbulence [5]. The existence of relaxation time or delay time is an important feature in reaction diffusion and convection diffusion systems [6–8]. Many methods have been used to find exact solutions of nonlinear partial differential equations, such as the tanh function method [9–13], Jacobi elliptic function method [14], simplest equation method [15], unified algebraic method [16]. Also, many methods have been used to find numerical solutions of nonlinear partial differential equations, such as the Adomian decomposition method [17,18]. Recently, factorization of second-order linear differential equations became a well established technique to find solutions in an algebraic manner [19–29]. Rosu and Cornejo find one particular solution once the nonlinear equation is factorized with the use of two first order differential operators [20]. They use the method for equations of types: u00 þ cu0 þ f ðuÞ ¼ 0;
ð1Þ
u00 þ gðuÞu0 þ f ðuÞ ¼ 0;
ð2Þ
and
E-mail address: [email protected] 0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.02.007
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E.S. Fahmy / Chaos, Solitons and Fractals 38 (2008) 1209–1216
where 0 means the derivative D ¼ dzd , g(u) and f(u) are polynomials in u. We concentrate our work in this paper on equation of the type (2). Now, Eq. (2) can be factorized as ½D u2 ðuÞ½D u1 ðuÞu ¼ 0;
ð3Þ
which leads to the equation du1 0 uu u1 u0 u2 u0 þ u1 u2 u ¼ 0; du
ð4Þ
du1 u u1 þ u2 þ u u0 þ u1 u2 u ¼ 0: du
ð5Þ
u00 or
00
Comparing (5) and (2) we find du1 gðuÞ ¼ u1 þ u2 þ u du
and
f ðuÞ ¼ u1 u2 u:
ð6Þ
If Eq. (2) can be factorized as in Eq. (3), then a first particular solution can be easily found by solving ½D u1 ðuÞu ¼ 0:
ð7Þ
If u1(u) is a linear function of the dependent variable u, Eq. (7) turns out to be a Riccati equation for this variable and in this case we can find the general solution [26].
2. The generalized time-delayed Burger’s Huxley equation The generalized time-delayed Burger’s Huxley equation is given as: df ut ¼ aud ux þ uxx þ f ðuÞ; f ðuÞ ¼ buð1 ud Þðud cÞ; sutt þ 1 s du
ð8Þ
where a; b; c are constants and s is a time-delayed constant. It is clear that when s ¼ 0, Eq. (8) reduces to the generalized Burger’s Huxley equation discussed by Wang et al. in [30]. Using the coordinate transformation z ¼ x ct (c is the propagation speed) in Eq. (8) we obtain the following nonlinear ordinary differential equation: ð1 c2 sÞu00 þ ½ðc þ csbcÞ þ ða csbðd þ 1Þðc þ 1Þud Þ csbð2d þ 1Þu2d þ buð1 ud Þðud cÞ ¼ 0;
1 c2 s > 0; ð9Þ
or u00 þ gðuÞu0 þ F ðuÞ ¼ 0;
ð10Þ
where gðuÞ ¼
1 ½ðc þ csbcÞ þ ða csbðd þ 1Þðc þ 1Þud Þ csbð2d þ 1Þu2d ; 1 c2 s
ð11Þ
F ðuÞ ¼
b uð1 ud Þðud cÞ: ð1 c2 sÞ
ð12Þ
Using operator notation, Eq. (10) takes the form F ðuÞ D2 þ gðuÞD þ u ¼ 0: u
ð13Þ
The factorization of (13) leads to ½D u2 ðuÞ½D u1 ðuÞu ¼ 0;
ð14Þ
and then
du u00 u2 þ u1 þ 1 u u0 þ u1 u2 u ¼ 0: du
ð15Þ
E.S. Fahmy / Chaos, Solitons and Fractals 38 (2008) 1209–1216
Comparing (15) and (10) we obtain the conditions on u1 and u2 as: du F ðuÞ u2 þ u1 þ 1 u ¼ gðuÞ; u1 u2 ¼ ; du u
1211
ð16Þ
therefore u1 u2 ¼
bð1 ud Þðud cÞ : ð1 c2 sÞ
ð17Þ
Now, we choose u1 and u2 such that sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b 1 b d að1 u Þ; u2 ðuÞ ¼ ðud cÞ: u1 ðuÞ ¼ ð1 c2 sÞ a ð1 c2 sÞ From (18) and (11) we get pffiffiffi c 1 1 þ a da ud ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½ðc þ csbcÞ þ ða csbðd þ 1Þðc þ 1Þud Þ csbð2d þ 1Þu2d b a a a ð1 c2 sÞ
ð18Þ
ð19Þ
which implies that a ¼
ða csbðd þ 1Þðc þ 1ÞÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ða csbðd þ 1Þðc þ 1ÞÞ2 þ 4bð1 c2 sÞð1 þ dÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð1 þ dÞ bð1 c2 sÞ
then sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b að1 ud Þ; u1 ðuÞ ¼ ð1 c2 sÞ
1 u2 ðuÞ ¼ a
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b ðud cÞ; ð1 c2 sÞ
and the corresponding factorization is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " #" # 1 b b d d ðu ð1 u cÞ D a Þ u ¼ 0; D a ð1 c2 sÞ ð1 c2 sÞ and the compatible first order differential equation is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b 0 ð1 ud Þu ¼ 0: u a ð1 c2 sÞ By direct integration of (22) we get " sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #!1d b dðz z0 Þ ; u ðzÞ ¼ 1 exp a ð1 c2 sÞ where z0 is the integration constant. The solution (23) in hyperbolic form is given as: " #!1d pffiffiffi 1 1 a1 d b þ tanh pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðz z0 Þ ; u ðzÞ ¼ 2 2 2 ð1 c2 sÞ " #!1d pffiffiffi 1 1 a1 d b coth pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðz z0 Þ : u ðzÞ ¼ 2 2 2 ð1 c2 sÞ Now, if we choose the factorization terms as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b 1 b d ðu cÞ; u2 ðuÞ ¼ ð1 ud Þ: u1 ðuÞ ¼ b ð1 c2 sÞ b ð1 c2 sÞ and use (26) and (11) we obtain
ð20Þ
ð21Þ
ð22Þ
ð23Þ
ð24Þ
ð25Þ
ð26Þ
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pffiffiffi b
1 1 1 cb þ db þ b ud ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½ðc þ csbcÞ þ ða csbðd þ 1Þðc þ 1Þud Þ csbð2d þ 1Þu2d : b b ð1 c2 sÞ ð27Þ
Then b ¼
ða csbðd þ 1Þðc þ 1ÞÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ða csbðd þ 1Þðc þ 1ÞÞ2 þ 4bð1 c2 sÞð1 þ dÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2ð1 þ dÞ bð1 c2 sÞ
and Eq. (13) is then factorized in the following different form sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " #" # 1 b b d d ð1 u ðu Þ D b cÞ u ¼ 0: D b ð1 c2 sÞ ð1 c2 sÞ The corresponding compatible first order equation is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b 0 bðud cÞu ¼ 0: u ð1 c2 sÞ
ð28Þ
ð29Þ
Direct integration of Eq. (29) gives a different first order particular solution of Eq. (8), 11d 0 c B h qffiffiffiffiffiffiffiffiffiffiffi iC u ðzÞ ¼ @ A: b 1 exp b ð1c2 sÞdcðz z0 Þ
ð30Þ
Putting s ¼ 0 in (23) and (30) we find an exact particular solution for the generalized Burger’s Huxley equation [20]. Following the work of Reyes and Rosu [26], we find a two-parameter solution when d ¼ 1. In this case qffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffi u1 ðuÞ ¼ a ð1cb 2 sÞð1 uÞ and u2 ðuÞ ¼ 1a ð1cb 2 sÞðu cÞ, one particular solution is obtained from (23) as: " sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #!1 b ðz z0 Þ ; ð31Þ u1 ðzÞ ¼ 1 þ exp a ð1 c2 sÞ and Eq. (22) is transformed to the following Riccati equation sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b ð1 uÞu ¼ 0: u0 a ð1 c2 sÞ The two-parameter solution is obtained from (31) and (32) as: h qffiffiffiffiffiffiffiffiffiffiffi i1 1 þ exp a ð1cb 2 sÞðz z0 Þ h qffiffiffiffiffiffiffiffiffiffiffi ih h qffiffiffiffiffiffiffiffiffiffiffi i i: uk ¼ u1 þ exp a ð1cb 2 sÞðz z0 Þ k 1 þ exp a ð1cb 2 sÞðz z0 Þ 1
ð32Þ
ð33Þ
It is clear that when jkj runs from zero to infinity, the above parametric solution goes from the trivial solution u ¼ 0 to the particular solution u ¼ u1 .
3. Time-delayed convective Fisher’s equation The convective Fisher’s equation given in the following form [31]: 1 ut ¼ uxx luux þ uð1 uÞ 2 and the time-delayed convective Fisher equation can be obtained as: df 1 ut ¼ uxx luux þ uð1 uÞ; sutt þ 1 s du 2 where l is a positive parameter that serves to tune the relative strength of convection.
ð34Þ
ð35Þ
E.S. Fahmy / Chaos, Solitons and Fractals 38 (2008) 1209–1216
1213
Using the coordinate transformation z ¼ x ct in Eq. (35) we obtain the following nonlinear ordinary differential equation u00 þ
2 2 ½cð1 sÞ þ ð2sc lÞuu0 þ uð1 uÞ ¼ 0; ð1 2c2 sÞ ð1 2c2 sÞ
1 > 2c2 s
ð36Þ
or u00 þ gðuÞu0 þ F ðuÞ ¼ 0;
ð37Þ
where gðuÞ ¼
2½cð1 sÞ þ ð2sc lÞu ð1 2c2 sÞ
and
F ðuÞ ¼
2 uð1 uÞ: ð1 2c2 sÞ
ð38Þ
Now, Eq. (37) can be factorized as ½D u2 ðuÞ½D u1 ðuÞu ¼ 0:
ð39Þ
Using operator notation, Eq. (37) takes the form F ðuÞ D2 þ gðuÞD þ u ¼ 0; u
ð40Þ
and then
du u00 u2 þ u1 þ 1 u u0 þ u1 u2 u ¼ 0: du
ð41Þ
Comparing (41) and (37) we obtain the conditions on u1 and u2 as: du F ðuÞ ; u2 þ u1 þ 1 u ¼ gðuÞ; u1 u2 ¼ u du
ð42Þ
therefore u1 u2 ¼
2 ð1 uÞ: ð1 2c2 sÞ
Now, choosing u1 and u2 such that sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 rð1 uÞ and u1 ðuÞ ¼ ð1 2c2 sÞ
ð43Þ
1 u2 ðuÞ ¼ r
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ; ð1 2c2 sÞ
and using (44) and (42) we get sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 ½cð1 sÞ þ ð2sc lÞu; þ r½1 2u ¼ r ð1 2c2 sÞ
r 6¼ 0;
ð44Þ
ð45Þ
hence ð2sc lÞ r ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2ð1 2c2 sÞ
1 2c2 s > 0;
and Eq. (39) is reduced to 2 ð2sc lÞ D D ð1 uÞ u ¼ 0: ð2sc lÞ ð1 2c2 sÞ
ð46Þ
ð47Þ
The compatible first order differential equation is u0
ð2sc lÞ ð1 uÞu ¼ 0: ð1 2c2 sÞ
By direct integration we get 1 ð2sc lÞðz z0 Þ u ¼ 1 exp ð1 2c2 sÞ
ð48Þ
ð49Þ
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E.S. Fahmy / Chaos, Solitons and Fractals 38 (2008) 1209–1216
or in hyperbolic form:
1 1 ð2sc lÞ ðz z Þ ; tanh 0 2 2 2ð1 2c2 sÞ 1 1 ð2sc lÞ ðz z0 Þ : u ðzÞ ¼ coth 2 2 2ð1 2c2 sÞ
uþ ðzÞ ¼
ð50Þ ð51Þ
Putting s ¼ 0 in (23) we find an exact particular solution for the convective Fisher equation [20]. Using (48) and (49), a two-parameter solution is obtained as: h i 1 ð2sclÞ ð1 þ exp 2ð12c 2 sÞ ðz z0 Þ Þ h ih h i i: uk ¼ uþ þ ð2sclÞ ð2sclÞ exp 2ð12c k 1 þ exp 2ð12c 1 2 sÞ ðz z0 Þ 2 sÞ ðz z0 Þ
ð52Þ
3.1. The generalized time-delayed Burgers–Fisher equation The generalized Burgers–Fisher equation is given in the following form: ut ¼ uxx pus ux þ quð1 us Þ
ð53Þ
and the generalized time-delayed Burgers–Fisher equation can be obtained as: df sutt þ 1 s ut ¼ uxx pus ux þ f ðuÞ; f ðuÞ ¼ quð1 us Þ; du
ð54Þ
where s, p, q are any real numbers and s 2 N. Eq. (54) is an extended form of the generalized Burger’s Fisher equation (53). When q ¼ 0, Eq. (54) is reduced to the time-delayed Burger’s Fisher equation [6]. Using the coordinate transformation z ¼ x ct in Eq. (54), we get ð1 c2 sÞu00 þ ½cðð1 qsÞ þ qsðs þ 1ÞÞus pus u0 þ f ðuÞ ¼ 0;
1 > c2 s;
ð55Þ
or u00 þ gðuÞu0 þ F ðuÞ ¼ 0;
ð56Þ
where gðuÞ ¼
cð1 qsÞ ½qsðs þ 1Þ p s þ u ð1 c2 sÞ ð1 c2 sÞ
and
F ðuÞ ¼
q uð1 us Þ: ð1 c2 sÞ
ð57Þ
Now, Eq. (56) can be factorized as ½D u2 ðuÞ½D u1 ðuÞu ¼ 0;
ð58Þ
where u1 ðuÞ ¼ a1
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q ð1 us Þ and ð1 c2 sÞ
u2 ðuÞ ¼
1 a1
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q ; ð1 c2 sÞ
a1 6¼ 0;
and using (59) and (56) we get rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q 1 cð1 qsÞ ½qsðs þ 1Þ p s s þ u; ¼ þ a ðs þ 1Þu 1 ð1 c2 sÞ a1 ð1 c2 sÞ ð1 c2 sÞ
ð59Þ
ð60Þ
and hence a1 ¼
½qsðs þ 1Þ p pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ðs þ 1Þ qð1 c2 sÞ
Using Eq. (61), Eq. (58) reduced to qðs þ 1Þ ½qsðs þ 1Þ pð1 us Þ D u ¼ 0: D qsðs þ 1Þ p ðs þ 1Þð1 c2 sÞ The compatible first order differential equation is
ð61Þ
ð62Þ
E.S. Fahmy / Chaos, Solitons and Fractals 38 (2008) 1209–1216
u0
½qsðs þ 1Þ p ð1 us Þu ¼ 0: ðs þ 1Þð1 c2 sÞ
1215
ð63Þ
By direct integration we get 1s ½qsðs þ 1Þ ps Þ ; ðz z u ¼ 1 exp 0 ðs þ 1Þð1 c2 sÞ
ð64Þ
or uþ ðzÞ ¼
1s 1 1 ½qsðs þ 1Þ ps tanh ðz z Þ ; 0 2 2 ðs þ 1Þð1 c2 sÞ 0 2 311s
B1 1 6 ½qsðs þ 1Þ ps 7C qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðz z0 Þ5A : u ðzÞ ¼ @ coth 4 2 2 2 2 ðs þ 1Þ ð1 c sÞ
ð65Þ
ð66Þ
Following the work of Reyes and Rosu [26,27], we find a two-parameter solution for the special case when s ¼ 1. In 2
2ð1c sÞ 2qsp this case u1 ðuÞ ¼ 2ð1c 2 sÞð1 uÞ and u2 ðuÞ ¼ 2qsp , one particular solution is obtained from (64) as
u1 ðzÞ ¼ ð1 þ exp½hðz z0 ÞÞ1
ð67Þ
and Eq. (63) is transformed to the following Riccati equation u0
2qs p ð1 uÞu ¼ 0: 2ð1 c2 sÞ
The two-parameter solution is obtained from (67) and (68) as: h i 1 2qsp ð1 þ exp 2ð1c 2 sÞ ðz z0 Þ Þ h i h i ul ðzÞ ¼ u1 þ : 2qsp 2qsp exp 2ð1c 1 2 sÞ ðz z0 Þ ½k 1 þ exp 2ð1c2 sÞ ðz z0 Þ
ð68Þ
ð69Þ
4. Conclusions In this paper, the efficient factorization method that was proposed by Rosu and Cornejo-Pe´erez [20] has been applied to some important time-delayed nonlinear partial differential equations: Generalized time-delayed Burger’s Huxley equation, time-delayed convective Fisher equation, and generalized time-delayed Burger’s–Fisher. Exact particular solutions have been obtained and a two-parameter solution have been obtained for the time-delayed convective Fisher equation, Generalized time-delayed Burger’s Huxley equation when d ¼ 1, and for the generalized time-delayed Burger’s–Fisher when s ¼ 1.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
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[12] Wazwaz AM. The tanh-method: solitons and periodic solutions for the Dodd–Bullough–Mikhailov and the Tzitzeica–Dodd– Bullough equations. Chaos, Solitons & Fractals 2005;25:55. [13] Abdusalam HA, Alabdullatif M. Exact and approximate solutions of the telegraph model of dispersive variability. Far East J Dyn Syst 2006;8(1):65. [14] Fu ZT, Liu SK, Liu SD, Zhao Q. New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations. Phys Lett A 2001;290:72. [15] Kudryashov NA. Simplest equation method to look for exact solutions of nonlinear differential equations. Chaos, Solitons & Fractals 2005;24:1217. [16] Vladimirov VA, Kutafina EV. Exact travelling wave solutions of some nonlinear evolutionary equations. Rep Math Phys 2004;54:261. [17] Adomian G. Solving frontier problems of physics: The decomposition method. Boston (MA): Kluwer Academic Publishers; 1994. [18] Bobolian E, Javadi S. New method for calculating Adomian polynomials. Appl Math Comput 2004;153:253. [19] Rosu HC, Cornejo-Pe´erez O. Supersymmetric pairing of kinks for polynomial nonlinearities. Phys Rev E 2005;71:046607. [20] Cornejo-Pe´erez O, Rosu HC. Nonlinear second order ODE’S: factorization and particular solutions. Prog Theor Phys 2005;114:533. [21] Cornejo-Pe´erez O, Negro J, Nieto LM, Rosu HC. Travelling wave solutions for Korteweg–De Vries–Burger equations through factorizations. arXIV.math-ph/0604004v1. [22] Smirnov Yu F. Factorization method: new aspects. Rev Mex Fis 1999;45(S2):1. [23] Infeld L, Hull TE. The factorization method. Rev Mod Phys 1951;23:21. [24] Zwillinger D. Handbook of differential equations. San Diego: Academic Press Inc.; 1992. [25] Wang XY. Exact and explicit solitary wave solutions for the generalized Fisher equation. Phys Lett A 1988;131:277. [26] Reyes MA, Rosu HC. Two-parameter solutions of nonlinear second order ode’s. arXIV.math-ph/0510072v2. [27] Mielnik B. Factorization method and new potentials with the oscillator spectrum. J Math Phys 1984;25:3387. [28] Abdusalam HA, Fahmy ES. Exact solution for the generalized Telegraph Fisher’s equation. Chaos, Solitons & Fractals [submitted]. [29] Abdusalam HA, Fahmy ES. Exact solution for the modified equation for spatial population dynamics. Chaos, Solitons & Fractals [submitted]. [30] Wang XY, Zhu ZS, Lu YK. Solitary wave solutions of the generalized Burgers–Huxley equation. J Phys A 1990;23:271. [31] Scho¨nborn O, Desai RC, Stauffer D. Nonlinear bias and the convective Fisher equation. J Phys A 1994;27:L251.
Chaos, Solitons and Fractals 38 (2008) 1209–1216 www.elsevier.com/locate/chaos
Travelling wave solutions for some time-delayed equations through factorizations E.S. Fahmy King Saud University, Women Students Medical Studies & Sciences Sections, Mathematics Department, P.O. Box 22452, Riyadh 11495, Saudi Arabia Accepted 19 February 2007
Abstract In this work, we use factorization method to find explicit particular travelling wave solutions for the following important nonlinear second-order partial differential equations: The generalized time-delayed Burgers–Huxley, timedelayed convective Fishers, and the generalized time-delayed Burgers–Fisher. Using the particular solutions for these equations we find the general solutions, two-parameter solution, as special cases. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction A number of nonlinear phenomena in many branches of sciences such as physical [1], chemical, economical [2] and biological processes [3,4] are described by reaction and diffusion or by the interaction between convection and diffusion. One of the well known partial differential equations which governs a wide variety of them is Burgers equation which provides the simplest nonlinear model of turbulence [5]. The existence of relaxation time or delay time is an important feature in reaction diffusion and convection diffusion systems [6–8]. Many methods have been used to find exact solutions of nonlinear partial differential equations, such as the tanh function method [9–13], Jacobi elliptic function method [14], simplest equation method [15], unified algebraic method [16]. Also, many methods have been used to find numerical solutions of nonlinear partial differential equations, such as the Adomian decomposition method [17,18]. Recently, factorization of second-order linear differential equations became a well established technique to find solutions in an algebraic manner [19–29]. Rosu and Cornejo find one particular solution once the nonlinear equation is factorized with the use of two first order differential operators [20]. They use the method for equations of types: u00 þ cu0 þ f ðuÞ ¼ 0;
ð1Þ
u00 þ gðuÞu0 þ f ðuÞ ¼ 0;
ð2Þ
and
E-mail address: [email protected] 0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.02.007
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E.S. Fahmy / Chaos, Solitons and Fractals 38 (2008) 1209–1216
where 0 means the derivative D ¼ dzd , g(u) and f(u) are polynomials in u. We concentrate our work in this paper on equation of the type (2). Now, Eq. (2) can be factorized as ½D u2 ðuÞ½D u1 ðuÞu ¼ 0;
ð3Þ
which leads to the equation du1 0 uu u1 u0 u2 u0 þ u1 u2 u ¼ 0; du
ð4Þ
du1 u u1 þ u2 þ u u0 þ u1 u2 u ¼ 0: du
ð5Þ
u00 or
00
Comparing (5) and (2) we find du1 gðuÞ ¼ u1 þ u2 þ u du
and
f ðuÞ ¼ u1 u2 u:
ð6Þ
If Eq. (2) can be factorized as in Eq. (3), then a first particular solution can be easily found by solving ½D u1 ðuÞu ¼ 0:
ð7Þ
If u1(u) is a linear function of the dependent variable u, Eq. (7) turns out to be a Riccati equation for this variable and in this case we can find the general solution [26].
2. The generalized time-delayed Burger’s Huxley equation The generalized time-delayed Burger’s Huxley equation is given as: df ut ¼ aud ux þ uxx þ f ðuÞ; f ðuÞ ¼ buð1 ud Þðud cÞ; sutt þ 1 s du
ð8Þ
where a; b; c are constants and s is a time-delayed constant. It is clear that when s ¼ 0, Eq. (8) reduces to the generalized Burger’s Huxley equation discussed by Wang et al. in [30]. Using the coordinate transformation z ¼ x ct (c is the propagation speed) in Eq. (8) we obtain the following nonlinear ordinary differential equation: ð1 c2 sÞu00 þ ½ðc þ csbcÞ þ ða csbðd þ 1Þðc þ 1Þud Þ csbð2d þ 1Þu2d þ buð1 ud Þðud cÞ ¼ 0;
1 c2 s > 0; ð9Þ
or u00 þ gðuÞu0 þ F ðuÞ ¼ 0;
ð10Þ
where gðuÞ ¼
1 ½ðc þ csbcÞ þ ða csbðd þ 1Þðc þ 1Þud Þ csbð2d þ 1Þu2d ; 1 c2 s
ð11Þ
F ðuÞ ¼
b uð1 ud Þðud cÞ: ð1 c2 sÞ
ð12Þ
Using operator notation, Eq. (10) takes the form F ðuÞ D2 þ gðuÞD þ u ¼ 0: u
ð13Þ
The factorization of (13) leads to ½D u2 ðuÞ½D u1 ðuÞu ¼ 0;
ð14Þ
and then
du u00 u2 þ u1 þ 1 u u0 þ u1 u2 u ¼ 0: du
ð15Þ
E.S. Fahmy / Chaos, Solitons and Fractals 38 (2008) 1209–1216
Comparing (15) and (10) we obtain the conditions on u1 and u2 as: du F ðuÞ u2 þ u1 þ 1 u ¼ gðuÞ; u1 u2 ¼ ; du u
1211
ð16Þ
therefore u1 u2 ¼
bð1 ud Þðud cÞ : ð1 c2 sÞ
ð17Þ
Now, we choose u1 and u2 such that sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b 1 b d að1 u Þ; u2 ðuÞ ¼ ðud cÞ: u1 ðuÞ ¼ ð1 c2 sÞ a ð1 c2 sÞ From (18) and (11) we get pffiffiffi c 1 1 þ a da ud ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½ðc þ csbcÞ þ ða csbðd þ 1Þðc þ 1Þud Þ csbð2d þ 1Þu2d b a a a ð1 c2 sÞ
ð18Þ
ð19Þ
which implies that a ¼
ða csbðd þ 1Þðc þ 1ÞÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ða csbðd þ 1Þðc þ 1ÞÞ2 þ 4bð1 c2 sÞð1 þ dÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð1 þ dÞ bð1 c2 sÞ
then sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b að1 ud Þ; u1 ðuÞ ¼ ð1 c2 sÞ
1 u2 ðuÞ ¼ a
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b ðud cÞ; ð1 c2 sÞ
and the corresponding factorization is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " #" # 1 b b d d ðu ð1 u cÞ D a Þ u ¼ 0; D a ð1 c2 sÞ ð1 c2 sÞ and the compatible first order differential equation is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b 0 ð1 ud Þu ¼ 0: u a ð1 c2 sÞ By direct integration of (22) we get " sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #!1d b dðz z0 Þ ; u ðzÞ ¼ 1 exp a ð1 c2 sÞ where z0 is the integration constant. The solution (23) in hyperbolic form is given as: " #!1d pffiffiffi 1 1 a1 d b þ tanh pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðz z0 Þ ; u ðzÞ ¼ 2 2 2 ð1 c2 sÞ " #!1d pffiffiffi 1 1 a1 d b coth pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðz z0 Þ : u ðzÞ ¼ 2 2 2 ð1 c2 sÞ Now, if we choose the factorization terms as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b 1 b d ðu cÞ; u2 ðuÞ ¼ ð1 ud Þ: u1 ðuÞ ¼ b ð1 c2 sÞ b ð1 c2 sÞ and use (26) and (11) we obtain
ð20Þ
ð21Þ
ð22Þ
ð23Þ
ð24Þ
ð25Þ
ð26Þ
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E.S. Fahmy / Chaos, Solitons and Fractals 38 (2008) 1209–1216
pffiffiffi b
1 1 1 cb þ db þ b ud ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½ðc þ csbcÞ þ ða csbðd þ 1Þðc þ 1Þud Þ csbð2d þ 1Þu2d : b b ð1 c2 sÞ ð27Þ
Then b ¼
ða csbðd þ 1Þðc þ 1ÞÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ða csbðd þ 1Þðc þ 1ÞÞ2 þ 4bð1 c2 sÞð1 þ dÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2ð1 þ dÞ bð1 c2 sÞ
and Eq. (13) is then factorized in the following different form sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " #" # 1 b b d d ð1 u ðu Þ D b cÞ u ¼ 0: D b ð1 c2 sÞ ð1 c2 sÞ The corresponding compatible first order equation is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b 0 bðud cÞu ¼ 0: u ð1 c2 sÞ
ð28Þ
ð29Þ
Direct integration of Eq. (29) gives a different first order particular solution of Eq. (8), 11d 0 c B h qffiffiffiffiffiffiffiffiffiffiffi iC u ðzÞ ¼ @ A: b 1 exp b ð1c2 sÞdcðz z0 Þ
ð30Þ
Putting s ¼ 0 in (23) and (30) we find an exact particular solution for the generalized Burger’s Huxley equation [20]. Following the work of Reyes and Rosu [26], we find a two-parameter solution when d ¼ 1. In this case qffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffi u1 ðuÞ ¼ a ð1cb 2 sÞð1 uÞ and u2 ðuÞ ¼ 1a ð1cb 2 sÞðu cÞ, one particular solution is obtained from (23) as: " sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #!1 b ðz z0 Þ ; ð31Þ u1 ðzÞ ¼ 1 þ exp a ð1 c2 sÞ and Eq. (22) is transformed to the following Riccati equation sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b ð1 uÞu ¼ 0: u0 a ð1 c2 sÞ The two-parameter solution is obtained from (31) and (32) as: h qffiffiffiffiffiffiffiffiffiffiffi i1 1 þ exp a ð1cb 2 sÞðz z0 Þ h qffiffiffiffiffiffiffiffiffiffiffi ih h qffiffiffiffiffiffiffiffiffiffiffi i i: uk ¼ u1 þ exp a ð1cb 2 sÞðz z0 Þ k 1 þ exp a ð1cb 2 sÞðz z0 Þ 1
ð32Þ
ð33Þ
It is clear that when jkj runs from zero to infinity, the above parametric solution goes from the trivial solution u ¼ 0 to the particular solution u ¼ u1 .
3. Time-delayed convective Fisher’s equation The convective Fisher’s equation given in the following form [31]: 1 ut ¼ uxx luux þ uð1 uÞ 2 and the time-delayed convective Fisher equation can be obtained as: df 1 ut ¼ uxx luux þ uð1 uÞ; sutt þ 1 s du 2 where l is a positive parameter that serves to tune the relative strength of convection.
ð34Þ
ð35Þ
E.S. Fahmy / Chaos, Solitons and Fractals 38 (2008) 1209–1216
1213
Using the coordinate transformation z ¼ x ct in Eq. (35) we obtain the following nonlinear ordinary differential equation u00 þ
2 2 ½cð1 sÞ þ ð2sc lÞuu0 þ uð1 uÞ ¼ 0; ð1 2c2 sÞ ð1 2c2 sÞ
1 > 2c2 s
ð36Þ
or u00 þ gðuÞu0 þ F ðuÞ ¼ 0;
ð37Þ
where gðuÞ ¼
2½cð1 sÞ þ ð2sc lÞu ð1 2c2 sÞ
and
F ðuÞ ¼
2 uð1 uÞ: ð1 2c2 sÞ
ð38Þ
Now, Eq. (37) can be factorized as ½D u2 ðuÞ½D u1 ðuÞu ¼ 0:
ð39Þ
Using operator notation, Eq. (37) takes the form F ðuÞ D2 þ gðuÞD þ u ¼ 0; u
ð40Þ
and then
du u00 u2 þ u1 þ 1 u u0 þ u1 u2 u ¼ 0: du
ð41Þ
Comparing (41) and (37) we obtain the conditions on u1 and u2 as: du F ðuÞ ; u2 þ u1 þ 1 u ¼ gðuÞ; u1 u2 ¼ u du
ð42Þ
therefore u1 u2 ¼
2 ð1 uÞ: ð1 2c2 sÞ
Now, choosing u1 and u2 such that sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 rð1 uÞ and u1 ðuÞ ¼ ð1 2c2 sÞ
ð43Þ
1 u2 ðuÞ ¼ r
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ; ð1 2c2 sÞ
and using (44) and (42) we get sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 ½cð1 sÞ þ ð2sc lÞu; þ r½1 2u ¼ r ð1 2c2 sÞ
r 6¼ 0;
ð44Þ
ð45Þ
hence ð2sc lÞ r ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2ð1 2c2 sÞ
1 2c2 s > 0;
and Eq. (39) is reduced to 2 ð2sc lÞ D D ð1 uÞ u ¼ 0: ð2sc lÞ ð1 2c2 sÞ
ð46Þ
ð47Þ
The compatible first order differential equation is u0
ð2sc lÞ ð1 uÞu ¼ 0: ð1 2c2 sÞ
By direct integration we get 1 ð2sc lÞðz z0 Þ u ¼ 1 exp ð1 2c2 sÞ
ð48Þ
ð49Þ
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E.S. Fahmy / Chaos, Solitons and Fractals 38 (2008) 1209–1216
or in hyperbolic form:
1 1 ð2sc lÞ ðz z Þ ; tanh 0 2 2 2ð1 2c2 sÞ 1 1 ð2sc lÞ ðz z0 Þ : u ðzÞ ¼ coth 2 2 2ð1 2c2 sÞ
uþ ðzÞ ¼
ð50Þ ð51Þ
Putting s ¼ 0 in (23) we find an exact particular solution for the convective Fisher equation [20]. Using (48) and (49), a two-parameter solution is obtained as: h i 1 ð2sclÞ ð1 þ exp 2ð12c 2 sÞ ðz z0 Þ Þ h ih h i i: uk ¼ uþ þ ð2sclÞ ð2sclÞ exp 2ð12c k 1 þ exp 2ð12c 1 2 sÞ ðz z0 Þ 2 sÞ ðz z0 Þ
ð52Þ
3.1. The generalized time-delayed Burgers–Fisher equation The generalized Burgers–Fisher equation is given in the following form: ut ¼ uxx pus ux þ quð1 us Þ
ð53Þ
and the generalized time-delayed Burgers–Fisher equation can be obtained as: df sutt þ 1 s ut ¼ uxx pus ux þ f ðuÞ; f ðuÞ ¼ quð1 us Þ; du
ð54Þ
where s, p, q are any real numbers and s 2 N. Eq. (54) is an extended form of the generalized Burger’s Fisher equation (53). When q ¼ 0, Eq. (54) is reduced to the time-delayed Burger’s Fisher equation [6]. Using the coordinate transformation z ¼ x ct in Eq. (54), we get ð1 c2 sÞu00 þ ½cðð1 qsÞ þ qsðs þ 1ÞÞus pus u0 þ f ðuÞ ¼ 0;
1 > c2 s;
ð55Þ
or u00 þ gðuÞu0 þ F ðuÞ ¼ 0;
ð56Þ
where gðuÞ ¼
cð1 qsÞ ½qsðs þ 1Þ p s þ u ð1 c2 sÞ ð1 c2 sÞ
and
F ðuÞ ¼
q uð1 us Þ: ð1 c2 sÞ
ð57Þ
Now, Eq. (56) can be factorized as ½D u2 ðuÞ½D u1 ðuÞu ¼ 0;
ð58Þ
where u1 ðuÞ ¼ a1
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q ð1 us Þ and ð1 c2 sÞ
u2 ðuÞ ¼
1 a1
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q ; ð1 c2 sÞ
a1 6¼ 0;
and using (59) and (56) we get rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q 1 cð1 qsÞ ½qsðs þ 1Þ p s s þ u; ¼ þ a ðs þ 1Þu 1 ð1 c2 sÞ a1 ð1 c2 sÞ ð1 c2 sÞ
ð59Þ
ð60Þ
and hence a1 ¼
½qsðs þ 1Þ p pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ðs þ 1Þ qð1 c2 sÞ
Using Eq. (61), Eq. (58) reduced to qðs þ 1Þ ½qsðs þ 1Þ pð1 us Þ D u ¼ 0: D qsðs þ 1Þ p ðs þ 1Þð1 c2 sÞ The compatible first order differential equation is
ð61Þ
ð62Þ
E.S. Fahmy / Chaos, Solitons and Fractals 38 (2008) 1209–1216
u0
½qsðs þ 1Þ p ð1 us Þu ¼ 0: ðs þ 1Þð1 c2 sÞ
1215
ð63Þ
By direct integration we get 1s ½qsðs þ 1Þ ps Þ ; ðz z u ¼ 1 exp 0 ðs þ 1Þð1 c2 sÞ
ð64Þ
or uþ ðzÞ ¼
1s 1 1 ½qsðs þ 1Þ ps tanh ðz z Þ ; 0 2 2 ðs þ 1Þð1 c2 sÞ 0 2 311s
B1 1 6 ½qsðs þ 1Þ ps 7C qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðz z0 Þ5A : u ðzÞ ¼ @ coth 4 2 2 2 2 ðs þ 1Þ ð1 c sÞ
ð65Þ
ð66Þ
Following the work of Reyes and Rosu [26,27], we find a two-parameter solution for the special case when s ¼ 1. In 2
2ð1c sÞ 2qsp this case u1 ðuÞ ¼ 2ð1c 2 sÞð1 uÞ and u2 ðuÞ ¼ 2qsp , one particular solution is obtained from (64) as
u1 ðzÞ ¼ ð1 þ exp½hðz z0 ÞÞ1
ð67Þ
and Eq. (63) is transformed to the following Riccati equation u0
2qs p ð1 uÞu ¼ 0: 2ð1 c2 sÞ
The two-parameter solution is obtained from (67) and (68) as: h i 1 2qsp ð1 þ exp 2ð1c 2 sÞ ðz z0 Þ Þ h i h i ul ðzÞ ¼ u1 þ : 2qsp 2qsp exp 2ð1c 1 2 sÞ ðz z0 Þ ½k 1 þ exp 2ð1c2 sÞ ðz z0 Þ
ð68Þ
ð69Þ
4. Conclusions In this paper, the efficient factorization method that was proposed by Rosu and Cornejo-Pe´erez [20] has been applied to some important time-delayed nonlinear partial differential equations: Generalized time-delayed Burger’s Huxley equation, time-delayed convective Fisher equation, and generalized time-delayed Burger’s–Fisher. Exact particular solutions have been obtained and a two-parameter solution have been obtained for the time-delayed convective Fisher equation, Generalized time-delayed Burger’s Huxley equation when d ¼ 1, and for the generalized time-delayed Burger’s–Fisher when s ¼ 1.
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