J. Res. Appl. Sci. Vol., 2(4): 101-107, 2015
Journal of Research in Applied Sciences. Vol., 2(4): 101-107, 2015
Available online at http://www.jrasjournal.com
ISSN 2148-6662 © Copyright 2015
ORIGINAL ARTICLE
An Infinite Sequence of Exact Solutions of
The
Reaction-convection-diffusion
Equation
According To A Riccati-Bernoulli Sub-ODE
Method
Emad H.M. Zahrana, Mostafa M. A. Khaterb*
a
Department of Engineering Mathematics and Physics, Faculty of Engineering Shubra, Benha University,
Egypt
b
Department of Mathematics, Faculty of Science, Mansoura University, 35516 Mansoura, Egypt
*Corresponding Author Email: [email protected]
Abstract: In this article, the exact solutions of the reaction-convection-diffusion equation which play an
important role in many branches of physics and biology is investigate as the first time in the framework of
Riccat-Bernoulli Sub-ODE method. The solutions obtained can be generating an infinite sequence of exact
solutions according to Backlund transformation. The proposed method also can be used for many other
nonlinear evolution equations.
Keywords: Riccati-Bernoulli Sub-ODE method, Bäcklund transformation of the Riccati-Bernoulli equation,
Reaction-convection-diffusion equation, Traveling wave solution, Solitary wave solution.
AMS subject classifications: 35A05, 35A20, 65K99, 65Z05, 76R50, 70K70.
Introduction
al., 2009), trigonometric function series method
The nonlinear equations of mathematical
physics are major subjects in physical science
(Ablowitz & Segur, 1981). Exact solutions for
these equations play an important role in many
phenomena in physics such as fluid mechanics,
hydrodynamics, Optics, Plasma physics and so
on. Recently many new approaches for finding
these solutions have been proposed, for example,
tanh - sech method (Maliet, 1992, 1996;
Wazwaz, 2004), extended tanh – method (ELWakil & Abdou, 2007; Fan, 2000; Wazwaz,
2007), sine - cosine method (Yan, 1996;
Wazwaz, 2004, 2005), homogeneous balance
method (Fan, 1998) and (Wang , 1996), Jacobi
elliptic function method (Liu et al., 2001; Fan &
Zhang, 2002; Dai & Zhang, 2006; Zhao et al.,
2006), F-expansion method (Ren & Zhang,
2006; Zhang et al., 2006; Abdou, 2007), expfunction method (He & Wu, 2006; Aminikhad et
(Zhang, 2008), ( ) - expansion method (Wang
𝐺
et al., 2008; Zhang et al., 2008; Zayed &
Gepreel, 2009; Zayed, 2009), the modified
simple equation method (Jawad et al., 2010;
Zayed, 2011; Zayed & Hoda Ibrahim, 2012,
2013; Zayed & Arnous, 2012) and so on.
In the present paper, we shall propose a new
method which is called the Riccati-Bernoulli
Sub-ODE method (Xiao-Feng et al., 2015) to
seek traveling wave solutions of nonlinear
evolution equations. We can obtain a new
infinite sequence of solutions of the NLPDEs by
using a Bäcklund transformation. The paper is
organized as follows: In section 2, we give the
description of the Riccati-Bernoulli Sub-ODE
method. In section 3, Bäcklund transformation of
the Riccati-Bernoulli equation. In section 4, we
use this method to find infinite sequence of exact
solutions for the perturbed nonlinear Schrodinger
101
𝐺′
J. Res. Appl. Sci. Vol., 2(4): 101-107, 2015
equation with Kerr law non linearity pointed out
above and some figures of our results are drawn.
In section 5, conclusion is given.
Description of the Riccati-Bernoulli Sub-ODE
method
Consider the following nonlinear evolution
equation
𝑃(𝑢, 𝑢𝑡 , 𝑢𝑥 , 𝑢𝑡𝑡 , 𝑢𝑥𝑥 , … . . ) = 0,
where P is in general a polynomial function of its
arguments, the subscripts denote the partial
derivatives. The Riccati-Bernoulli Sub-ODE
method consists of three steps.
with
(2.1)
Step 1. Combining the independent variables x
and t into one variable
 = 𝑘(𝑥 + 𝑉𝑡),
(2.2)
𝑢(𝑥, 𝑡) = 𝑢(),
(2.3)
where the localized wave solution 𝑢() travels with speed 𝑉, by using Eqs.(2.2) and (2.3), one can
transform Eq.(2.1) to an ODE
𝑃(𝑢, 𝑢′ , 𝑢′′ , 𝑢′′′ , … . . ) = 0,
(2.4)
𝑑𝑢
where 𝑢′ denotes 𝑑
Step 2. Suppose that the solution of Eq.(2.4) is the solution of the Riccati-Bernoulli equation
𝑢′ = 𝑎𝑢2−𝑚 + 𝑏𝑢 + 𝑐𝑢𝑚 ,
(2.5)
where a, b, c, and m are constants to be determined later. From Eq.(2.5) and by directly calculating, we get
2−𝑚
3−2𝑚
2𝑚−1
𝑢′′ = 𝑎𝑏(3 − 𝑚)(𝑢(𝑥))
+ 𝑎2 (2 − 𝑚)(𝑢(𝑥))
+ 𝑚𝑐 2 (𝑢(𝑥))
𝑚
𝑏𝑐(𝑚 + 1)(𝑢(𝑥)) + (2𝑎𝑐 + 𝑏 2 )𝑢,
+
𝑢′′′ = (𝑎𝑏(3 − 𝑚)(2 − 𝑚)𝑢1−𝑚 + 𝑎2 (2 − 𝑚)(3 − 2𝑚)𝑢2−2𝑚 + 𝑚(2𝑚 − 1)𝑐 2 𝑢2𝑚−2 +
𝑏𝑐(𝑚 + 1)𝑢𝑚−1 + 2𝑎𝑐 + 𝑏 2 )𝑢′ ,
Remark: When ac  0and m = 0, Eq.(2.5) is a
Riccati equation. When a  0, c = 0, and m  1,
Eq.(2.5) is a Bernoulli equation. Obviously, the
Riccati equation and Bernoulli equation are
special cases of Eq.(2.5). Because Eq.(2.5) is
(2.6)
(2.7)
firstly proposed, we call Eq.(2.5) the RiccatiBernoulli equation in order to avoid introducing
new terminology. Equation (2.5) has solutions as
follows:
Case 1. When m = 1, the solution of Eq. (2.5) is
𝑢() = 𝐶𝑒 (𝑎+𝑏+𝑐) ,
(2.8)
Case 2. When m  1, b = 0 and c = 0, the solution of Eq. (2.5) is
1
𝑢() = (𝑎(𝑚 − 1)( + 𝐶))(1−𝑚) ,
(2.9)
Case 3. When m  1, b  0 and c = 0, the solution of Eq. (2.5) is
1
𝑢() =
102
𝑎
(−
𝑏
+ 𝐶𝑒
𝑏(𝑚−1) (𝑚−1)
)
,
(2.10)
J. Res. Appl. Sci. Vol., 2(4): 101-107, 2015
Case 4. When m  1, a  0 and 𝑏 2 − 4𝑎𝑐 < 0, the solution of Eq. (2.5) is
1
−𝑏
( 2𝑎
𝑢() =
+
√4𝑎𝑐−𝑏2 𝑡𝑎𝑛(1/2(1−𝑚)√4𝑎𝑐−𝑏2 (+𝐶)) (1−𝑚)
)
2𝑎
(2.11)
,
and
1
−𝑏
𝑢() = ( 2𝑎 +
√4𝑎𝑐−𝑏2 𝑐𝑜𝑡(1/2(1−𝑚)√4𝑎𝑐−𝑏2 (+𝐶)) (1−𝑚)
)
2𝑎
(2.12)
,
Case 5. When m  1, a  0 and 𝑏 2 − 4𝑎𝑐 > 0, the solution of Eq. (2.5) is
1
𝑢() =
−𝑏
( 2𝑎
+
√𝑏2 −4𝑎𝑐𝑐𝑜𝑡ℎ(1/2(1−𝑚)√𝑏2 −4𝑎𝑐(+𝐶)) (1−𝑚)
)
2𝑎
(2.13)
,
and
1
𝑢() =
−𝑏
( 2𝑎
+
√𝑏2 −4𝑎𝑐𝑡𝑎𝑛ℎ(1/2(1−𝑚)√𝑏2 −4𝑎𝑐(+𝐶)) (1−𝑚)
)
2𝑎
(2.14)
,
Case 6. When m  1, a  0 and 𝑏 2 − 4𝑎𝑐 = 0 the solution of Eq. (2.5) is
1
(2.15)
1
𝑏 (1−𝑚)
− )
,
𝑎(𝑚−1)(+𝐶)
2𝑎
𝑢() = (
where C is an arbitrary constant.
Step 3. Substituting the derivatives of 𝑢 into
Eq.(2.4) yields an algebraic equation of 𝑢.
Noticing the symmetry of the right-hand item of
Eq.(2.5) and setting the highest power exponents
of 𝑢 to equivalence in Eq.(2.4), m can be
determined. Comparing the coefficients of 𝑢𝑖
yields a set of algebraic equations for a, b, c, and
𝑉. Solving the set of algebraic equations and
substituting m, a, b, c, 𝑉, and  = 𝑘(𝑥 + 𝑉𝑡) into
since,
𝑢𝑛′ =
Eq.(2.8)-(2.15), we can get traveling wave
solutions of Eq.(2.1).
Bäcklund transformation of the RiccatiBernoulli equation
When 𝑢𝑛−1 ()and 𝑢𝑛 () are solutions of
Eq.(2.5), then
𝑢𝑛′ = 𝑎𝑢𝑛2−𝑚 + 𝑏𝑢𝑛 + 𝑐𝑢𝑛𝑚 ,
(3.1)
′
2−𝑚
𝑚
𝑢𝑛−1
= 𝑎𝑢𝑛−1
+ 𝑏𝑢𝑛−1 + 𝑐𝑢𝑛−1
,
(3.2)
𝑑𝑢𝑛 ()
𝑑𝑢𝑛 () 𝑑𝑢𝑛−1 ()
𝑑𝑢𝑛 ()
2−𝑚
𝑚 ]
[𝑎𝑢𝑛−1
=
=
+ 𝑏𝑢𝑛−1 + 𝑐𝑢𝑛−1
𝑑
𝑑𝑢𝑛−1 ()
𝑑
𝑑𝑢𝑛−1 ()
Now, from Eq.(3.1) and (3.3), we obtain
𝑑𝑢𝑛 ()
2−𝑚
[𝑎𝑢𝑛−1
𝑛−1 ()
𝑎𝑢𝑛2−𝑚 + 𝑏𝑢𝑛 + 𝑐𝑢𝑛𝑚 = 𝑑𝑢
i. e.
𝑎𝑢𝑛2−𝑚
𝑚 ],
+ 𝑏𝑢𝑛−1 + 𝑐𝑢𝑛−1
𝑑𝑢𝑛 ()
𝑑𝑢𝑛−1 ()
2−𝑚
𝑚
𝑚 =
+ 𝑏𝑢𝑛 + 𝑐𝑢𝑛
𝑎𝑢𝑛−1 + 𝑏𝑢𝑛−1 + 𝑐𝑢𝑛−1
(3.3)
(3.4)
(3.5)
Integrating above equation once with respect to , we get
1−𝑚
1
1−𝑚
−𝑐𝐴1 + 𝑎𝐴2 (𝑢𝑛−1 ())
𝑢𝑛 () = (
)
1−𝑚
𝑏𝐴2 + 𝑎𝐴1 (𝑢𝑛−1 ())
103
(3.6)
J. Res. Appl. Sci. Vol., 2(4): 101-107, 2015
Application
Here, we will apply A Riccati-Bernoulli SubODE method described in sec.2 to find the exact
traveling wave solutions and then the solitary
wave solutions of reaction-convection-diffusion
equation. Consider the following reactionconvection-diffusion equation of the form
where 𝐴1 and 𝐴2 are arbitrary constants.
According to Eq.(3.6), we can get infinite
sequence of solutions of Eq.(2.5) and hence we
can get infinite sequence of solutions of Eq.(2.1).
𝑢𝑡 = ( + 0 𝑢)𝑢𝑥𝑥 + 1 𝑢𝑢𝑥 + 2 𝑢 − 3 𝑢3,
(4.1)
where , 0 , 1 , 2 and 3 are real constants (Zayed & Arnous, 2012). In the particular case  = 1and 0 =
0, this equation coincides with the Murray equation
𝑢𝑡 = 𝑢𝑥𝑥 + 𝑢𝑢𝑥 + 2 𝑢 − 3 𝑢2 ,
(4.2)
which itself is a generalization of the well-known Fisher equation when 1 = 0. When both 2 and 1 are
zero, it is reduced to the classical Burgers equation. We introduce the traveling wave variable 𝑢(𝑥, 𝑡) =
𝑢();  = 𝑘(𝑥 + 𝑡) into Eq. (4.2) to find
𝑘 2 𝑢′′ − 1 𝑘𝑢𝑢′ + 2 𝑢 − 3 𝑢2 − 𝑘𝑢′ = 0,
(4.3)
where prime denotes the derivatives with respect to . Substituting Eq.(2.5) and its derivatives into
Eq.(4.3), we get
𝑘 2 𝑎𝑏(3 − 𝑚)𝑢2−𝑚 + 𝑘 2 𝑎2 (2 − 𝑚)𝑢3−2𝑚 + 𝑚𝑘 2 𝑐 2 𝑢2𝑚−1 + 𝑘 2 𝑏𝑐(𝑚 + 1)𝑢𝑚 +
𝑘
+ 𝑏 2 )𝑢 + 1 𝑘(𝑎𝑢3−𝑚 + 𝑏𝑢2 + 𝑐𝑢𝑚+1 ) − 𝑘(𝑎𝑢2−𝑚 + 𝑏𝑢 + 𝑐𝑢𝑚 ) + 2 𝑢 −
3 𝑢2 = 0,
(4.4)
2 (2𝑎𝑐
Setting m = 2 and c = 0, we get
𝑘 2 𝑎𝑏 + 𝑘 2 𝑏2 𝑢 + 1 𝑘𝑎𝑢 + 1 𝑘𝑏𝑢2 − 𝑘𝑏𝑢 − 𝑘𝑎 + 2 𝑢 − 3 𝑢2 = 0,
(4.5)
setting the coefficient of 𝑢𝑖 , i = 0, 1, 2 to zero, we get
𝑢2 : 1 𝑘𝑏 − 3 = 0,
(4.6)
𝑢1 : 𝑘 2 𝑏 2 + 1 𝑘𝑎 + 2 − 𝑘𝑏 = 0,
(4.7)
𝑢0 : 𝑘 2 𝑎𝑏 − 𝑘𝑎 = 0,
(4.8)
𝑘𝑏 = ,
(4.9)
Solving (4.6)-(4.8), we get
𝑘𝑎 =
− 2
(4.10)
1
Case A. When m  1, b  0 and c = 0, the solution of Eq. (4.3) is
2
1
𝑢() = (
+ 𝐶𝑒 (𝑥+𝑡) ),
Case B. When m  1, a  0 and 𝑏 2 − 4𝑎𝑐 > 0, the solution of Eq. (4.3) is
104
(4.11)
J. Res. Appl. Sci. Vol., 2(4): 101-107, 2015
𝑢() =

( 1
22
+
−1
√𝑏2 (𝑘(𝑥+𝑡)+𝐶))
2
√𝑏2 𝑐𝑜𝑡ℎ(
−1
(4.12)
) ,
2𝑎
and
𝑢() =

(21
2
+
−1
√𝑏2 (𝑘(𝑥+𝑡)+𝐶))
2
√𝑏2 𝑡𝑎𝑛ℎ(
−1
(4.13)
) ,
2𝑎
a
b
Eq.(4.11
)
Eq.(4.12
)
c
Eq.(4.13
)
Figure 1. Solarity wave solutions of Eqs.(4.11),(4.12) and (4.13).
Conclusion
In this paper, we note that the Riccati-Bernoulli
Sub- ODE method is given a more accurate and a
wide range of solutions of nonlinear partial
105
differential equations where the physical
meaning of figures: when the parameters takes
the values (x= -5:5, t= -5:5) give the bell singular
solution, kink shape solution and dark periodic
solution of the above figures respectively and
J. Res. Appl. Sci. Vol., 2(4): 101-107, 2015
furthermore infinite sequence of exact solutions
of the reaction-convection-diffusion equation can
be obtained according to a Backlund
transformation of the Riccati-Bernoulli equation.
Also, the new method can be used for many
other nonlinear evolution equations.
Conflict of interest
The authors declare no conflict of interest.
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