An infinite sequence of exact solutions of the reactionconvection-diffusion equation according to A RiccatiBernoulli Sub-ODE method
Emad H.M. Zahran and Mostafa M. A. Khater*
*Department of Engineering Mathematics and Physics, Faculty of Engineering Shubra,
Benha University, Egypt
Department of Mathematics, Faculty of Science, Mansoura University, 35516 Mansoura, Egypt.
*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: The Riccati-Bernoulli Sub-ODE method; Bäcklund transformation of the
Riccati-Bernoulli equation; The reaction-convection-diffusion equation; traveling wave solution;
solitary wave solution.
AMS subject classifications: 35A05, 35A20, 65K99, 65Z05, 76R50, 70K70
1 Introduction
The nonlinear equations of mathematical physics are major subjects in physical science [1]. 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 [2]-[4], extended
tanh - method [5]-[7], sine - cosine method [8]-[10], homogeneous balance method [11]and [12],
Jacobi elliptic function method [13]-[16], F-expansion method [17]-[19], exp-function method
𝐺′
[20]-[21], trigonometric function series method [22], ( 𝐺 ) - expansion method [23]-[26], the
modified simple equation method [27]-[32] and so on.
In the present paper, we shall propose a new method which is called the Riccati-Bernoulli SubODE method [33] 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 equation with Kerr law non linearity pointed out above and some figures
of our results are drawn. In section 5, conclusion is given.
2 Description of the Riccati-Bernoulli Sub-ODE method
Consider the following nonlinear evolution equation
𝑃(𝑢, 𝑢𝑡 , 𝑢𝑥 , 𝑢𝑡𝑡 , 𝑢𝑥𝑥 , … . . ) = 0,
(2.1)
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.
Step 1. Combining the independent variables x and t into one variable
with
 = 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,
where 𝑢′ denotes
(2.4)
𝑑𝑢
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′ ,
(2.6)
(2.7)
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 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
(2.8)
𝑢() = 𝐶𝑒 (𝑎+𝑏+𝑐) ,
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() =
𝑎
(− 𝑏
+ 𝐶𝑒
𝑏(𝑚−1) (𝑚−1)
)
,
(2.10)
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 Eq.(2.8)-(2.15), we can get traveling wave solutions of Eq.(2.1).
3 Bäcklund transformation of the Riccati-Bernoulli equation
When 𝑢𝑛−1 ()and 𝑢𝑛 () are solutions of Eq.(2.5), then
since,
𝑢𝑛′ =
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 ()
(3.3)
Now, from Eq.(3.1) and (3.3), we obtain
𝑎𝑢𝑛2−𝑚 + 𝑏𝑢𝑛 + 𝑐𝑢𝑛𝑚 =
i. e.
𝑎𝑢𝑛2−𝑚
𝑑𝑢𝑛 ()
2−𝑚
[𝑎𝑢𝑛−1
𝑑𝑢𝑛−1 ()
𝑚 ],
+ 𝑏𝑢𝑛−1 + 𝑐𝑢𝑛−1
𝑑𝑢𝑛 ()
𝑑𝑢𝑛−1 ()
2−𝑚
𝑚
𝑚 =
+ 𝑏𝑢𝑛 + 𝑐𝑢𝑛
𝑎𝑢𝑛−1 + 𝑏𝑢𝑛−1 + 𝑐𝑢𝑛−1
(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 ())
(3.6)
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).
4 Application
Here, we will apply A Riccati-Bernoulli Sub-ODE 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 reaction-convection-diffusion equation of the form
(4.1)
𝑢𝑡 = ( + 0 𝑢)𝑢𝑥𝑥 + 1 𝑢𝑢𝑥 + 2 𝑢 − 3 𝑢3,
where , 0 , 1 , 2 and 3 are real constants [30]. 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 2 (2𝑎𝑐 + 𝑏 2 )𝑢 + 1 𝑘(𝑎𝑢3−𝑚 + 𝑏𝑢2 + 𝑐𝑢𝑚+1 ) − 𝑘(𝑎𝑢2−𝑚 + 𝑏𝑢 + 𝑐𝑢𝑚 ) + 2 𝑢 −
3 𝑢2 = 0,
(4.4)
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) ),
(4.11)
Case B. When m  1, a  0 and 𝑏 2 − 4𝑎𝑐 > 0, the solution of Eq. (4.3) is
𝑢() =

(21
2
𝑢() =

( 1
22
+
−1
−1
√𝑏2 (𝑘(𝑥+𝑡)+𝐶))
2
√𝑏2 𝑐𝑜𝑡ℎ(
2𝑎
(4.12)
) ,
and
+
−1
√𝑏2 (𝑘(𝑥+𝑡)+𝐶))
2
√𝑏2 𝑡𝑎𝑛ℎ(
2𝑎
−1
) ,
(4.13)
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)
5 Conclusions
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 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 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.
6 Competing interests
This research received no specific grant from any funding agency in the public, commercial, or
Not-for-profit sectors. The author did not have any competing interests in this research.
7 Author's contributions
All parts contained in the research carried out by the researcher through hard work and a review
of the various references and contributions in the field of mathematics and the physical Applied
References
[1] M. J. Ablowitz, H. Segur, Solitions and Inverse Scattering Transform, SIAM, Philadelphia
1981.
[2] W. Maliet, Solitary wave solutions of nonlinear wave equation, Am. J. Phys., 60 (1992)
650-654.
[3] W. Maliet, W. Hereman, The tanh method: Exact solutions of nonlinear evolution and
wave equations, Phys.Scr., 54 (1996) 563-568.
[4] A. M. Wazwaz, The tanh method for travelling wave solutions of nonlinear equations,
Appl. Math. Comput., 154 (2004) 714-723.
[5] S. A. EL-Wakil, M.A.Abdou, New exact travelling wave solutions using modified
extended tanh-function method, Chaos Solitons Fractals, 31 (2007) 840-852.
[6] E. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys.
Lett. A 277 (2000) 212-218.
[7] A. M.Wazwaz, The extended tanh method for abundant solitary wave solutions of
nonlinear wave equations, Appl. Math. Comput., 187 (2007) 1131-1142.
[8] A. M. Wazwaz, Exact solutions to the double sinh-Gordon equation by the tanh method
and a variable separated ODE. method, Comput. Math. Appl., 50 (2005) 1685-1696.
[9] A. M.Wazwaz, A sine-cosine method for handling nonlinear wave equations, Math.
Comput. Modelling, 40 (2004) 499-508.
[10] C. Yan, A simple transformation for nonlinear waves, Phys. Lett. A 224 (1996) 77-84.
[11] E. Fan, H.Zhang, A note on the homogeneous balance method, Phys. Lett. A 246 (1998)
403-406.
[12] M. L. Wang, Exact solutions for a compound KdV-Burgers equation, Phys. Lett. A 213
(1996) 279-287.
[13] C. Q. Dai , J. F. Zhang, Jacobian elliptic function method for nonlinear differential
?difference equations, Chaos Solutions Fractals, 27 (2006) 1042-1049.
[14] E. Fan , J .Zhang, Applications of the Jacobi elliptic function method to special-type
nonlinear equations, Phys. Lett. A 305 (2002) 383-392.
[15] S. Liu, Z. Fu, S. Liu, Q.Zhao, Jacobi elliptic function expansion method and periodic wave
solutions of nonlinear wave equations, Phys. Lett. A 289 (2001) 69-74.
[16] X. Q. Zhao, H.Y.Zhi, H.Q.Zhang, Improved Jacobi-function method with symbolic
computation to construct new double-periodic solutions for the generalized Ito system,
Chaos Solitons Fractals, 28 (2006) 112-126.
[17] M. A. Abdou, The extended F-expansion method and its application for a class of nonlinear
evolution equations, Chaos Solitons Fractals, 31 (2007) 95-104.
[18] Y. J. Ren, H. Q. Zhang, A generalized F-expansion method to find abundant families of
Jacobi elliptic function solutions of the (2+1)-dimensional Nizhnik-Novikov-Veselov
equation, Chaos Solitons Fractals, 27 (2006) 959-979.
[19] J. L. Zhang, M. L. Wang, Y. M. Wang, Z. D. Fang, The improved F-expansion method and
its applications, Phys.Lett.A 350 (2006) 103-109.
[20] J. H. He, X. H. Wu, Exp-function method for nonlinear wave equations, Chaos Solitons
Fractals 30 (2006) 700-708.
[21] H. Aminikhad, H. Moosaei, M. Hajipour, Exact solutions for nonlinear partial differential
equations via Exp-function method, Numer. Methods Partial Differ. Equations, 26 (2009)
1427-1433.
[22] Z. Y. Zhang, New exact traveling wave solutions for the nonlinear Klein-Gordon equation,
Turk. J. Phys., 32 (2008) 235-240.
𝐺′
[23] M. L. Wang, J. L. Zhang, X. Z. Li, The ( 𝐺 ) - expansion method and travelling wave
solutions of nonlinear evolutions equations in mathematical physics, Phys. Lett. A 372
(2008) 417-423.
𝐺′
[24] S. Zhang, J. L. Tong, W.Wang, A generalized ( ) - expansion method for the mKdv
𝐺
equation with variable coefficients, Phys. Lett. A 372 (2008) 2254-2257.
𝐺′
[25] E. M. E. Zayed and K. A. Gepreel, The ( 𝐺 ) - expansion method for finding traveling wave
solutions of nonlinear partial differential equations in mathematical physics, J. Math. Phys.,
50 (2009) 013502-013513.
𝐺′
[26] E. M. E. Zayed, The ( 𝐺 ) - expansion method and its applications to some nonlinear
evolution equations in mathematical physics, J. Appl. Math. Computing, 30 (2009) 89-103.
[27] A. J. M . Jawad, M. D. Petkovic and A. Biswas, Modified simple equation method for
nonlinear evolution equations, Appl. Math. Comput., 217 (2010) 869-877.
[28] E. M. E. Zayed, A note on the modified simple equation method applied to Sharam-TassoOlver equation, Appl. Math. Comput., 218 (2011) 3962-3964.
[29] E. M. E. Zayed and S. A. Hoda Ibrahim, Exact solutions of nonlinear evolution equation in
mathematical physics using the modified simple equation method, Chin. Phys. Lett., 29
(2012) 060201-4.
[30] E. M. E. Zayed and A. H. Arnous, Exact solutions of the nonlinear ZK-MEW and the
potential YTSF equations using the modified simple equation method, AIP Conf. Proc.,
1479 (2012) 2044-2048.
[31] E. M. E. Zayed and S. A. Hoda Ibrahim, Modified simple equation method and its
applications for some nonlinear evolution equations in mathematical physics, Int. J.
Computer Appl., 67 (2013) 39-44.
[32] E. M. E. Zayed and S. A. Hoda Ibrahim, Exact solutions of Kolmogorov-Petrovskii
Piskunov equation using the modified simple equation method, Acta Math. Appl. Sinica,
English series, In press.
[33] Yang, Xiao-Feng, Zi-Chen Deng, and Yi Wei. A Riccati-Bernoulli sub-ODE method for
nonlinear partial differential equations and its application. Advances in Difference
Equations 2015.1 (2015): 1-17.