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G)-expansion and extended tanh methods for (2 + 1)-dimensional Calogero–Bogoyavlenskii–Schiff equation
Alexandria Engineering Journal (2015) 54, 27–33
H O S T E D BY
Alexandria University
Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com
ORIGINAL ARTICLE
Improved (G0/G)-expansion and extended tanh methods for (2 + 1)-dimensional Calogero–Bogoyavlenskii–Schiff equation Muhammad Shakeel, Syed Tauseef Mohyud-Din
*
Department of Mathematics, Faculty of Sciences, HITEC University, Taxila Cantt, Pakistan Received 18 December 2013; revised 1 October 2014; accepted 13 November 2014 Available online 4 December 2014
KEYWORDS Calogero–Bogoyavlenskii– Schiff equation; Extended tanh-function method; Improved (G0 /G)-expansion method; Auxiliary equation; Traveling wave solutions
Abstract Traveling wave solutions of nonlinear (2 + 1)-dimensional Calogero–Bogoyavlenskii– Schiff equation are constructed by using improved (G0 /G)-expansion and extended tanh-function methods. We demonstrate that some of the obtained exact solutions of these two methods are analogous, but they are not one and the same. The solutions obtained in this article may be imperative and significant for the explanation of some practical physical phenomena. It is observed that improved (G0 /G)-expansion method gives a variety of exact solutions. ª 2014 Production and hosting by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/3.0/).
1. Introduction Nonlinear evolution equations play a noteworthy role in various scientific and engineering fields, such as, optical fibers, solid state physics, fluid mechanics, plasma physics, the heat flow and the wave propagation phenomena, quantum mechanics, propagation of shallow water waves etc. Nonlinear wave phenomena of diffusion, reaction, dispersion, dissipation, and convection are very important in nonlinear wave equations. The rapid development of nonlinear sciences witnesses a wide range of reliable and efficient techniques which are of great help in tackling physical problems even of highly complex nature. After the observation of solitonary phenomena by John Scott Russell [1] in 1834 and since the Korteweg–de * Corresponding author. E-mail address: [email protected] (S.T. Mohyud-Din). Peer review under responsibility of Faculty of Engineering, Alexandria University.
Vries (KdV) equation was solved by Gardner et al. [2] by the inverse scattering method, finding exact solutions of nonlinear evolution equations has turned out to be one of the most exciting and particularly active areas of research. The appearance of solitary wave solutions in nature is quite common; Bellshaped sech-solutions and kink-shaped tanh-solutions model wave phenomena in elastic media, plasmas, solid state physics, condensed matter physics, electrical circuits, optical fibers, chemical kinematics, fluids, bio-genetics etc. The traveling wave solutions of the KdV equation and the Boussinesq equation which describe water waves are well-known examples. There are many different methods to look for the exact solutions of these equations. In soliton theory, there are many methods and techniques to deal with the problem of solitary wave solutions for nonlinear evolution equations such as; Backlund Transformation [3], Variational Iteration [4], Homogeneous Balance [5], Tanh-function [6–11], F-expansion [12], Homotopy Perturbation [13], First Integration [14], Exp-function [15–20], Truncated Painleve expansion [21], Weierstrass
http://dx.doi.org/10.1016/j.aej.2014.11.003 1110-0168 ª 2014 Production and hosting by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).
28
M. Shakeel, S.T. Mohyud-Din
elliptic function [22] and Jacobi elliptic function expansion [23–26], Local Fractional Variational Iteration and Local Fractional Decomposition methods [27–29]. There are other methods which can be found in [30–35]. For integrable nonlinear differential equations, Inverse Scattering transform [36] and Hirota methods [37] are used for searching the exact solutions. Wang et al. [38] introduced a direct and concise method, called (G0 /G)-expansion method to look for traveling wave solutions of nonlinear partial differential equations, where G = G(n) satisfies the second order linear ordinary differential equation G00 ðnÞ þ k G0 ðnÞ þ l GðnÞ ¼ 0; k and l are arbitrary constants. For additional references see the articles [39,40,8,41–44]. In this work, we use an alternative approach, called improved (G0 /G)-expansion method to find the exact traveling wave solutions of (2 + 1)-dimensional Calogero–Bogoyavlenskii–Schiff equation, where G = G(n) satisfies the auxiliary 2 ordinary differential equation ½G0 ðnÞ ¼ p G2 ðnÞ þ q G4 ðnÞþ 6 r G ðnÞ; p, q and r are constants. Recently El-Wakil et al. [8] and Parkes [41] have shown that the extended tanh-function method proposed by Fan [9] and the basic (G0 /G)-expansion method proposed by Wang et al. [38] are entirely equivalent as they deliver exactly the same set of solutions to a given nonlinear evolution equation. This observation has also been pointed out recently by Kudryashov [45]. In this work, we assert even though the basic (G0 /G)-expansion method is equivalent to the extended tanh-function method, the improved (G0 /G)-expansion method presented in this work is not equivalent to the extended tanh-function method. The method projected in this work is varied to some extent from the extended (G0 /G)-expansion method. The objective of this work is to show that the improved (G0 /G)-expansion method and the celebrated extended tanhfunction method are not identical. Further many new solutions are achieved via the offered improved (G0 /G)-expansion method. This approach will play an imperative role in constructing many exact traveling wave solutions for the nonlinear PDEs. 2. The improved (G0 /G)-expansion method Suppose we have the following nonlinear partial differential equation: Pðu; ut ; ux ; uy ; uz ; utt ; uy y ; uz z ; . . .Þ ¼ 0;
ð1Þ
where u ¼ uðx; y; z; tÞ is an unknown function, P is a polynomial in u ¼ uðx; y; z; tÞ and its partial derivatives in which the highest order derivatives and the nonlinear terms are involved. The main steps of the further improved (G0 /G)-expansion method are as follows: Step 1: The traveling wave variable, uðx; z; tÞ ¼ uðnÞ;
n ¼ x þ z V t;
ð2Þ
where V is the speed of the traveling wave, which converts the Eq. (1) into an ODE in the form, Qðu; u0 ; u00 ; u000 ; . . .Þ ¼ 0;
ð3Þ
where prime denotes the derivative with respect to n. Step 2: Suppose that the solution of the Eq. (3) can be expressed by means of a polynomial in (G0 /G) as follows:
0 i n X G ai G i¼0
uðnÞ ¼
ð4Þ
where ai (i ¼ 1; 2; 3; . . .) are constants provided an „ 0 and G = G(n) satisfies the following nonlinear auxiliary equation: 2
½G0 ðnÞ ¼ p G2 ðnÞ þ q G4 ðnÞ þ r G6 ðnÞ;
ð5Þ
where p, q and r are random constants to be determined later. The general solutions of Eq. (5) are as follows [46,47]: No. G(n) 1
pffiffi p q sec h2 ð p nÞ 2 pffiffi q2 p r ð1tanh ð pnÞÞ
No. G(n) 6
12
12 pffiffiffiffiffi p sec2 ð pnÞ pffiffiffiffiffiffiffi pffiffiffiffiffi q2 p r tan ð pnÞ
or
12 pffiffi p q csc h2 ð p nÞ ; 2 pffiffi 2 q p r ð1coth ð p nÞÞ
2
3
4
5
12
2p pffiffiffi pffiffi D cosh ð2 pnÞq
12 pffiffiffiffiffi p csc2 ð pnÞ ; pffiffiffiffiffiffiffi pffiffiffiffiffi q2 p r cot ð p nÞ
p>0 7
;
p>0D>0 12 2p pffiffiffi or pffiffiffiffiffi D cos ð2 p nÞq 12 2p pffiffiffi ; pffiffiffiffiffi D sin ð2 p nÞq p < 0; D > 0 12 pffiffiffiffiffi 2 p pffiffi ; D sinh ð2 p nÞq p > 0; D < 0 12 pffiffi p sec h2 ð p nÞ or pffiffiffiffi pffiffi q2 p r tanh ð pnÞ 1 pffiffi 2 p csc h2 ð pnÞ ; pffiffiffiffi pffiffi q2 p r coth ð p nÞ p > 0; r > 0
or
p < 0; r > 0 pffi p e2 pn pffi 2 2 pn ðe
8
12
4 qÞ 64 p r
; p>0
h
1 pffiffiffi i2 pq 1 tanhð12 pnÞ or h i12 p ffiffi ffi pq 1 cothð12 p nÞ ;
p > 0; D ¼ 0 9
h
10
p1ffiffiq n ; p ¼ 0; r ¼ 0
p
ffi
p e2 pnpffi 164 p r e4 pn
i12
; p > 0; q ¼ 0
where D ¼ q2 4 p r. Step 3: In Eq. (4), n is a positive integer which is usually obtained by balancing the highest order linear term(s) with the nonlinear term(s) of the highest order come out in Eq. (3). Step 4: Substituting Eq. (4), into Eq. (3) and utilizing Eq. (5), we obtain polynomials in Gi(n) and G0 ðnÞ Gi ðnÞ ði ¼ 0; 1; 2; 3; . . .Þ: Vanishing each coefficient of the resulted polynomials to zero, yields a set of algebraic equations for an ; p; q; r; V and constant(s) of integration, if applicable. Suppose with the aid of symbolic computation software such as Maple, the unknown constants an ; p; q; r and V can be found by solving these set of algebraic equations and substituting these values into Eq. (4), new and more general exact traveling wave solutions of the nonlinear partial differential Eq. (1) can be found. 3. Solution procedure In this section, we apply the improved (G0 /G)-expansion and extended tanh-function methods to the (2 + 1)-dimensional Calogero–Bogoyavlenskii–Schiff equation which is an extremely important nonlinear evolution equation in mathematical physics and have been paid attention by a lot of researchers.
Improved (G0 /G)-expansion and extended tanh methods
29 Family 1. If p > 0, the solution of Eq. (5) has the form,
3.1. The (2 + 1)-dimensional Calogero–Bogoyavlenskii–Schiff equation
"
We start with the (2 + 1)-dimensional Calogero–Bogoyavlenskii–Schiff (CBS) equation in the form ux t þ ux x x z þ 4 ux ux z þ 2 ux x uz ¼ 0;
or GðnÞ ¼
#12 pffiffiffi p q csc h2 p n pffiffiffi 2 q2 p r 1 coth pn
ð13Þ
"
ð6Þ
The CBS equation was first investigated by Bogoyavlenskii and Schiff by many different ways. Bogoyavlenskii used the modified Lax formalism, whereas Schiff derived the same G0 ¼ G
#12 pffiffiffi p q sec h2 p n GðnÞ ¼ pffiffiffi 2 q2 p r 1 tanh pn
ð14Þ
In these cases we have the ratio,
pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi 2 pffiffiffi p q sinh p n cosh p n 2p r sinh p n coshð p nÞ 2 p r cosh2 p n p r pffiffiffi pffiffiffi pffiffiffi pffiffiffi ; q2 cosh2 p n þ 2 p r cosh2 p n 2p r sinh p n cosh p n p r
ð15Þ
pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi 2 p q sinh p n cosh p n 2p r sinh p n cosh p n 2 p r cosh2 p n p r pffiffiffi pffiffiffi pffiffiffi pffiffiffi q2 cosh2 p n þ 2 p r cosh2 p n 2p r sinh p n cosh p n þ q2 p r
ð16Þ
and G0 ¼ G
equation by reducing the self-dual Yang–Mills equation [48– 51]. Tanh-Coth method was used in [52] to find kink solutions. Let us now solve the Eq. (6) by the proposed improved (G0 /G)expansion method. Using the traveling wave variable (2) in Eq. (6) that converts the given PDE into an ODE and on integrating once and setting constant of integration as zero yields: 2
Vu0 þ 3 ðu0 Þ þ u000 ¼ 0;
ð7Þ
respectively. pffiffiffiffiffi Since q ¼ 2 p r, subsequently, we obtain the following traveling wave solutions, pffiffiffi pffiffiffi 8 p sec h2 2 p n pffiffiffi pffiffiffi pffiffiffi uðnÞ ¼ a0 4 p ð17Þ sec h2 2 p n 2 tanh 2 p n þ 2 and
Balancing u000 with ðu0 Þ2 in (7) we find n = 1. Therefore, the solution (4) can be written as 0 G : ð8Þ uðnÞ ¼ a0 þ a1 G
pffiffiffi uðnÞ ¼ a0 þ 4 p
Substituting (8) and (5) into (7), we obtain the following polynomial equation in G as follows:
n ¼ x þ z 4 p t:
ð18Þ where
Family 2. If p > 0,r > 0, the solution of Eq. (5) has the form,
G2 : V a1 q þ 4 a1 q p ¼ 0; G4 : 3 a21 q2 þ 32 a1 r p 2V a1 r þ 6 a1 q2 ¼ 0; G6 : 12 a21 r q þ 48a1 r q ¼ 0;
ð9Þ
Solving the above system of equations using the symbolic computation software such as Maple and obtain the following two sets of solutions: The set 1. pffiffiffiffiffi ð10Þ a0 ¼ a0 ; a1 ¼ 4; q ¼ 2 p r; V ¼ 4p; The set 2.
where a0, p and r are arbitrary constants. Now for the set 1, we have the following solution: 0 G ; uðnÞ ¼ a0 4 G
"
pffiffiffi #12 p sec h2 p n pffiffiffi or pffiffiffiffiffi q 2 p r tanh pn " #12 pffiffiffi p csc h2 p n pffiffiffi : GðnÞ ¼ pffiffiffiffiffi q 2 p r coth pn GðnÞ ¼
G8 : 12 a21 r2 þ 48 a1 r2 ¼ 0:
a0 ¼ a0 ; a1 ¼ 4; q ¼ 0; V ¼ 16p;
pffiffiffi pffiffiffi 8 p sec h2 2 p n pffiffiffi pffiffiffi ; 3 sec h2 2 p n þ 2 tanh 2 p n 2
ð19Þ
Then we have the ratio, pffiffiffi pffiffiffi pffiffiffi pffiffiffiffiffi pffiffiffi pffiffiffiffiffi p q sinh p n cosh p n 2 p r cosh2 p n p r G0 pffiffiffi pffiffiffi pffiffiffi ¼ pffiffiffiffiffi G cosh p n q cosh p n 2 p r sinh p n
ð11Þ or
pffiffiffi pffiffiffi pffiffiffi pffiffiffiffiffi pffiffiffiffiffi 2 pffiffiffi G0 p q sinh p n cosh p n 2 p r cosh ð p nÞ p r pffiffiffi pffiffiffi ¼ : pffiffiffiffiffi pffiffiffi G sinh p n q sinh p n 2 p r cosh p n ð20Þ
ð12Þ
where n ¼ x þ z 4 p t: Also, according to the step 2 of Section 2, we have the subsequent families of exact solutions:
pffiffiffiffiffi Since q ¼ 2 p r, subsequently, we obtain the following traveling wave solutions: pffiffiffi pffiffiffi pffiffiffi uðnÞ ¼ a0 2 p þ 2 p tanh p n pffiffiffi pffiffiffi pffiffiffi or uðnÞ ¼ a0 2 p þ 2 p coth p n ; where
ð21Þ
30
M. Shakeel, S.T. Mohyud-Din
n ¼ x þ z 4 p t: Family 3. If p < 0, r > 0, the solution of Eq. (5) has the form, "
#12 pffiffiffiffiffiffiffi p sec2 p n pffiffiffiffiffiffiffi GðnÞ ¼ or GðnÞ pffiffiffiffiffiffiffiffiffi q 2 p r tan pn " #12 pffiffiffiffiffiffiffi p csc2 p n pffiffiffiffiffiffiffi : ¼ pffiffiffiffiffiffiffiffiffi q 2 p r cot pn
ð22Þ
or
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi p p r 2 p r cos2 p n q sin p n cos p n pffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi cos p n 2 p r sin p n q cos p n
G0 ¼ G
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi p pr 2 pr cos2 p n q sin p n cos p n pffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi : sin p n 2 pr cos p n q sin p n ð23Þ
pffiffiffiffiffi Since q ¼ 2 p r, subsequently, we obtain the following traveling wave solutions: pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 4 p tan 2 p n 1 pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi uðnÞ ¼ a0 þ 1 tan 2 p n þ sec 2 p n pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 4 p tan 2 p n 1 pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi ; uðnÞ ¼ a0 þ 1 tan 2 p n sec 2 p n
ð24Þ
Family 4. If p > 0, D = 0, the solution of Eq. (5) has the form, 12 p 1 pffiffiffi pn GðnÞ ¼ 1 tanh q 2 12 p 1 pffiffiffi pn 1 coth GðnÞ ¼ q 2 Then we have the ratio,
pffiffiffi p G0 1 pffiffiffi pn ¼ 1 tanh 2 G 4
pffiffiffi p G0 1 pffiffiffi pn ¼ 1 coth 2 4 G
ð25Þ
"
#12 2p pffiffiffiffi GðnÞ ¼ pffiffiffi D cosh 2 pn q
or
2
Family 7. If p > 0, D < 0, the solution of Eq. (5) has the form, ð35Þ
Since q = 0, then r > 0. In this case we have the ratio, ð26Þ
) 12
4 qÞ 64 p r
ð33Þ
Therefore, we obtain the following traveling wave solution, pffiffiffi pffiffiffi ð34Þ uðnÞ ¼ a0 þ 4 p tanh 2 p n ;
#12 2p pffiffiffiffiffiffiffi GðnÞ ¼ : pffiffiffi D sinh 2 pn q
Family 5. If p > 0, the solution of Eq. (5) has the form, pffiffi pn
ð32Þ
"
n ¼ x þ z 4 p t:
pffiffi pn
According to the step 2 of Section 2, we obtain the subsequent families of exact solutions: Family 6. If p > 0, D > 0, the solution of Eq. (5) has the form,
n ¼ x þ z 16 p t:
where
ðe 2
n ¼ x þ z 16 p t:
where or
Subsequently, we obtain the following traveling wave solutions:
1 pffiffiffi pffiffiffi pn or uðnÞ uðnÞ ¼ a0 p 1 tanh 2
1 pffiffiffi pffiffiffi ¼ a0 p 1 coth pn ; ð27Þ 2
p e 2
ð31Þ
where
pffiffiffi G0 pffiffiffi ¼ p tanh 2 p n : G
n ¼ x þ z 4 p t:
GðnÞ ¼
For the set 2, we have the following solution: 0 G ; uðnÞ ¼ a0 4 G
Since q = 0, then r < 0. In this case we have the ratio,
where
(
where n ¼ x þ z 4 p t:
Then we have the ratio, G0 ¼ G
Then we have the ratio, pffiffiffi 4pffiffip n pðe 16 q2 þ 64 p rÞ G0 pffiffi ¼ 4pffiffip n ð29Þ 2 G e 8 qe p n 16 q2 64 p r pffiffiffiffiffi Since q ¼ 2 p r, subsequently, we obtain the following traveling wave solutions: pffiffi pffiffiffi 4 p e 2 p n pffiffi ; uðnÞ ¼ a0 þ ð30Þ pffiffiffiffiffi 16 p r e 2 p n
ð28Þ
pffiffiffi G0 pffiffiffi ¼ p coth 2 p n : G
ð36Þ
Therefore, we obtain the following traveling wave solutions: pffiffiffi pffiffiffi uðnÞ ¼ a0 þ 4 p coth 2 p n ; ð37Þ where n ¼ x þ z 16 p t: Family 8. If p < 0, D > 0, the solutions of Eq. (5) has the form, "
#12 2p pffiffiffiffi GðnÞ ¼ or pffiffiffiffiffiffiffi D cos 2 pn q " #12 2p pffiffiffiffi GðnÞ ¼ : pffiffiffiffiffiffiffi D sin 2 pn q
ð38Þ
Improved (G0 /G)-expansion and extended tanh methods
31
Since q = 0, then r > 0. Thus we have the ratio, 0
pffiffiffiffiffiffiffi G pffiffiffiffiffiffiffi ¼ p tan 2 p n or G pffiffiffiffiffiffiffi G0 pffiffiffiffiffiffiffi ¼ p cot 2 p n : G
ð39Þ
where
n ¼ x þ z 16 p t: Family 12. If p = 0, r = 0, then the solution of Eq. (5) has the form, 1 GðnÞ ¼ pffiffiffi : qn
n ¼ x þ z 16 p t: Family 9. If p > 0, r > 0, the solutions of Eq. (5) has the form, pffiffiffi #12 p sec h pn pffiffiffi GðnÞ ¼ or pffiffiffiffiffi q 2 p r tanh pn " #12 pffiffiffi p csc h2 p n pffiffiffi : GðnÞ ¼ pffiffiffiffiffi q 2 p r coth pn 2
ð41Þ
G0 1 ¼ ; n G
ð42Þ
where n ¼ x þ z 16 p t: Family 10. If p < 0, r > 0, the solutions of Eq. (5) has the form, "
4 uðnÞ ¼ a0 þ ; n
3.2. Extended tanh-function method for (2 + 1)-dimensional CBS equation With reference to the well-known extended tanh-function method [7–11], the solution of the (2 + 1)-dimensional Calogero–Bogoyavlenskii–Schiff (CBS) equation Eq. (6) can be represented as,
ð44Þ
ð45Þ
where n ¼ x þ z 16 p t:
u0 ðnÞ ¼ A þ u2 ðnÞ:
ð54Þ
Eq. (54) has the following solutions: (i) If A < 0, then pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi An or uðnÞ ¼ A tanh pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi An : uðnÞ ¼ A coth
ð55Þ
(ii) If A > 0, then pffiffiffiffi pffiffiffiffi An or uðnÞ ¼ A tan pffiffiffiffi pffiffiffiffi An : uðnÞ ¼ A cot
ð56Þ
Family 11. If p > 0, q = 0, the solution of Eq. (5) has the form,
GðnÞ ¼
p e
pffiffiffiffi pn
Then we have the ratio, G0 1 n ¼ pffiffi coth pffiffi ; G 8 r 4 r
(iii) If A = 0, then 1 uðnÞ ¼ : n
#12
pffiffiffiffi : 1 64 p r e4 p n
ð53Þ
where uðnÞ satisfy the Riccati equation
Therefore, we obtain the following traveling wave solutions: pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi uðnÞ ¼ a0 þ 2 p cot p n tan p n ; ð46Þ
2
ð52Þ
These are the exact solutions of the (2 + 1)-dimensional Calogero–Bogoyavlenskii–Schiff (CBS) equation obtained by the improved (G0 /G)-expansion method.
uðnÞ ¼ a0 þ a1 uðnÞ;
Since q = 0, then we have the ratio,
"
ð51Þ
n ¼ x þ z 16 p t:
Therefore, we obtain the following traveling wave solution: pffiffiffi pffiffiffi pffiffiffi ð43Þ uðnÞ ¼ a0 þ 2 p tanh p n þ coth p n ;
pffiffiffiffiffiffiffi G0 1 pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi p cot p n tan p n : ¼ G 2
Then we have the ratio,
where
0
#12 pffiffiffiffiffiffiffi p sec2 p n pffiffiffiffiffiffiffi GðnÞ ¼ or pffiffiffiffiffiffiffiffiffi q 2 p r tan pn " #12 pffiffiffiffiffiffiffi p csc2 p n pffiffiffiffiffiffiffi : GðnÞ ¼ pffiffiffiffiffiffiffiffiffi q 2 p r cot pn
ð50Þ
Therefore, we have the solution:
Since q = 0, we have the ratio, pffiffiffi pffiffiffi G 1 pffiffiffi p tanh p n þ coth p n : ¼ 2 G
ð49Þ
where
Therefore, we obtain the following traveling wave solutions: pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi uðnÞ ¼ a0 4 p tan 2 p n or pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi uðnÞ ¼ a0 þ 4 p cot 2 p n ; ð40Þ
"
where 64 p r ¼ 1. Therefore, we have the solution: 1 n uðnÞ ¼ a0 pffiffi coth pffiffi ; 2 r 4 r
ð47Þ
Substituting (53) and (54) into (7), we obtain the following polynomial equation in u as follows:
ð48Þ
ð57Þ
3a21 þ 6a1 u4 ðnÞ þ a1 V þ 8a1 A þ 6a21 A u2 ðnÞ þ a1 A V þ 2 a1 A2 þ 3a21 A2 ¼ 0
ð58Þ
32
M. Shakeel, S.T. Mohyud-Din
Equating to zero the coefficients of the polynomial and solving the set of algebraic equations with the help of Maple, we obtain the following solution: a0 ¼ a0 ; a1 ¼ 2; V ¼ 4A: where a0 and A are arbitrary constants. So, the exact solutions of Eq. (6) are: When A < 0, the solution takes the form, pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi A n uðnÞ ¼ a0 þ 2 A tanh or pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi A n ; uðnÞ ¼ a0 þ 2 A coth
ð59Þ
5. Conclusions
ð60Þ
where n ¼ x þ z þ 4A t: When A > 0, the solution takes the form, pffiffiffiffi pffiffiffiffi uðnÞ ¼ a0 2 A tanð A nÞ or pffiffiffiffi pffiffiffiffi An ; uðnÞ ¼ a0 þ 2 A cot
the selection of initial approximation and the appearance of the dependent variable as transcendental function (i.e. sin u; log u; eu etc) which cannot be perturbed are the main disadvantages of HPM. Similarly, we can write advantages and disadvantages of other methods as well.
ð61Þ
where
An improved (G0 /G)-expansion method is suggested and applied to the (2 + 1)-dimensional Calogero–Bogoyavlenskii–Schiff (CBS) equation. The results obtained by the suggested method have been compared with those obtained by the extended tanh-function method. From this study, we observe that the improved (G0 /G)-expansion method and the extended tanh-function method are not equivalent. We see that all the results obtained by the extended tanh-function are found by the suggested method and in addition some new solutions are attained. From this analysis we can conclude that the proposed method is quite resourceful and practically well suited to be used in finding exact solutions of NLEEs.
n ¼ x þ z þ 4A t: When A = 0, the solution takes the form, 2 uðnÞ ¼ a0 þ ; n where
References ð62Þ
n ¼ x þ z þ 4A t: 4. Discussion From the above results obtained by the two methods, we can draw the following remarks: If we put A ¼ 4 p where p > 0, the results obtained in (60) are identical to the results (34) and (37) respectively. If we put A ¼ 4 p where p < 0 then the results obtained in (61) are identical to the results given in (40). From these remarks we have the following assessments: ‘‘The exact solutions of the Calogero–Bogoyavlenskii– Schiff equation obtained by means of the extended tanh-function method can also be obtained by the improved (G0 /G)expansion method and in appendage some new solutions are obtained. The calculations of the projected method are also easier than the extended tanh-function method as well as the basic (G0 /G)-expansion method’’. The advantages and disadvantages of some of the methods are 4.1. Advantages and disadvantages of VIM This method produces the solutions in terms of convergent series and does not require linearization or small perturbation is the main advantage, whereas, calculation of exact Lagrange multiplier is very difficult to calculate for higher order differential equation and the restriction of the nonlinearity term are disadvantages of VIM. 4.2. Advantages and disadvantages of HPM The HPM solves nonlinear problems without using Adomian’s polynomials is the main advantage of this method, whereas,
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H O S T E D BY
Alexandria University
Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com
ORIGINAL ARTICLE
Improved (G0/G)-expansion and extended tanh methods for (2 + 1)-dimensional Calogero–Bogoyavlenskii–Schiff equation Muhammad Shakeel, Syed Tauseef Mohyud-Din
*
Department of Mathematics, Faculty of Sciences, HITEC University, Taxila Cantt, Pakistan Received 18 December 2013; revised 1 October 2014; accepted 13 November 2014 Available online 4 December 2014
KEYWORDS Calogero–Bogoyavlenskii– Schiff equation; Extended tanh-function method; Improved (G0 /G)-expansion method; Auxiliary equation; Traveling wave solutions
Abstract Traveling wave solutions of nonlinear (2 + 1)-dimensional Calogero–Bogoyavlenskii– Schiff equation are constructed by using improved (G0 /G)-expansion and extended tanh-function methods. We demonstrate that some of the obtained exact solutions of these two methods are analogous, but they are not one and the same. The solutions obtained in this article may be imperative and significant for the explanation of some practical physical phenomena. It is observed that improved (G0 /G)-expansion method gives a variety of exact solutions. ª 2014 Production and hosting by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/3.0/).
1. Introduction Nonlinear evolution equations play a noteworthy role in various scientific and engineering fields, such as, optical fibers, solid state physics, fluid mechanics, plasma physics, the heat flow and the wave propagation phenomena, quantum mechanics, propagation of shallow water waves etc. Nonlinear wave phenomena of diffusion, reaction, dispersion, dissipation, and convection are very important in nonlinear wave equations. The rapid development of nonlinear sciences witnesses a wide range of reliable and efficient techniques which are of great help in tackling physical problems even of highly complex nature. After the observation of solitonary phenomena by John Scott Russell [1] in 1834 and since the Korteweg–de * Corresponding author. E-mail address: [email protected] (S.T. Mohyud-Din). Peer review under responsibility of Faculty of Engineering, Alexandria University.
Vries (KdV) equation was solved by Gardner et al. [2] by the inverse scattering method, finding exact solutions of nonlinear evolution equations has turned out to be one of the most exciting and particularly active areas of research. The appearance of solitary wave solutions in nature is quite common; Bellshaped sech-solutions and kink-shaped tanh-solutions model wave phenomena in elastic media, plasmas, solid state physics, condensed matter physics, electrical circuits, optical fibers, chemical kinematics, fluids, bio-genetics etc. The traveling wave solutions of the KdV equation and the Boussinesq equation which describe water waves are well-known examples. There are many different methods to look for the exact solutions of these equations. In soliton theory, there are many methods and techniques to deal with the problem of solitary wave solutions for nonlinear evolution equations such as; Backlund Transformation [3], Variational Iteration [4], Homogeneous Balance [5], Tanh-function [6–11], F-expansion [12], Homotopy Perturbation [13], First Integration [14], Exp-function [15–20], Truncated Painleve expansion [21], Weierstrass
http://dx.doi.org/10.1016/j.aej.2014.11.003 1110-0168 ª 2014 Production and hosting by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).
28
M. Shakeel, S.T. Mohyud-Din
elliptic function [22] and Jacobi elliptic function expansion [23–26], Local Fractional Variational Iteration and Local Fractional Decomposition methods [27–29]. There are other methods which can be found in [30–35]. For integrable nonlinear differential equations, Inverse Scattering transform [36] and Hirota methods [37] are used for searching the exact solutions. Wang et al. [38] introduced a direct and concise method, called (G0 /G)-expansion method to look for traveling wave solutions of nonlinear partial differential equations, where G = G(n) satisfies the second order linear ordinary differential equation G00 ðnÞ þ k G0 ðnÞ þ l GðnÞ ¼ 0; k and l are arbitrary constants. For additional references see the articles [39,40,8,41–44]. In this work, we use an alternative approach, called improved (G0 /G)-expansion method to find the exact traveling wave solutions of (2 + 1)-dimensional Calogero–Bogoyavlenskii–Schiff equation, where G = G(n) satisfies the auxiliary 2 ordinary differential equation ½G0 ðnÞ ¼ p G2 ðnÞ þ q G4 ðnÞþ 6 r G ðnÞ; p, q and r are constants. Recently El-Wakil et al. [8] and Parkes [41] have shown that the extended tanh-function method proposed by Fan [9] and the basic (G0 /G)-expansion method proposed by Wang et al. [38] are entirely equivalent as they deliver exactly the same set of solutions to a given nonlinear evolution equation. This observation has also been pointed out recently by Kudryashov [45]. In this work, we assert even though the basic (G0 /G)-expansion method is equivalent to the extended tanh-function method, the improved (G0 /G)-expansion method presented in this work is not equivalent to the extended tanh-function method. The method projected in this work is varied to some extent from the extended (G0 /G)-expansion method. The objective of this work is to show that the improved (G0 /G)-expansion method and the celebrated extended tanhfunction method are not identical. Further many new solutions are achieved via the offered improved (G0 /G)-expansion method. This approach will play an imperative role in constructing many exact traveling wave solutions for the nonlinear PDEs. 2. The improved (G0 /G)-expansion method Suppose we have the following nonlinear partial differential equation: Pðu; ut ; ux ; uy ; uz ; utt ; uy y ; uz z ; . . .Þ ¼ 0;
ð1Þ
where u ¼ uðx; y; z; tÞ is an unknown function, P is a polynomial in u ¼ uðx; y; z; tÞ and its partial derivatives in which the highest order derivatives and the nonlinear terms are involved. The main steps of the further improved (G0 /G)-expansion method are as follows: Step 1: The traveling wave variable, uðx; z; tÞ ¼ uðnÞ;
n ¼ x þ z V t;
ð2Þ
where V is the speed of the traveling wave, which converts the Eq. (1) into an ODE in the form, Qðu; u0 ; u00 ; u000 ; . . .Þ ¼ 0;
ð3Þ
where prime denotes the derivative with respect to n. Step 2: Suppose that the solution of the Eq. (3) can be expressed by means of a polynomial in (G0 /G) as follows:
0 i n X G ai G i¼0
uðnÞ ¼
ð4Þ
where ai (i ¼ 1; 2; 3; . . .) are constants provided an „ 0 and G = G(n) satisfies the following nonlinear auxiliary equation: 2
½G0 ðnÞ ¼ p G2 ðnÞ þ q G4 ðnÞ þ r G6 ðnÞ;
ð5Þ
where p, q and r are random constants to be determined later. The general solutions of Eq. (5) are as follows [46,47]: No. G(n) 1
pffiffi p q sec h2 ð p nÞ 2 pffiffi q2 p r ð1tanh ð pnÞÞ
No. G(n) 6
12
12 pffiffiffiffiffi p sec2 ð pnÞ pffiffiffiffiffiffiffi pffiffiffiffiffi q2 p r tan ð pnÞ
or
12 pffiffi p q csc h2 ð p nÞ ; 2 pffiffi 2 q p r ð1coth ð p nÞÞ
2
3
4
5
12
2p pffiffiffi pffiffi D cosh ð2 pnÞq
12 pffiffiffiffiffi p csc2 ð pnÞ ; pffiffiffiffiffiffiffi pffiffiffiffiffi q2 p r cot ð p nÞ
p>0 7
;
p>0D>0 12 2p pffiffiffi or pffiffiffiffiffi D cos ð2 p nÞq 12 2p pffiffiffi ; pffiffiffiffiffi D sin ð2 p nÞq p < 0; D > 0 12 pffiffiffiffiffi 2 p pffiffi ; D sinh ð2 p nÞq p > 0; D < 0 12 pffiffi p sec h2 ð p nÞ or pffiffiffiffi pffiffi q2 p r tanh ð pnÞ 1 pffiffi 2 p csc h2 ð pnÞ ; pffiffiffiffi pffiffi q2 p r coth ð p nÞ p > 0; r > 0
or
p < 0; r > 0 pffi p e2 pn pffi 2 2 pn ðe
8
12
4 qÞ 64 p r
; p>0
h
1 pffiffiffi i2 pq 1 tanhð12 pnÞ or h i12 p ffiffi ffi pq 1 cothð12 p nÞ ;
p > 0; D ¼ 0 9
h
10
p1ffiffiq n ; p ¼ 0; r ¼ 0
p
ffi
p e2 pnpffi 164 p r e4 pn
i12
; p > 0; q ¼ 0
where D ¼ q2 4 p r. Step 3: In Eq. (4), n is a positive integer which is usually obtained by balancing the highest order linear term(s) with the nonlinear term(s) of the highest order come out in Eq. (3). Step 4: Substituting Eq. (4), into Eq. (3) and utilizing Eq. (5), we obtain polynomials in Gi(n) and G0 ðnÞ Gi ðnÞ ði ¼ 0; 1; 2; 3; . . .Þ: Vanishing each coefficient of the resulted polynomials to zero, yields a set of algebraic equations for an ; p; q; r; V and constant(s) of integration, if applicable. Suppose with the aid of symbolic computation software such as Maple, the unknown constants an ; p; q; r and V can be found by solving these set of algebraic equations and substituting these values into Eq. (4), new and more general exact traveling wave solutions of the nonlinear partial differential Eq. (1) can be found. 3. Solution procedure In this section, we apply the improved (G0 /G)-expansion and extended tanh-function methods to the (2 + 1)-dimensional Calogero–Bogoyavlenskii–Schiff equation which is an extremely important nonlinear evolution equation in mathematical physics and have been paid attention by a lot of researchers.
Improved (G0 /G)-expansion and extended tanh methods
29 Family 1. If p > 0, the solution of Eq. (5) has the form,
3.1. The (2 + 1)-dimensional Calogero–Bogoyavlenskii–Schiff equation
"
We start with the (2 + 1)-dimensional Calogero–Bogoyavlenskii–Schiff (CBS) equation in the form ux t þ ux x x z þ 4 ux ux z þ 2 ux x uz ¼ 0;
or GðnÞ ¼
#12 pffiffiffi p q csc h2 p n pffiffiffi 2 q2 p r 1 coth pn
ð13Þ
"
ð6Þ
The CBS equation was first investigated by Bogoyavlenskii and Schiff by many different ways. Bogoyavlenskii used the modified Lax formalism, whereas Schiff derived the same G0 ¼ G
#12 pffiffiffi p q sec h2 p n GðnÞ ¼ pffiffiffi 2 q2 p r 1 tanh pn
ð14Þ
In these cases we have the ratio,
pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi 2 pffiffiffi p q sinh p n cosh p n 2p r sinh p n coshð p nÞ 2 p r cosh2 p n p r pffiffiffi pffiffiffi pffiffiffi pffiffiffi ; q2 cosh2 p n þ 2 p r cosh2 p n 2p r sinh p n cosh p n p r
ð15Þ
pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi 2 p q sinh p n cosh p n 2p r sinh p n cosh p n 2 p r cosh2 p n p r pffiffiffi pffiffiffi pffiffiffi pffiffiffi q2 cosh2 p n þ 2 p r cosh2 p n 2p r sinh p n cosh p n þ q2 p r
ð16Þ
and G0 ¼ G
equation by reducing the self-dual Yang–Mills equation [48– 51]. Tanh-Coth method was used in [52] to find kink solutions. Let us now solve the Eq. (6) by the proposed improved (G0 /G)expansion method. Using the traveling wave variable (2) in Eq. (6) that converts the given PDE into an ODE and on integrating once and setting constant of integration as zero yields: 2
Vu0 þ 3 ðu0 Þ þ u000 ¼ 0;
ð7Þ
respectively. pffiffiffiffiffi Since q ¼ 2 p r, subsequently, we obtain the following traveling wave solutions, pffiffiffi pffiffiffi 8 p sec h2 2 p n pffiffiffi pffiffiffi pffiffiffi uðnÞ ¼ a0 4 p ð17Þ sec h2 2 p n 2 tanh 2 p n þ 2 and
Balancing u000 with ðu0 Þ2 in (7) we find n = 1. Therefore, the solution (4) can be written as 0 G : ð8Þ uðnÞ ¼ a0 þ a1 G
pffiffiffi uðnÞ ¼ a0 þ 4 p
Substituting (8) and (5) into (7), we obtain the following polynomial equation in G as follows:
n ¼ x þ z 4 p t:
ð18Þ where
Family 2. If p > 0,r > 0, the solution of Eq. (5) has the form,
G2 : V a1 q þ 4 a1 q p ¼ 0; G4 : 3 a21 q2 þ 32 a1 r p 2V a1 r þ 6 a1 q2 ¼ 0; G6 : 12 a21 r q þ 48a1 r q ¼ 0;
ð9Þ
Solving the above system of equations using the symbolic computation software such as Maple and obtain the following two sets of solutions: The set 1. pffiffiffiffiffi ð10Þ a0 ¼ a0 ; a1 ¼ 4; q ¼ 2 p r; V ¼ 4p; The set 2.
where a0, p and r are arbitrary constants. Now for the set 1, we have the following solution: 0 G ; uðnÞ ¼ a0 4 G
"
pffiffiffi #12 p sec h2 p n pffiffiffi or pffiffiffiffiffi q 2 p r tanh pn " #12 pffiffiffi p csc h2 p n pffiffiffi : GðnÞ ¼ pffiffiffiffiffi q 2 p r coth pn GðnÞ ¼
G8 : 12 a21 r2 þ 48 a1 r2 ¼ 0:
a0 ¼ a0 ; a1 ¼ 4; q ¼ 0; V ¼ 16p;
pffiffiffi pffiffiffi 8 p sec h2 2 p n pffiffiffi pffiffiffi ; 3 sec h2 2 p n þ 2 tanh 2 p n 2
ð19Þ
Then we have the ratio, pffiffiffi pffiffiffi pffiffiffi pffiffiffiffiffi pffiffiffi pffiffiffiffiffi p q sinh p n cosh p n 2 p r cosh2 p n p r G0 pffiffiffi pffiffiffi pffiffiffi ¼ pffiffiffiffiffi G cosh p n q cosh p n 2 p r sinh p n
ð11Þ or
pffiffiffi pffiffiffi pffiffiffi pffiffiffiffiffi pffiffiffiffiffi 2 pffiffiffi G0 p q sinh p n cosh p n 2 p r cosh ð p nÞ p r pffiffiffi pffiffiffi ¼ : pffiffiffiffiffi pffiffiffi G sinh p n q sinh p n 2 p r cosh p n ð20Þ
ð12Þ
where n ¼ x þ z 4 p t: Also, according to the step 2 of Section 2, we have the subsequent families of exact solutions:
pffiffiffiffiffi Since q ¼ 2 p r, subsequently, we obtain the following traveling wave solutions: pffiffiffi pffiffiffi pffiffiffi uðnÞ ¼ a0 2 p þ 2 p tanh p n pffiffiffi pffiffiffi pffiffiffi or uðnÞ ¼ a0 2 p þ 2 p coth p n ; where
ð21Þ
30
M. Shakeel, S.T. Mohyud-Din
n ¼ x þ z 4 p t: Family 3. If p < 0, r > 0, the solution of Eq. (5) has the form, "
#12 pffiffiffiffiffiffiffi p sec2 p n pffiffiffiffiffiffiffi GðnÞ ¼ or GðnÞ pffiffiffiffiffiffiffiffiffi q 2 p r tan pn " #12 pffiffiffiffiffiffiffi p csc2 p n pffiffiffiffiffiffiffi : ¼ pffiffiffiffiffiffiffiffiffi q 2 p r cot pn
ð22Þ
or
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi p p r 2 p r cos2 p n q sin p n cos p n pffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi cos p n 2 p r sin p n q cos p n
G0 ¼ G
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi p pr 2 pr cos2 p n q sin p n cos p n pffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi : sin p n 2 pr cos p n q sin p n ð23Þ
pffiffiffiffiffi Since q ¼ 2 p r, subsequently, we obtain the following traveling wave solutions: pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 4 p tan 2 p n 1 pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi uðnÞ ¼ a0 þ 1 tan 2 p n þ sec 2 p n pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 4 p tan 2 p n 1 pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi ; uðnÞ ¼ a0 þ 1 tan 2 p n sec 2 p n
ð24Þ
Family 4. If p > 0, D = 0, the solution of Eq. (5) has the form, 12 p 1 pffiffiffi pn GðnÞ ¼ 1 tanh q 2 12 p 1 pffiffiffi pn 1 coth GðnÞ ¼ q 2 Then we have the ratio,
pffiffiffi p G0 1 pffiffiffi pn ¼ 1 tanh 2 G 4
pffiffiffi p G0 1 pffiffiffi pn ¼ 1 coth 2 4 G
ð25Þ
"
#12 2p pffiffiffiffi GðnÞ ¼ pffiffiffi D cosh 2 pn q
or
2
Family 7. If p > 0, D < 0, the solution of Eq. (5) has the form, ð35Þ
Since q = 0, then r > 0. In this case we have the ratio, ð26Þ
) 12
4 qÞ 64 p r
ð33Þ
Therefore, we obtain the following traveling wave solution, pffiffiffi pffiffiffi ð34Þ uðnÞ ¼ a0 þ 4 p tanh 2 p n ;
#12 2p pffiffiffiffiffiffiffi GðnÞ ¼ : pffiffiffi D sinh 2 pn q
Family 5. If p > 0, the solution of Eq. (5) has the form, pffiffi pn
ð32Þ
"
n ¼ x þ z 4 p t:
pffiffi pn
According to the step 2 of Section 2, we obtain the subsequent families of exact solutions: Family 6. If p > 0, D > 0, the solution of Eq. (5) has the form,
n ¼ x þ z 16 p t:
where
ðe 2
n ¼ x þ z 16 p t:
where or
Subsequently, we obtain the following traveling wave solutions:
1 pffiffiffi pffiffiffi pn or uðnÞ uðnÞ ¼ a0 p 1 tanh 2
1 pffiffiffi pffiffiffi ¼ a0 p 1 coth pn ; ð27Þ 2
p e 2
ð31Þ
where
pffiffiffi G0 pffiffiffi ¼ p tanh 2 p n : G
n ¼ x þ z 4 p t:
GðnÞ ¼
For the set 2, we have the following solution: 0 G ; uðnÞ ¼ a0 4 G
Since q = 0, then r < 0. In this case we have the ratio,
where
(
where n ¼ x þ z 4 p t:
Then we have the ratio, G0 ¼ G
Then we have the ratio, pffiffiffi 4pffiffip n pðe 16 q2 þ 64 p rÞ G0 pffiffi ¼ 4pffiffip n ð29Þ 2 G e 8 qe p n 16 q2 64 p r pffiffiffiffiffi Since q ¼ 2 p r, subsequently, we obtain the following traveling wave solutions: pffiffi pffiffiffi 4 p e 2 p n pffiffi ; uðnÞ ¼ a0 þ ð30Þ pffiffiffiffiffi 16 p r e 2 p n
ð28Þ
pffiffiffi G0 pffiffiffi ¼ p coth 2 p n : G
ð36Þ
Therefore, we obtain the following traveling wave solutions: pffiffiffi pffiffiffi uðnÞ ¼ a0 þ 4 p coth 2 p n ; ð37Þ where n ¼ x þ z 16 p t: Family 8. If p < 0, D > 0, the solutions of Eq. (5) has the form, "
#12 2p pffiffiffiffi GðnÞ ¼ or pffiffiffiffiffiffiffi D cos 2 pn q " #12 2p pffiffiffiffi GðnÞ ¼ : pffiffiffiffiffiffiffi D sin 2 pn q
ð38Þ
Improved (G0 /G)-expansion and extended tanh methods
31
Since q = 0, then r > 0. Thus we have the ratio, 0
pffiffiffiffiffiffiffi G pffiffiffiffiffiffiffi ¼ p tan 2 p n or G pffiffiffiffiffiffiffi G0 pffiffiffiffiffiffiffi ¼ p cot 2 p n : G
ð39Þ
where
n ¼ x þ z 16 p t: Family 12. If p = 0, r = 0, then the solution of Eq. (5) has the form, 1 GðnÞ ¼ pffiffiffi : qn
n ¼ x þ z 16 p t: Family 9. If p > 0, r > 0, the solutions of Eq. (5) has the form, pffiffiffi #12 p sec h pn pffiffiffi GðnÞ ¼ or pffiffiffiffiffi q 2 p r tanh pn " #12 pffiffiffi p csc h2 p n pffiffiffi : GðnÞ ¼ pffiffiffiffiffi q 2 p r coth pn 2
ð41Þ
G0 1 ¼ ; n G
ð42Þ
where n ¼ x þ z 16 p t: Family 10. If p < 0, r > 0, the solutions of Eq. (5) has the form, "
4 uðnÞ ¼ a0 þ ; n
3.2. Extended tanh-function method for (2 + 1)-dimensional CBS equation With reference to the well-known extended tanh-function method [7–11], the solution of the (2 + 1)-dimensional Calogero–Bogoyavlenskii–Schiff (CBS) equation Eq. (6) can be represented as,
ð44Þ
ð45Þ
where n ¼ x þ z 16 p t:
u0 ðnÞ ¼ A þ u2 ðnÞ:
ð54Þ
Eq. (54) has the following solutions: (i) If A < 0, then pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi An or uðnÞ ¼ A tanh pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi An : uðnÞ ¼ A coth
ð55Þ
(ii) If A > 0, then pffiffiffiffi pffiffiffiffi An or uðnÞ ¼ A tan pffiffiffiffi pffiffiffiffi An : uðnÞ ¼ A cot
ð56Þ
Family 11. If p > 0, q = 0, the solution of Eq. (5) has the form,
GðnÞ ¼
p e
pffiffiffiffi pn
Then we have the ratio, G0 1 n ¼ pffiffi coth pffiffi ; G 8 r 4 r
(iii) If A = 0, then 1 uðnÞ ¼ : n
#12
pffiffiffiffi : 1 64 p r e4 p n
ð53Þ
where uðnÞ satisfy the Riccati equation
Therefore, we obtain the following traveling wave solutions: pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi uðnÞ ¼ a0 þ 2 p cot p n tan p n ; ð46Þ
2
ð52Þ
These are the exact solutions of the (2 + 1)-dimensional Calogero–Bogoyavlenskii–Schiff (CBS) equation obtained by the improved (G0 /G)-expansion method.
uðnÞ ¼ a0 þ a1 uðnÞ;
Since q = 0, then we have the ratio,
"
ð51Þ
n ¼ x þ z 16 p t:
Therefore, we obtain the following traveling wave solution: pffiffiffi pffiffiffi pffiffiffi ð43Þ uðnÞ ¼ a0 þ 2 p tanh p n þ coth p n ;
pffiffiffiffiffiffiffi G0 1 pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi p cot p n tan p n : ¼ G 2
Then we have the ratio,
where
0
#12 pffiffiffiffiffiffiffi p sec2 p n pffiffiffiffiffiffiffi GðnÞ ¼ or pffiffiffiffiffiffiffiffiffi q 2 p r tan pn " #12 pffiffiffiffiffiffiffi p csc2 p n pffiffiffiffiffiffiffi : GðnÞ ¼ pffiffiffiffiffiffiffiffiffi q 2 p r cot pn
ð50Þ
Therefore, we have the solution:
Since q = 0, we have the ratio, pffiffiffi pffiffiffi G 1 pffiffiffi p tanh p n þ coth p n : ¼ 2 G
ð49Þ
where
Therefore, we obtain the following traveling wave solutions: pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi uðnÞ ¼ a0 4 p tan 2 p n or pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi uðnÞ ¼ a0 þ 4 p cot 2 p n ; ð40Þ
"
where 64 p r ¼ 1. Therefore, we have the solution: 1 n uðnÞ ¼ a0 pffiffi coth pffiffi ; 2 r 4 r
ð47Þ
Substituting (53) and (54) into (7), we obtain the following polynomial equation in u as follows:
ð48Þ
ð57Þ
3a21 þ 6a1 u4 ðnÞ þ a1 V þ 8a1 A þ 6a21 A u2 ðnÞ þ a1 A V þ 2 a1 A2 þ 3a21 A2 ¼ 0
ð58Þ
32
M. Shakeel, S.T. Mohyud-Din
Equating to zero the coefficients of the polynomial and solving the set of algebraic equations with the help of Maple, we obtain the following solution: a0 ¼ a0 ; a1 ¼ 2; V ¼ 4A: where a0 and A are arbitrary constants. So, the exact solutions of Eq. (6) are: When A < 0, the solution takes the form, pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi A n uðnÞ ¼ a0 þ 2 A tanh or pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi A n ; uðnÞ ¼ a0 þ 2 A coth
ð59Þ
5. Conclusions
ð60Þ
where n ¼ x þ z þ 4A t: When A > 0, the solution takes the form, pffiffiffiffi pffiffiffiffi uðnÞ ¼ a0 2 A tanð A nÞ or pffiffiffiffi pffiffiffiffi An ; uðnÞ ¼ a0 þ 2 A cot
the selection of initial approximation and the appearance of the dependent variable as transcendental function (i.e. sin u; log u; eu etc) which cannot be perturbed are the main disadvantages of HPM. Similarly, we can write advantages and disadvantages of other methods as well.
ð61Þ
where
An improved (G0 /G)-expansion method is suggested and applied to the (2 + 1)-dimensional Calogero–Bogoyavlenskii–Schiff (CBS) equation. The results obtained by the suggested method have been compared with those obtained by the extended tanh-function method. From this study, we observe that the improved (G0 /G)-expansion method and the extended tanh-function method are not equivalent. We see that all the results obtained by the extended tanh-function are found by the suggested method and in addition some new solutions are attained. From this analysis we can conclude that the proposed method is quite resourceful and practically well suited to be used in finding exact solutions of NLEEs.
n ¼ x þ z þ 4A t: When A = 0, the solution takes the form, 2 uðnÞ ¼ a0 þ ; n where
References ð62Þ
n ¼ x þ z þ 4A t: 4. Discussion From the above results obtained by the two methods, we can draw the following remarks: If we put A ¼ 4 p where p > 0, the results obtained in (60) are identical to the results (34) and (37) respectively. If we put A ¼ 4 p where p < 0 then the results obtained in (61) are identical to the results given in (40). From these remarks we have the following assessments: ‘‘The exact solutions of the Calogero–Bogoyavlenskii– Schiff equation obtained by means of the extended tanh-function method can also be obtained by the improved (G0 /G)expansion method and in appendage some new solutions are obtained. The calculations of the projected method are also easier than the extended tanh-function method as well as the basic (G0 /G)-expansion method’’. The advantages and disadvantages of some of the methods are 4.1. Advantages and disadvantages of VIM This method produces the solutions in terms of convergent series and does not require linearization or small perturbation is the main advantage, whereas, calculation of exact Lagrange multiplier is very difficult to calculate for higher order differential equation and the restriction of the nonlinearity term are disadvantages of VIM. 4.2. Advantages and disadvantages of HPM The HPM solves nonlinear problems without using Adomian’s polynomials is the main advantage of this method, whereas,
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