Selçuk J. Appl. Math. Selçuk Journal of Vol. 14. No. 1. pp. 37-45, 2013 Applied Mathematics
The Functional Variable Method for Some Nonlinear (2+1)-Dimensional Equations
Orkun Tasbozan, N. Murat Yagmurlu, Alaattin Esen
Inonu University, Faculty of Arts and Sciences, Department of Mathematics, 44280 Malatya, Turkiye.
e-mail:aesen@ inonu.edu.tr,orkun.tasb ozan@ inonu.edu.tr,murat.yagmurlu@ inonu.edu.tr
Received Date: March 31, 2012 Accepted Date: March 10, 2013
Abstract. In this paper, the functional variable method is applied to the (2+1)-dimensional Boussinesq, (2+1)-dimensional KdV and (2+1)-dimensional Nizhnik-Novikov-Veselov equations to obtain their some solitary and periodic wave solutions.
Key words: The functional variable method; (2+1)-dimensional Boussinesq equation; (2+1)-dimensional KdV equation; (2+1)-dimensional Nizhnik-Novikov-Veselov equation.
AMS Classi…cation: 35C07,76B25. 1. Introduction
When many physical phenomena in di¤erent …elds of pysics and engineering are mathematically modelled, they generally result in nonlinear ordinary or mostly partial di¤erential equations. In applied mathematics, it is of great impor-tance to obtain and investigate the exact solutions of these equations. Thus, recently, numerous e¢ cient and accurate methods such as sine-cosine method[1], tanh function method[2], variational iteration method[3], homotopy perturba-tion method[4,5,6], Exp-funcperturba-tion method[7,8], F-expansion method[9,10], (G0 =G)-expansion method[11,12] and others have been presented and successfully ap-plied to obtain exact solutions of nonlinear partial di¤erential equations. These methods vary greatly from each other in terms of the initial approximation, transformations, etc. For example, some of them use transformations to con-vert nonlinear equations into more easily handled simple equations, some others use trial functions in an iterative scheme to converge rapidly to the exact so-lution, and still others look for the solution of nonlinear evolution equations (NLEEs) viewed as polinomial in variable satisfying a subsidiary nonlinear or-dinary di¤erential equation. Most recently, A. Zerarka et al.[13] have presented
functional variable method (FVM) to obtain exact travelling wave solutions. They have successfully applied the method to obtain the travelling wave so-lutions including the parameters of the one dimensional Boussinesq and the PHI-four equations. In the present study, before applying the method, we will give the solution procedure which will be followed in the rest of the study. 2. General formulation
The general characteristics of the FVM can be outlined as follows. A nonlinear partial di¤erential equation with several independent variables can be written in the form of
(1) P (u; ut; ux; uy; uz; utt; uxt; uxx; :::) = 0
where P is a function, the subscripts denote partial derivatives, and u(t; x; y; z; :::) is the unknown function to be determined. First of all, the new wave variable can be written as (2) = p X i=0 i i+
where i’s are the independent variables, and and i’s are free parameters.
Next, we can introduce the following transformation for a travelling wave solu-tion of Eq. (1),
(3) u( 0; 1; :::) = U ( )
and the chain rule
(4) @ @ i(:) = i @ @ (:); @ @ i@ j(:) = i j @2 @ 2(:); ::::
Using Eq. (3) and Eq. (4), the nonlinear partial di¤erential equation (1) can be transformed into an ordinary di¤erential equation of the form
(5) Q(U; U ; U ; U ; U ; :::) = 0:
Finally, using a transformation where the unknown function U is considered as a functional variable of the form
(6) U = F (U )
and the successive derivatives of U of the forms
(7) U =12(F2)0; U = 1 2(F 2)00p F2; U = 12[(F2)000F2+ (F2)00(F2)0]; .. .
where "0" stands for dUd ; we proceed as follows. When expressions given in Eq. (7) are used in Eq. (5), the ordinary di¤erential equation in Eq. (5) can be written in terms of U and F as follows
(8) R(U; F; F0; F00; F000; F(4); :::) = 0
Since the newly obtained form (8) allows the analytical solutions of many non-linear wave type equations, the main idea behind this particular form is of great interest. By taking the integral, the Eq. (8) becomes an expression of F , and the obtained result together with the Eq. (6) result in an appropriate solution of the original problem.
To show how e¤ective and convenient the method is, we will apply it to the dimensional Boussinesq, the dimensional KdV and the (2+1)-dimensional Nizhnik-Novikov-Veselov equations in the rest of the paper. 3. Applications
3.1. (2+1)-dimensional Boussinesq equation
The (2+1)-dimensional Boussinesq equation is of the form[14,15] (9) utt uxx uyy (u2)xx u4x= 0:
By taking the new wave variable in Eq. (2) as = 0t + 1x + 2y + and
using the transformation given in Eq. (3) and applying the chain rule (4), we can rewrite the Eq. (9) as follows
(10) ( 20 21 22)U 21(U2) 41U4 = 0:
When Eq. (10) is integrated twice with respect to ; the constants of integration are set to zero, Eq. (11) can be rewritten in a practical form as follows
(11) ( 20 21 22)U 21U2 41U = 0: By using U =1
2(F 2)0
in Eq. (7), we obtain F (U ) as follows from Eq. (11)
(12) F (U ) = p 2 0 21 22 2 1 U s 1 2 2 1 3( 2 0 21 22) U :
With the help of Eq.(12), it is obviously seen that the di¤erential equation (6) is completely integrable because of the fact that its solutions are deduced directly from the integral as follows
(13)
Z dy
yp1 y = ln
1 p1 y 1 +p1 y :
By taking the Eq.(6) and the relation (13) into account, the following quadratic equation can easily be obtained
A2Z2+ 4(1 A2)Z 4(1 A2) = 0 where A = 2tanh( p 2 0 21 22 2 2 1 ) 1 + tanh2(p 20 21 22 2 2 1 ) and Z = 2 2 1 3( 2 0 21 22) U:
Finally, after some simple algebraic manipulations, we obtain two solutions of the problem as follow
(14) U1( ) = 3( 2 0 21 22) 2 2 1 sech2( p 2 0 21 22 2 2 1 ) and (15) U2( ) = 3( 2 1+ 22 20) 2 2 1 csch2( p 2 0 21 22 2 2 1 ): In the case 2
0> 21+ 22, the Eqs. (14) and (15) become
u1(x; y; t) = 3( 2 0 21 22) 2 2 1 sech2( p 2 0 21 22 2 2 1 ( 0t + 1x + 2y + )); u2(x; y; t) = 3( 2 1+ 22 20) 2 2 1 csch2( p 2 0 21 22 2 2 1 ( 0t + 1x + 2y + ))
respectively, which are solitary solutions. For 2
0< 21+ 22, the Eqs. (14) and (15) become
u3(x; y; t) = 3( 2 0 21 22) 2 2 1 sec2( p 2 1+ 22 20 2 2 1 ( 0t + 1x + 2y + )); u4(x; y; t) = 3( 2 0 21 22) 2 2 1 csc2( p 2 1+ 22 20 2 2 1 ( 0t + 1x + 2y + ))
respectively, which are periodic solutions. If we take 0= ; 1 = ; 2 =
and = 0 then we get the solutions identical to the Eqs. (22)-(23) given in Ref[16].
3.2. (2+1)-dimensional KdV equation
The (2+1)-dimensional KdV equation can be written of the form[17,18] (16) ut+ uxxx 3vxu 3vux= 0;
By taking the new wave variable in Eq. (2) as = 0t+ 1x+ 2y+ ; u(x; y; t) =
U ( ) and v(x; y; t) = V ( ); using the transformation (3) and appliying the chain rule (4), we can rewrite the Eq. (16) as follows
(17) 0U +
3
1U 3 1V U 3 1V U = 0; 1U = 2V :
When Eq. (17) is integrated once with respect to ; and the constants of in-tegration are set to zero, then Eq. (17) can be written in a practical form as follows (18) 0U + 31U 3 21 2 2 U2= 0; V = 1 2 U:
When the equality U = 12(F2)0 is taken into account and used in Eq. (18), the function F (U ) can be rewritten as follows
(19) F (U ) = r 0 3 1 U s 1 2 2 1 0 2 U :
With the help of Eq. (19), it can easily be seen that the di¤erential equation (6) is completely integrable, since its solutions are deduced directly from the integral given in Eq.(13). By using Eq.(6) and the relation (13), the following quadratic equation can easily be obtained
A2Z2+ 4(1 A2)Z 4(1 A2) = 0 where A = 2tanh(12q 0 3 1 ) 1 + tanh2(1 2 q 0 3 1 ) and Z = 2 2 1 0 2 U:
Finally, after some simple algebraic manipulation, the following solutions are obtained (20) U1( ) = 0 2 2 2 1 sech2(1 2 r 0 3 1 ) and (21) U2( ) = 0 2 2 2 1 csch2(1 2 r 0 3 1 ): In the case 0 3
1 < 0, the following solitary solutions u1(x; y; t) and v1(x; y; t) from
u1(x; y; t) = 0 2 2 2 1 sech2(1 2 r 0 3 1 ( 0t + 1x + 2y + )); u2(x; y; t) = 0 2 2 2 1 csch2(1 2 r 0 3 1 ( 0t + 1x + 2y + )); v1(x; y; t) = 0 2 1 sech2(1 2 r 0 3 1 ( 0t + 1x + 2y + )); v2(x; y; t) = 0 2 1 csch2(1 2 r 0 3 1 ( 0t + 1x + 2y + )): When 0 3
1 > 0; the following periodic solutions u3(x; y; t) and v3(x; y; t) from Eq.
(20) and u4(x; y; t) and v4(x; y; t) from (21) are obtained
u3(x; t) = 0 2 2 2 1 sec2(1 2 r 0 3 1 ( 0t + 1x + 2y + )); u4(x; t) = 0 2 2 2 1 csc2(1 2 r 0 3 1 ( 0t + 1x + 2y + )); v3(x; t) = 0 2 1 sec2(1 2 r 0 3 1 ( 0t + 1x + 2y + )); v4(x; t) = 0 2 1 csc2(1 2 r 0 3 1 ( 0t + 1x + 2y + )):
3.3. (2+1)-dimensional Nizhnik-Novikov-Veselov(NNV) equation The (2+1)-dimensional NNV equation is written of the form[19,20]
(22)
ut= uxxx+ uyyy+ 3(vu)x+ 3(uw)y ;
ux= vy ;
uy= wx :
By taking the new wave variable in Eq. (2) as = 0t+ 1x+ 2y+ ; u(x; y; t) =
U ( ); v(x; y; t) = V ( ) and w(x; y; t) = W ( ); using the transformation (3) and appliying the chain rule (4), we can rewrite the Eq. (22) as follows
(23)
0U = ( 31+ 32)U + 3 1(V U ) + 3 2(U W ) ; 1U = 2V ;
2U = 1W :
When the Eq. (23) is integrated once with respect to and the constants of integration are set to zero, Eq. (23) can be rewritten in a practical form as follows (24) 0U = ( 31+ 32)U + 3 3 1+ 32 1 2 U2= 0; V = 1 2 U; W = 2 1 U:
Again, when the equality U = 12(F2)0is pluged into Eq. (24), the following expression for the function F (U ) is obtained
(25) F (U ) = r 0 3 1+ 32 U s 1 2( 3 1+ 32) 0 1 2 U :
With the help of Eq. (25), it can easily be seen that the di¤erential equation (6) is completely integrable, since its solutions are deduced directly from the integral given in Eq.(13) . By using Eq.(6) and the relation (13), the following quadratic equation can easily be obtained
A2Z2+ 4(1 A2)Z 4(1 A2) = 0 where A = 2tanh(1 2 q 0 3 1+ 32 ) 1 + tanh2(1 2 q 0 3 1+ 32 ) and Z = 2( 3 1+ 32) 0 1 2 U:
After appying some simple algebraic manipulation, …nally, the following two solutions of the problem are obtained
(26) U1( ) = 0 1 2 2( 3 1+ 32) sech2(1 2 r 0 3 1+ 32 ) and (27) U2( ) = 0 1 2 2( 3 1+ 32) csch2(1 2 r 0 3 1+ 32 ): In the case of 0 3
1+ 32 > 0, the following solitary solutions u1(x; y; t); v1(x; y; t)
and w1(x; y; t) from Eq. (26) and u2(x; y; t); v2(x; y; t) and w2(x; y; t) from Eq.
(27) are obtained u1(x; y; t) = 0 1 2 2( 3 1+ 32) sech2(1 2 r 0 3 1+ 32 ( 0t + 1x + 2y + )); u2(x; y; t) = 0 1 2 2( 3 1+ 32) csch2(1 2 r 0 3 1+ 32 ( 0t + 1x + 2y + )); v1(x; y; t) = 0 2 1 2( 3 1+ 32) sech2(1 2 r 0 3 1+ 32 ( 0t + 1x + 2y + )); v2(x; y; t) = 0 2 1 2( 3 1+ 32) csch2(1 2 r 0 3 1+ 32 ( 0t + 1x + 2y + ));
w1(x; y; t) = 0 2 2 2( 3 1+ 32) sech2(1 2 r 0 3 1+ 32 ( 0t + 1x + 2y + )); w2(x; y; t) = 0 2 2 2( 3 1+ 32) csch2(1 2 r 0 3 1+ 32 ( 0t + 1x + 2y + )): When 0 3
1+ 32 < 0, the following periodic solutions are obtained for u3(x; y; t);
v3(x; y; t) and w3(x; y; t) from Eq. (26) and u2(x; y; t); v2(x; y; t) and w2(x; y; t)
from (27) are obtained u3(x; y; t) = 0 1 2 2( 3 1+ 32) sec2(1 2 r 0 3 1+ 32 ( 0t + 1x + 2y + )); u4(x; y; t) = 0 1 2 2( 3 1+ 32) csc2(1 2 r 0 3 1+ 32 ( 0t + 1x + 2y + )); v3(x; y; t) = 0 2 1 2( 3 1+ 32) sec2(1 2 r 0 3 1+ 32 ( 0t + 1x + 2y + )); v4(x; y; t) = 0 2 1 2( 3 1+ 32) csc2(1 2 r 0 3 1+ 32 ( 0t + 1x + 2y + )); w3(x; y; t) = 0 2 2 2( 3 1+ 32) sec2(1 2 r 0 3 1+ 32 ( 0t + 1x + 2y + )); w4(x; y; t) = 0 22 2( 3 1+ 32) csc2(1 2 r 0 3 1+ 32 ( 0t + 1x + 2y + )):
If we take 0= c; 1= 2= 1; and = 0 then we get the solutions identical
to the Eqs. (19)-(22) given in Ref[21]. 4. Conclusion
In this paper, the functional variable method has been successfully used to obtain some exact travelling wave solutions for the (2+1)-dimensional Boussi-nesq, the (2+1)-dimensional KdV and the (2+1)-dimensional Nizhnik-Novikov-Veselov equations. It is clearly seen from the obtained results that the present method is straightforward, concise and a promising tool for various general NLLEs arising in di¤erent …elds of science and physics.
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