General Solution of Two Generalized Form of Burgers Equation by Using the (G'/G) Expansion Method

General Solution of Two Generalized Form of Burgers

Equation by Using the



_







G

G





-Expansion Method

Abdollah Borhanifar1*, Reza Abazari2

1

Department of Mathematics, University of Mohaghegh Ardabili, Ardabil, Iran 2

Department of Mathematics, Islamic Azad University, Ardabil Branch, Ardabil, Iran Email: *[email protected], [email protected]

Received May 11, 2011; revised June 11, 2011; accepted June 18, 2011

ABSTRACT

In this work, the G

G     



-expansion method is proposed for constructing more general exact solutions of two general

form of Burgers type equation arising in fluid mechanics namely, Burgers-Korteweg-de Vries (Burgers-KdV) and

Bur-ger-Fisher equations. Our work is motivated by the fact that the G

G     

 -expansion method provides not only more

gen-eral forms of solutions but also periodic and solitary waves. If we set the parameters in the obtained wider set of solu-tions as special values, then some previously known solusolu-tions can be recovered. The method appears to be easier and faster by means of a symbolic computation system.

Keywords: G

G     

 -Expansion Method; Generalized Burgers-KdV Equation; Generalized Burgers-Fisher Equation;

Hyperbolic Function Solutions; Trigonometric Function Solutions

1. Introduction

Nonlinear evolution equations (NLEEs) have been the subject of study in various branches of mathematical- physical sciences such as physics, biology, chemistry, etc. The analytical solutions of such equations are of funda- mental importance since a lot of mathematical-physical models are described by NLEEs. Among the possible solutions to NLEEs, certain special form solutions may depend only on a single combination of variables such as traveling wave variables. In the literature, there is a wide variety of approaches to nonlinear problems for con- structing traveling wave solutions. Some of these appro- aches are the Jacobi elliptic function method [1], inverse scattering method [2], Hirotas bilinear method [3], homo- geneous balance method [4], homotopy perturbation me- thod [5], Weierstrass function method [6], symmetry me- thod [7], Adomian decomposition method [8], sine/cosine method [9], tanh/coth method [10], the Exp-function me- thod [11-16] and so on. But, most of the methods may sometimes fail or can only lead to a kind of special so- lution and the solution procedures become very com-

plex as the degree of nonlinearity increases.

Recently, the G

G     

 -expansion method, firstly intro-

duced by Wang et al. [17], has become widely used to

search for various exact solutions of NLEEs [17-27]. The

G G     

 -expansion method is based on the assumptions

that the travelling wave solutions can be expressed by a

polynomial in G

G    

 , and that G G=

 

 , satisfies a

second order linear ordinary differential equation

 

2

2

d d

= 0 d

d

G G

G

 _  _{ }



   (1)

where  , are arbitrary constants and  =kxt.

method. The G-expansion method is direct, concise,

G    

elementary and effective, and can be used for nonlinear

, much work has been done on developing

an

partial differential equation involving higher-order non- linear terms.

As we know

d extending the G-expansion method for solving

G    

nonlinear evolution equations (see, for example [17-27]

(2) and their references). But in generalized cases, a small amount of work has been done (see, for example [28,29]). In this paper, we restrict our attention to the study of the following generalized forms of Burgers equation,

= 0,

n

  

g-Burgers-KdV

t x xx xxx

u pu u qu ru



1



=

g-Burgers-Fisher

n n

t x xx

u pu u qu ru u 0,

(3)

where and and are constants. These

ns are t al m

(4)

Equation (4) is the lowest order approx on

> 1,

n p q, r

equatio he gener ized co bined forms of Burgers

equation

= 0

t x xx

u pu qu

imation for the e-dimensional propagation of weak shock waves in a fluid [30,31]. It is also used in the description of the variation in vehicle density in highway traffic [32]. It is one of the fundamental model equations in fluid mecha- nics. The Burgers equation demonstrates the coupling be- tween diffusion uxx and the convection process uux.

Burgers introduce his equation to capture some of

features of turbulent fluid in a channel caused by the interaction of the opposite effects of convection and di- ffusion [33]. It is also used to describe the structure of shock waves, traffic flow, and acoustic transmission [34]. The generalized Burgers-KdV Equation (2) are models

d t the

for the propagation of waves on an elastic tube (see [28, 29,35,36] and their references). It can be regarded as a combination of the Burgers equation (p0, q0, r0)

and the KdV equation



p0, = 0,q



ec ,

Ismail Aslan in [26] applied the 0

r . Esp ially

G

 

G

   -exp

thod for the Burgers-KdV equation ( and in [36]

tion

terest in the present work is in implement-

in

ansion me-

= 1

n )

to m

Ahmet Bekir applied the tanh method odified form

of Burgers-KdV equation (n= 2). In addition, the gene-

ralized Burgers-Fisher Equa (3), see [37], has a wide

range of applications in plasma physics, fluid physics, capillary-gravity waves, nonlinear optics and chemical physics [38].

Our first in

g the G

G

  

-expansion method to stress its power in

   

handling nonlinear equations, so that one can apply it to models of various types of nonlinearity. The next interest is in the determination of exact travelling wave solutions for the generalized form of Equations (2) and (3) espe-

cially the modified form of them n= 3. Searching

2

 

 

for exact solutions of nonlinear problems has attracted a

2. Description of the

considerable amount of research work where computer symbolic systems facilitate the computational work.













G

G



-Expansion

Method

The objective of this section is to outline the use of the

G G

  

 -expansion method for solving certain nonlinear

 

partial differential equations (PDEs). Suppose we have a nonlinear PDE for u x t

 

, , in the form



, ,



= 0

P u,u u u_x, _t, _xx u_xt  (5)

where P is a polynomial in its argum

no

ents, which in- cludes nlinear terms and the highest order derivatives. The transformation u x t

 

, =U

 

 , =kxt, reduces Equation (5) to the ordinary differential equation (ODE)



2



, , , , , = 0

P U kUU k U k U    (6)

where U=U

 

 , and prime denotes derivative with

respect sume that the solution of Equation (6)

can be expressed by a polynomial in to ξ. We as

G

 

  as follows:

G

 

 

0

=1

= ,

i m

i m

i

G U

G

  _  _   

 



0 (7)

where 0, and i are constants to be determined later,

and G

 

 satisfies the second order linear ordinary di- fferential Equation (1):

 

 

_{ }

2

d G  dG 

2 _d = 0

d    G (8)

where  and  are arbitrary constants. It follows

) a th

from (7 nd (8), at

1 1

=1 =

i i i

i i

G

U i

G G G

    

m  _{G G  } 

          

     

 

 





















2 1

=1

1 2

2 2

= 1 2 1

2 2 1

1

i i

m i i

i i

i

G G

U i i i

G G

G G

i i

G G

G i

G

 

  



 





    

        

   



 

   

  _{ }   _{ }

   

 

 

  _{ } 

 _{ }



(10)

and so on, here the prime denotes the derivative with respective to .

To determine explicitly, we take the following

er ne the integer by substituting Equ-

at a

ar term(s) and the highest order derivative. al

u

three steps:

Step 1. Det mi m

ion (7) into Equ tion (6) and balancing the highest order nonline

Step 2. Substitute Equation (7) ong with Equation (8) into Equation (6) and collect all terms with the same

order of G

G  

  together, the left-hand side of Equation

verte

(6) is con d into a polynomial in G

G  

 . Then set

each coefficient of this polynomial to zero to derive a set of algebraic equations for k, , , ,   0 and i.

Step 3. Solve the system of algebraic equations ob- tained in Step 2, for k, , , ,   0 and i by use of

Maple, along with the general solutions of Equation (8), w

demonstrate the

e can have more travelling wave solutions of Equation (5).

3. Application

In this section, we will G

G     

 -ex-

pansion method on the generalized equations listed in (2)

the generalized Burgers-KdV equation with and (3).

3.1. Generalized Burgers-KdV Equation

Considering

higher-order nonlinear terms

= 0

n

t x xx xxx

u pu u qu ru (11)

Now,using the wave variable u x t

 

, =U

 

 ,

=kx t,

 

ation and ne

in (11) and in

glecting the constant find

tegrating the

of integration, we resulting equ-

1 2 3

= 0 1

n

pk

U U qk U rk U

n

 _{ }

  

 (12)



To achieve our goal, we use the transformation

 

= 1n

 

,

U  V 



 





2

2 2 pkn 3 2

   

2 3

1 = 0

n V V qk nVV

rk nVV n V

  

(13)

According to Step 1, we get

We then suppose that E has the

following formal solutions: 1

n

3m= 2m2

quation (13)

, hence

= 2.

m

2

2 1 0 2

= G G , 0

V

G G

 _ _  _ _  

    (14)

where  2, 1, and 0 are constants which are un-

known termin later.

Substituting Equation (14) into Equation (13) and co-

all term

to be de ed

llecting s with the same order of _G_, together,

G  

the left-hand sides of Equation (13) are converted into a

polynomial in G

G   

 . Setting each coe t of each

  fficien

polynomial to zero, we derive a set of algebraic equations for k, ,   , 0, 1, and 2, (collecting the

coefficients of , = 0,1, , 6

i G  

d solving them e o

eneral result:

i G  

   and setting it to zero,

Appendix I) an by us f Maple, we get

the following g

































2 2

1 1

2 3 2

2 4 6 4

= ,

2 7

n pn qkn pn qkn kq rk n n

 



2 2 3

1 1 1

2 4 5 4 3 2

2 2 2

1 0

2 2 2

2 2

2 2

14 8

4 4 12 8

= ,

16 10 37 64 52 16

2 2

= , = ,

8 2 3

4

2 2 3

=

n

p qkn pn pn qkn kq n

r k n n n n n

kq n pn

rk n n r n

rk n n pn

  





 





 

    

    





  



  _

 _



(15)

Substitute the above general case in (10), we get

   

 _



 









2 2 2

2

2 2 1

1 _{2 2}

2 2 3

=

8 2 3

rk n n _G

V

G

pn  

pn G

G rk n n

 

  _{ }

 _{ }



 

 _{ }

 

 

(16)

then use the transformation

 

 

1 = n ,

U  V  the hyper-

 





2

2 2 _{1 2}

1 2

2 1

1 2

1

2 1

1 1

sinh cosh

2 2 3 _{2 2}

,

2

1 1

2 sinh cosh

2 2

1 1

sinh cosh

2 2

1 1

2 sinh cosh

2 2

C C

rk n n u x t

pn

C C

C C

C C

 



 

 



 

 _{  }

_{   } _{ }  _ _  _ _

   

 _   _

_ 

 _{    } 

         

       



    

_{ }  _ _  _

   

 



 _ _  _ 

  

   





1

2 2

1

2 2

2 8 2 3

n

pn rk n n

 



 



 



 _ _



 _{ }  _{ }



 _ _  _

  

(17)

and the trigonometric function solutions of Equation (11), will be:

 





2

2 2 _{1 2}

2 2

2 1

1 2

1

2 1

sin i

i C

 _{  }

 _    cos _

2 2 3 _{2 2}

,

2

2 sin cos

2 2

sin cos

2 2

2 sin cos

2 2

i C rk n n

u x t

pn _C i _C i

i i

i C C

i i

C C

 



 

 



 

 

_  _ 

   

   _{   }

 _   _

_ 

      

         

       



    

 _{ }  _ _  _

   

 



  _ _  _ 

   

 _{   }



 





1

2 2

1

2 2

2 8 2 3

n

pn rk n n

 



 



 



 _ _



 _  _{ }



 _ 

 

  

(18)

where





















2

2

2 2

2 2 2

2 2

1 1

2 3 2

2 2

= ,

4

= ,

4

2 4 6 4

=

2 7 14 8

kq n

kx t

r n q n k r n

n pn qkn pn qkn kq

rk n n n



 



 







   



  

and C C k1, 2, , and 1

ke

are arbitrary constants and

ous

known results in the literature can be rediscovered, for

instance, setting then the general solutions (17)-

2_{= 1.}

C and

i

If 1 C2 ta up special values, the vari

2

(18) reduces, respectively,

= 0,

C

 

























2 2 2

 _

2

2 2 2 2 _{2 2}

1,1 2 2 2 2

1 2 2

2

2 2 2 2 2 2

1

1 2 2 2 2 2

2 2 3 4 ₁

, tanh

2 2 ₄ 2

4 1

tanh

2 2 4 2 8 2 3

n

q n

rk n n k r n q n

u x t

pn k r n

q n

k r n _{q n pn}

rk n n

k r n

 

 

 



 _{ }

       

 



 _  _

 

  _  _ 

 _{ }

 _{ }





 



 

  _



 

 

  _{ }  

 

 

 _{ } 



 



  

 

























2 2 2

2

2 2 2 2 _{2 2}

2,1 2 2 2 2

1 2 2

2

2 2 _{2 2 2 2}

1

1 2 2 2 2 2

2 2 3 4

, tan

2 2 ₄ 2

4 tan

2 2 4 2 8 2 3

n

q n

rk n n k r n i q n

u x t i

pn k r n

q n

k r n i q n pn

i

rk n n

k r n

 

 

 

 _{ }

 _{ }

       

 



 _  _

 

  _  _ 

 _{ }

 _{ }





 



 

  _



 

 

  _{ }  

 

 

 _{ } 



 



  



and when then we deduce from general solutions (17)-(18) that, respectively

(20)

1= 0,

C

 









2 2 2

2 2 _{2 2 2 2 2}

1,2 2 2 2

1 2 2

2 2 2 2 2 2 2

1

1 2 2 2 2 2

2 2 3 _{( 4) 1}

, coth

2 2 ( 4) 2

( 4) 1

coth

2 2 ( 4) 2 8 2 3

n

q n

rk n n _{k r n q n}

u x t

pn k r n

q n

k r n q n pn

k r n rk n n

 

 

 

 _{ }

 _{ }

 

  

   _{ } 

_{ }  _

 _ 

 _{   }

  

 





 



 _{ } 

 

 _{ } 

 _{  } _ _

  

  

 

  _

  

2

(21)

 









2 2 2

2 2 _{2 2 2 2 2}

2,2 2 2 2

1 2 2

2 2 2 2 2 2 2

1

1 2 2 2 2 2

2 2 3 _{( 4)}

, cot

2 2 ( 4) 2

( 4) cot

2 2 ( 4) 2 8 2 3

n

q n

rk n n _{k r n i q n}

u x t i

pn k r n

q n

k r n i q n pn

i

k r n rk n n

 

 

 

2



 _{ }

 

 

   _{ } 

_{ }  _

 _ 

    

 _ _





 



 _{ } 

 

 _{ } 

 _{  } _ _

  

  

 

  _

  

(22)

where

 _

















2 2

2

1 1

2 2 3 2

2 4 6 4

2 2

= , =

2 7 14 8

4

n pn qkn pn qkn kq

kq n

kx t

rk n n n

r n

 

        

  



are arbitrary constants and i2= 1.

and k, and ₁

3 = ,

2

n and 1= 0 then

In particulars cases, if

 

2 2 ₃

2 2 2 2

1 2 2 2 2

70 3 3 28 3

, = tanh

9 22 22 121 22

rk q q kq t q

u x t kx

p k r k r r rk

 _{    } 

_{       } 

  _{  }_  

 

 

 

 

2 2 ₃

2 2 2 2

2 2 2 2 2

70 3 3 28 3

, coth

9 22 22 121 22

rk q q kq t q

u x t kx

p k r k r r rk

 _{    } 

 _{   } 

  _  _ 

  _{  }_  

 

 

and

 

 

2 2 ₃

2 2 2 2

3 , = ta

9 22

u x t

p

_{ } 

 

 2 2 2 2 

2 2 ₃

2 2 2 2

4 2 2 2 2

70 3 3 28 3

n

22 121 22

70 3 3 28 3

, cot

9 22 22 121 22

rk q i q _kx kq t q

r rk

k r k r

rk q i q kq t q

u x t kx

p k r k r r rk

 _{    } 

 _{  }

 

  _

 _{ } 

 

 

 

 _{    } 



 _{   } 

  _  _ 

  _{  }_  

 

 

 

are the four solitary wave solutions of the particular

Burgers-KdV equation







 





2 2 3 2

2

2 _{1 =}

rn V V n VV pknV V

qk n V nVV

  

  

 

   0

(26)

According to Step 1, we get 2, hence

We then suppose that following formal solutions:

3

2 _{= 0}

t x xx xxx

u pu u qu ru (23)

where is arbitrary constant and

nt real solutions of th urgers-KdV

Equation (2) were obtained by Ahmet Bekir in [36] using the extended tanh method.

3.2. Generalized Burgers-Fisher Equation

Now considering the generalized Burgers-Fisher equ- ation with higher-order nonlinear terms

0 (24)

3m1 = 2m

Equation (26) has the

= 1.

m

k

ere

2 = 1.

i 

e general B Diff

1 0 1

= G ,

V

G

 _  _   

  0 (27)

where and 0

ed later.



1



=

n n

t x xx

u pu u qu ru u

Using the wave variable u x t

 

, =U

 

 , =kxt, in (24), we find





2

1 =

n n

U pkU U qk U rU U

     0 (25)

hieve nsfor

To ac our goal, we use the tra mation

 

=_Vn

 

,

1

U   that will carry (25) into the ODE

1

 are constants hich are un- known

to be determin

Similar on previous section, substituting Equation (29) into Equation (26) and collecting all terms with the same

order of

w

G G     

 , together, we derive a set of algebraic

equations for k, , ,  0, and 1, (col

coefficients of

lecting the

i

, = 0,1, , 4

G _i  

   and setting it to zero,

Appen

G  

dix II) and solving them by use of Maple, we get the following general result:













2 2

2 2

0

( 1) 1 ( 1) 1

1

k qr n p qk n

n p qk n

p n 



  _{  }



2

 



1

, = , =

2 2pn  pn _ (28)

2 2 2

= , =

4 4k q n( 1)

   

 _



Substitute the above general case in (25), we get

then use the transformation

 

 

1 = n ,

U  V  the hyper-



1



1

=

2 2

bk n G

V

an  G



  _{  }

  

   (29) bolic and _{tion (11), will be:} the trigonometric function solutions of Equa-

1 2

 

2 2

2 2 2 2

2 2

2 2 2 1 2 2

)

2 sinh

2(1 ) 2(1 )

n p

n k q

U

p

C C

n k q n k q

     

 

_{ }

  

 _{      }_ 

      

 

1

2 C sinh ₂₍₁

qk p

  _



  _ cosh

n

C

   

1 = 2 2

2p k q _n

    



 

2(1 ) ₁

cosh

n k q

n p

 

    

  _{ }

 

 

(30)

n p

2

1 2 2 2 2 2

2

2

2 2

2 2 2 1 2 2

sin cos

2(1 ) 2(1 ) ₁

2

sin cos

2(1 ) 2(1 )

ni p ni p

C C

n k q n k q

qki p

ni p ni p

C C

n k q n k q

 

 

      

  _ _ _ _ 

 

  _{   } 



    

   

 _{      } _ 

       

 

(31)

 

2 = ₂ 2 2

U

p k q

    



where









2 2

( 1) =kx k qr n

   , C1,C k2, are

1

p t p n



 and

arbitrary constants and

e literature can be rediscovered, for then the general solutions

(30) reduc y,

2 = 1

i  .

If C1 and C2 take up special values, the various

known results in th instance, setti

-(31)

ng es, re

2= 0,

C

spectivel

 



_

_



1 2 2

2 2

1,1 2 2 2 2

( 1) ₁

, = tanh

2 2(1 ) 1

n

k qr n p

qk p n p

u x t kx t

p k q n k q p n

   _{ }  

     

   _  _ 

 

 2



(32)

 



_

_



1 2 2

2 2

2,1 2 2 2 2

( 1) ₁

, = tan

2 2(1 ) 1

n

k qr n p

qki p ni p

u x t kx t

p k q n k q p n

   _{ }  

     

   _   

 

 2

 

(33)

and when C1= 0, then we deduce from general solutions (30)-(31) that, respectively

 

2 2



_

2

_

2



1,1 2 2 2 2

( 1) ₁

1 2

k q n p

p

kx t

p n k q

 



 



  

 

(34)

, = coth

)

r

qk p n

u x t

q

    

    

 

2p k 2(1n 

 

2 p2  k qr n



(

_

1)2

_

p2



2,1 2 2 2 2

1

, = cot

2 2(1 ) 1 2

qki p ni

u x t kx t

p k q n k q p n

   

     

   _  _ 

 

 

(35)

where









2 2

( 1)

= ,

1

k qr n p

kx t

p n

   

 k, is arbitrary constant and

2

= 1

i  .

3

= ,

2

n

In particular, if then

 

 

2 3 2

2 2

25

3 2k 4 qr p

qk p p

 

1 2 2 2 2

2 3 2

2 2

2 2 2 2 2

1

, tanh

2 10 5 2

25

3 2 4 1

, tan

2 10 5 2

u x t kx t

p k q k q p

k qr p

qki p i p

u x t kx t

p k q k q p

     

 

    

  

    

    

 

 

 

    _   

   _ _  

    

   

    

  _ _ 

 

 



   _ _ 

and

 

 

2 3 2

2 2

3 2 2 2 2

2 3 2

2 2

4 2 2 2 2

25

3 2 4

, coth

2 10 5

25

3 2 4

, cot

2 10 5

k qr p

qk p p

u x t kx t

p k q k q p

k qr p

qki p i p

u x t kx t

p k q k q p

    _   

   _ _  

    

 

    

  _ _ 

 

 

    _   

   _ _  

    

 

    

  _ _ 

 

 

1 2

1 2 



3 3

2 ₁ 2 ₌

t x xx

u pu u qu ru_ u _

  0 (36)

is arbitrary constant and

Different real solutions o neral Burgers-Fisher

Equation (3) were obtained by Wazwaz in [39] using the tanh method and by El-Wakil in [38] using a modified tanh-function method and recently by Luwai Wazzan in [28] using a modified tanh-coth method.

4. Conclusions

This study shows that the

,

k _i2_{= 1}

. f the ge

G G     

 -expansion method is

quite efficient and practically well suited for use in find- ing exact solutions for the generalized form of Burgers- Korteweg-de Vries (Burgers-KdV) and Burger-Fisher equa- tions. The results show that this method is a powerful Mathematical tool for obtaining exact solutions for the general B-Fisher and B-KdV equations. With the aid of Maple, we have assured the correctness of the obtained solutions by putting them back into the original equation. It is also a promising method to solve other nonlinear partial differential equations arising in engineering sci-

5. Acknowledgements

This work is partially supported by Grant-in-Aid fro

the rdabil, Iran.

1-166. doi:10.1016/j.physleta.2005.07.034

ences.

m university of Mohaghegh Ardabili, A

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Appendix I

0

3 2 2 2 2 3 3 2 2 2 2

0 0 0 1

3 3 2 2 3 2

0 1 0 2

1 :

2

G

n n pkn rk qk n

G

rk n rk n rk n

G

      

      



  _{   }

   

  

  

2 3 2 2 2 3 2

0 1 qk n 0 1 rk n 1 2rk n 0 2

           

0 1

2 2

3 2 2 2 2 2 2 2

0 2 0 1 1

= 0,

qk n qk n qk n qk n

 

        

1 0 1 0 1 0

: 2 n 2 n 3pkn 2

G           

   

2 1

2 3 2 3 2 3 2

1 1 2 1

2rk 2rk n rk n



         

2 2 2 2 3

0 2 0 1 1 2

3 3

0 1

3 2 2

2 4

2

qk n qk n rk

rk n rk n

       

     

  

 

 

2 3

0 1 3 2

3 2 2 2 3

6rk

   

2 2 2

2 0

kn

k

 

3 2 2 3 2 2 3 2

0 2 2 1 2 1 2 0 1

n   rk n   rk n  rk n  

3 2 2 3 2 3 2 2 0 4rk 2 2rk 1 rk 1 3p

      

0 1 0 2

2

6 = 0,

rk n rk n

G

  

  

1 1 2 0

: n n 2 n

G        

 

  2

2 2 2 2

2

3 3 2

n

pkn qk n qk n

 

2 2 2 2 2 2 2 2

1 0 1 1 0 1

2 3 2 3 2 2 3

1 2 2 1

2 2 8

qk n qk n qk n qk n

rk n rk n rk

1 0 2 1 0 2

2 2 2 2 3 2

2 1 0 2 1

3 0 2

3 2

8 4

qk n qk n rk n

rk n r

      



  

   

        

      

   

  

       

  

  

  3 2 3

3 2 2

0 2

4rk n   3

 

3 3 2 2 2 3 2

1 7rk n 2 1 2rk n 2 8rk n 0 2

          

0 2 0

3 2 3 2

0 1 2 1

3

3 2 2 3 2 2 2 2 2 3 2 2 2

2 1 2 1 1 1 1 1 2 1 0 2 1

2 2 2 2

0 2 2

3

= 0,

: 2 2 2 6 3

2 2

n rk n

rk n rk n

G _{n n pkn qk n qk n rk pkn qk n}

G

qk n qk n

   

     

                

  

  

  _{      }

   

  2 2 2 2 2 2 3 2 3 2

0 2 2 2 1 1 2

3 3 2 3 3 2 3 3 2

2 1 2 1 0 1 2 2 1 2 1

3 3 2 2 3 2 3 2 2 3 2 3 2 2

0 2 1 0 1 2 2 1 2 1

3 2 0

2 2 3 3 8

8 4 2 6 10 5

10 2 2 2

10

qk n qk n qk n rk n rk

rk rk rk n rk n rk n rk n

rk n rk n rk n rk n rk n rk n

rk n

           

               

              

 

    

     

     

 ₂

4

3 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2

2 2 1 2 0 2 1 2 2 1 2

2 3 3 2 2 3 2 3 3 2 2 3 2

2 1 0 2 2 2 2 1 1 2

3 2 2 3 2 3 3 2

2 1 2 1 0

= 0,

: 3 3 2 3 2

3 6 4 8 8 8

4 2 13 6

G

n n rk pkn pkn qk n qk n qk n

G

qk n rk n rk n rk n rk rk n rk

rk rk n rk n rk n



              

             

       



  _{      }

   

      

    3 2

2 2 1

5

2 2 2 2 2 2 2 3 3 2 3 2 3

2 1 2 2 2 1 2 2 2 1

3 2 3 2 2

2 1 2

6

3 2 2 3 3 2 2 3 2

2 2 2 2

5 = 0,

: 3 2 2 8 10 8 4

4 2 = 0,

: 4 2 6 = 0.

rk n G

pkn qk n qk n rk n rk n rk rk

G

rk n rk n G

rk pkn rk n rk n

G

  

           

   

   

 

  _{     }

   

 



  _{  }

Appendix II

0

2 2 2 2 2 2 2 3 2 2 2 2

1 1 0 1 0 0 0 1 0 1

1

2 2 2 2 2 2 2

1 1 0 1 0 1 0 1 1

2 2 2 2 2 2

1 0 1 0 1 0 1 0

2 2

: = 0,

: 2 2

2 2 3 = 0,

:

G

qk n pkn rn rn qk n qk n

G G

n n pkn pkn qk qk n

G

qk n qk n rn rn

G

rn G

              

              

         



  _{     }

    

  _{     }

   

   

      

2 2 3 2 2 2 2

1 1 1 0 1 1 0 1 0 1

2 2 2 2 2 2

1 1 0 1 0

3

2 2 3 3 2 2 2 2 2 2

1 1 1 1 0 1 1 1 0

4

2 2 3 2 2

1 1 1

2 2

3 3 = 0,

: 2 2 2

: = 0.

n n pkn pkn pkn qk

qk qk n rn

G

n rn pkn pkn qk qk n qk n

G

G _{qk pkn qk n}

G

             

      

           

  

     

  



  _{      }

    

  _{ }

   