EXACT SOLUTIONS FOR NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS BY USING COSINE-FUNCTION METHOD Mahmoud M. El-Borai

(1)

361

EXACT SOLUTIONS FOR NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS BY USING COSINE-FUNCTION METHOD

Mahmoud M. El-Borai¹, Afaf A. Zaghrout ²& Amal M. Elshaer³

Department of Mathematics, Faculty of science, Alexandria University, Egypt

1

2Department of Mathematics, Faculty of science(girls), El- Azhar University, Egypt

3 Department of Mathematics, Faculty of science(girls), El- Azhar University, Egypt Email: [email protected]; [email protected] ; [email protected]

ABSTRACT

In this paper, we establish exact solutions for nonlinear partial differential equations. The cosine method is used to construct periodic and solitary wave solutions.

Keywords: Nonlinear PDEs, Exact Solutions, Nonlinear Waves, Cosine-function method.

1. INTRODUCTION

The study of numerical methods for the solution of nonlinear partial differential equations has enjoyed an intense period of activity over the last 40 years from both theoretical and practical points of view. Improvements in numerical techniques, together with the rapid advances in computer technology, have meant that many of the partial differential equations arising from engineering and scientific applications, which were previously intractable, can now, be routinely solved [1]. Finite difference methods approximate the differential operators and hence, solve the difference equations.

In finite element method the continuous domain is represented as a collection of a finite number N of subdomains known as elements. The collection of elements is called the finite element mesh. For time dependent problems, the differential equations are approximated by the finite element method to obtain a set of ordinary differential equations in time. These differential equations are solved approximately by finite difference methods or other methods. In all finite difference and finite elements it is necessary to have a boundary and initial conditions. But the Adomian decomposition method, which has been developed by George Adomian (Solving Frontier Problem of Physics: The Decomposition Method, Kluwer Academic Publishers, Boston, MA, 1994) depends only on the initial conditions to obtain solution in series form which almost converges to the exact solutions of the problem. In recently years, some other ansatz methods have been developed, such as, the tanh method [2,3], the extended tanh-function method [4,5], the modified extended tanh- function method [6–11], variational iteration method [12–17], the generalized hyperbolic-function [18,19], the separation of variables method [20,21] and the sine–cosine method [22,23].

The sine-cosine method (see e.g. [1-4]) has been used to to solve different types of nonlinear systems of PDEs.

The higher-dimensional nonlinear wave fields have richer phenomena than one-dimensional ones, since various localized solitons may be considered in

2. THE COSINE FUNCTION METHOD

Consider the nonlinear partial differential equation in the form

F  u ^, u t ^, u x ^, u y ^, u tt , u xx , u xy ^, u yy ^,... 

, (1) where u(x, y, t) is a traveling wave solution of nonlinear partial differential equation Eq. (1). We use the

transformations,

u



x,y,t



 f

  

, (2) where

  x  y  ct .

This enables us to use the following changes:

         

.

 

., ...

y ,

. x .

, .

.   d

d d

c d

t 



 



 

 

 (3)

Using Eq. (3) to transfer the nonlinear partial differential equation Eq. (1) to nonlinear ordinary differential equation

Q  f ^, f ^' ^, f ^' ^' ^, f ^' ^' ^'

,...

  ⁰ ^.

(4) The ordinary differential equation (4) is then integrated as long as all terms contain derivatives, where we neglect the integration constants. The solutions of many nonlinear equations can be expressed in the form

(2)

362

   





 



otherwise.

0

2 ,

cos

   

 

^

f (5)

where

 , 

^and



are parameters to be determined, μ and c are the wave number and the wave speed, respectively [2]. We use

     cos

^

   , f

       

   

cos

  

1



cos

  

1



cos

 

. ,

sin cos

2 2

2 ' 2

'

1 '















 

 



 



















d f f d

d

f df

(6)

Substituting Eq. (6) into the nonlinear ordinary differential equation Eq. (4) gives a trigonometric equation of

  

cos terms. To determine the parameters first balancing the exponents of each pair of cosine to determine α.

Then we collect all terms with the same power in cos^

  

and put to zero their coefficients to get a system of algebraic equations among the unknowns β, λ and μ. Now, the problem is reduced to a system of algebraic equations that can be solved to obtain the unknown parameters β, λ and μ. Hence, the solution considered in Eq. (5) is obtained. The above analysis yields the following theorem:

Theorem. The exact analytical solution of the nonlinear partial differential equation (1) can be determined in the form Eq. (5) where all constants found from the algebraic equations after its solutions.

3. APPLICATIONS

In order to illustrate the effectiveness of the proposed method eight examples in mathematical are chosen as follows 3.1. Test problem

3.1.1. The nonlinear partial differential equations

Consider the following problem: Find functions

u ,   x t

satisfying the equation in the form,

 ^ ^

p x

^

xxx



x

^

xxxt

^ ⁰ ^.

x

t

u u u u u

u   

(7)

By using the wave variable

  x  ct

^and

u   x , t  f   

then the equation (7) becomes

  ¹ ^ ^c  ^f

^'

^ ^ ^f

^p

^f

^'

^ ^ ^f

^'^''

  ^ ^c ^ ^f

^''^''

^ ⁰ ^.

(8) by twice time integrating (8) with respect to



and setting the constant of integration equal to zero, we find

   

0.

1 1 ¹  ^'^' 

 

 f ^ c f

f p

c



^p

 

(9) Seeking solutions of the form (5), we get

    cos ^ ^     cos    1 

cos 1

1

¹ ¹



²

 

²



 

   

^ ^

       



^ ^ c ^

c p ^p ^p

   c   cos

^^²

    

²

    1    c   cos

^

    0 .

(10) Equating the exponents and the coefficients of each pair of the cosine functions we find the following algebraic system:

 

 1     1   0 ,

2 1

2

    







v c v

c c

p



















1

 

0. 1

2

1   



    

 

p c

p (11)

(3)

363 By solving the algebraic system (11), we get,

¹ _, ₂



¹

 

³ ²



_,

2 ip

2,

1 1 2

p c p p p

c c

p 









   

 

 

  ^

 



 

  (12)

Then by substituting Eq. (12) into Eq. (5) then, the exact soliton solution of equation (7) can be written in the form:

      _ _

 

 







 







 



   



^ ^

1 x - ct

cos 2 2

3 2 1

,

²

2 1 1







c

c ip p

p t c

x

u ^p

p



(13)

3.1.2. The nonlinear partial differential equations The traveling solution for equation takes the form

( )14

  ^u ^o ^.

k bu u u a

u

_t



^p _x



_xxt



^m _x

 ct

x 

 

^and

u   x , t  f   

By using the wave variable

then the equation (14) becomes

)15)

  ^' ⁰ ^.

'' ' '

'    

 cf af

^p

f bcf k f

^m

one time, and let the integral constant is zero, we have



Integrate (15) with respect to

) 16)

Substituting Eq.(5)into Eq (16), the expansion is obtained as follows

 

^ ^

 

 1  cos    1  cos   cos   0 . cos

) ( 1 cos

cos

2 2

2

2 1

1









 









































m m p

p

k bc

bc

p bc c a

)17) Equating the exponents and the coefficients of each pair of the cosine functions and

m  p  1 .

we find the following algebraic system:

   



1



.

0 1

, 1 bc

- -c 0 , 2 1

1 2

1

2 2



   

 











p

p bc k

p a

bc m

p





















(18)

By solving the algebraic system (18), we get,

_, ₂



³ ²



_.

b 2 ip

2, _-₁_p ² ¹^p

k pk a

p p

c

p 

 







 

 



 



 (19)

substituting Eq. (19) into Eq. (5) we obtain the exact soliton solution of the Eq. (14) in the form

    _ _ _.

cos 2 2

2 3 ,

1 2 2

1





 



 

 

 









^



^

x ct

b ip k

pk a

p p

t c x

u

^p

p



(20)

where a is constant

3.1.3. The generalized coupled Hirota-Satsuma KdV system

Consider the following problem: Find functions

u   x , t and    x, t

are satisfying the generalized coupled Hirota-Satsuma KdV system

 

  , 3   . , 6 3

,

x xxx

t

x x

xxx t

uv bv

t x v

vv uu

au t x u











(21) where a ,b are real parameters.

  ⁰ ^.

1 ''

1

  

 



^p^

bcf k f

^m

p

cf af

(4)

364

Using the wave variable

u   x , t  f        , v x , t  g  . where   x - ct,

the system (21) is carried to a system of ODEs

 

  ' ' ' 3   '.

'

' 6 ' 3 ' ' ' '

g f g

b g

c

g g f f f

a f

c















(22)

Integrating (22) with respect to



 

  ' ' 3   . 2 3

' 3

'

² ²

g f g

b g

c

g f

f a f

c















(23)

The solutions of (23) can be expressed in the form

    

₁

cos

^¹

   , g     

₂

cos

^²

  

f

(24)

Substitution Eq. (24) into Eq. (23), the expansion is obtained as follows

) 25)

From Eq. (25), we can get following equations:

, 2 2

2 ,

2 ₁ ₂ ₁ ₁ ₂

2

    



     

 1  , 0  c 

₁

 a 

₁



₁



²

 a 

₁



²



₁



₁



` (27)

 

 1  3 ,

0 , 1 0

, 2 3

1 3 0

2 1 2

2 2 2

2 2 2 2 2

2 2 2

2 2 2 1 1

1 2 1













































b

b b

c a

Solving these algebraic equations. we obtain,

(28)

Combining (28) with (24), we obtain the exact solution to system (21) and then the exact solution to The generalized coupled Hirota-Satsuma KdV system can be written as

   

 

.

2 1 cos 2 3 t 2

x,

,

2 1 cos 2 ,

2 2



 



 







 



 





ct a x

c c

ct a x

c t c

x u



(29)

3.1.4. The generalized coupled Drinfeld –sokolov –wilso system

In this case we consider the Drinfeld –sokolov –wilso equations, in the normalized form

       

 1  cos   3 cos   cos   .

cos 1 cos

cos 0

, cos

3 2 cos

cos 3 1

cos 1 cos

cos 0

2 1

2

2 2

2

2 1

1

1 1

1

2 1 2

2 2 2

2 2

2 2 2 2

2 2 2

2 2 2 2

2 1 1

1 2 1

2 1

1 2 1 2

1 1 1































































































b

b b

c a

a a

c

2 3 2

2 , 2 ,

1 , - b ,

2

₁ ₂

2 1

c c

a

 

c

  







    



(5)

365

  ^.

3 , 3

xxx x t

x t

uw a w

w

w w u





(30)

By making the transformation the

u   x , t  f        , w x , t  g  . where   x - ct,

 

'.

' ' ' 3 '

, ' 3 '

g f a g cg

g g f

c









 (31)

,then we have



Integrating Equation (31) once with respect to

)32) g

f a g g c

c g f

' ' 3 0

2 , 3

₂





 

Substitution first equation into second Eq. (32), the expansion is obtained as follows

(33)

Substituting Eq. (5) into (33) gives:

       

         

























3 3 2

2 2

2

2 cos cos 3

1 3

cos 1 3

cos 3

cos 0

c a c







 ^

(34)

Equating the exponents and the coefficients of each pair of the cos -functions we find the following algebraic system:

(35)

By solving the algebraic system (35), we get,

(36)

.

3 2

3, c

,

1 a

 c











 



So we obtain the exact solution of the Drinfeld –sokolov –wilso equation in the form

)37)

   

    .

cos 3 , 2

3 , 3 cos

, 2

2 1

 

 



  



 

 



  







ct c x

a t c

x u

ct c x

a t c

x w



3.1.5. The nonlinear partial differential equations

Suppose we are given nonlinear partial differential equations in the form

)38)

 

  0 .

, 0









xy x t

x xxxy y x

yt

u uv v

bu uu

v u

By using the

u   x , t  f        , v x , t  g  . where   x  y - ct,

3

2 ' 3 ' 3

0

g

c g a

g

c

 



 

³

2

2 2

2 1 3 3

0

1 3

3 0

, 3 2























c a c











(6)

366

( )39

 

 ' '  ' 0 . '

, 0 ' ' ' ' ' ' ' ' '













f g f cg

bf f f g

cf

by integrating (39) with respect to



(40)

 



''



0. , 0 ' ' ' '

' '













f g f cg

bf f f g cf

    

₁

cos

^¹

   , g     

₂

cos

^²

  

f

(41)

           

          





    

 

















































sin cos

1

sin cos

2 1

sin cos

0

1 1

2 1 3 1

3 1

1 1 3 1 1

3 2 1 1

1 2 1 2 1 1

2 2 1

1 1

1

1 1

1 2

1











 b

b b

c

(42)

From Eq. (42), we can get following Equations:

(43)

Solving the algebraic equations above, yields

)44)

By using above, expression (41) can be written as

  ,

cos 8 2

3

2

 

 



   



^

x y ct

b i u b

(45)

Using the transformation:

   

  ^ix ^sec ^, ^sec   ^ix ^sec ^.

sech

, cos ix

cosh ,

cosh cos

x h x

x x

ix



Then Eq. (45) become:

  

 



1



0

, 1 0

, 2 1

0

, 2

, 1 2 3

1 1 2 1 2 1 1 2

1 1 2 1 2 1

1 2 1 3 1 3 2 1 1 2

2 1

1

1 1

1 3 1 1

2 1

1 2 1

1 1



























































































c

b b

c

b

  .

cos 8 4

3

₂

2





 



   



^

x y ct

b i v b

     

 ¹  ^cos    ¹  ^cos   ^.

cos cos

cos 0

1 1

1 2

1 1 2 1 2

1 1 2 1

2 1 1 2

1 2

















































c 

b . c 1 , 2

8 , b i

4 , 3 2 ,

3

2 2 1

2

1

           

 b b

(7)

367

 

  .

8 ech 1

4 3

, 8

ech 1 2

3

2 2

2

 

 



  



 

 



  



ct y b x

b s v

ct y b x

b s u

( )45 3.1.6. The nonlinear partial differential equations

To find the traveling solution for equation take the form

)48)

 ^  ^ ^ ⁰ ^.



_xxx ^p _x _x _yy

xt

u bu u au

u

In the same manner to exact soliton solution of this soliton equation can be deduced to the form:

(49)

     





 



  



 



   

 c x y ct

b p p

c

u a c-a

2 os ip

2

1 -2p

p

1

where a,b are constant.

3.1.7. The kadomtsev-petviashvili equation

In a similar way, when we consider the general improved The kadomtsev-petviashvili equation in the form (50)

 ^

x

^

xxx



x

^

yy

^ ⁰ ^.

p

t

u u u u

u  

The exact soliton solution of this soliton equation can be deduced to the form:

      _ _

_.

cos 2 2

2 3 ,

, ²

2 1

1 

 



   



 



   

 ^ c p p ^ ip c x y ct

t y x

u ^p

p







(51) 3.1.8. The nonlinear partial differential equations

To find the traveling solution for equation take the form

(52)

     

⁰

2 1 2

1

, 0













xxy yyy

xyy xxx

y x

t

yyy y

x t

xxx y

x t

v v

u u

L vw uw

w

w vv uv v

w vu uu

u



x

^,

y

^,

t

 

f

    ^,

v x

^,

y

^,

t

 

g

   ^, ^w  ^x, ^y, ^t  

q

   ^,

, ct y x  

 

where

The system (52) is carried to a system of ODEs

(53)

    ^'  ² ^' ^' ^' ² ^' ^' ^'  ⁰

2 ' 1 2

' 1

, 0 ' ' ' ' ' '



















g f

L f g q

f q c

q g g g f g c

q f g f f f c

  





₁cos^¹

  

, g

  





₂cos^²

  

, q

  





₃cos^³

  

.

f (54)

       

      

        

















































 



sin cos

1 sin cos

2 1

sin cos

0

1 3

2 3 3 3

3 3

3 3 3 3

1 3

2 3 3 1

1 2 1

2 1 1

1 1

3

3 2

1

1 1

















c

(8)

368

       

               

























sin 2 cos

sin 1 2 cos

1

sin 2 cos

sin 1 cos

0

1 2

3 2 1

1 3 1

1 3

3 1 1

3 3

3 2 3

1

3 1 3















c

(55)

From Eq. (55), we can get following Equations :

 

  

 

 1  2 

0 , 1 0

, 2 1

0 , 1 0

, 3 3

1 1

, 3 1

2 ,

3 1

2

3 3

3 3 3 2

2 2 2

2 1

3 2 3 3 3 3 2 3 3 2

2

3 3

3 3 3 1

2 1 1 2 1

3 2 3 3 3 3 2 3 3 1

1

2 1

3 2 3

1

3 2

3 1



























































































































c c

     

 1  2 2  1  .

2 2

0 , 2 1

2 2 1

2 2 1 2

1 2

0 1

2 2 2 3 2 3

2 2 2 1

2 1 3 1 3

2 1 1 3

3

2 2

2 3 2 1

1 1 3 1

3 3 2 2

3 2 1

3 1 3

3 1

















































































L L

c

L L

(56) Solving the algebraic equations above, yields two case

Case 1-

, 2 ,

2 c 2

, 3 2 , 3

3 2 2 1

3 1 2

1

             

 L

L i c

c

(57) Substituting Eq. (57) into Eq. (54), we get kinks solutions for Eq. (52) of the form

 

  .

2 2 sec 1 3

2 , 2 sech 1 2

3 2 , 2 sech 1 2

3

2 1 2

 

 



  



 

 



  



 

 



  



ct y L x

h c L c w

ct y L x

c c v

ct y L x

c c u

(58)

       

      

        

















































sin cos

1 sin cos

2 1

sin cos

0

1 3

2 3 3 3

3 3

3 3 3 3

1 3

2 3 3 1

2 2

1 2

2 1 1

2 2

3

3 2

2 1 2

















c

       

          

        

                 

















































sin cos

1 2

sin cos

2 1

2 sin

cos 2

sin

cos 1 2

sin cos

2 1

2

sin cos

2 sin

2 cos 1

1 2

2 2 3 2 3

2 2

2 3 2 1

3 2 2 2

1 1

2 1 3 1 3

1 1

1 3 1

1 3

2 1 1 1

3 3 2

2 2

2

1 1

1 3

2























L

L L

L

(9)

369

Case 2- , 2.

2 c 2 1 , 3

2 , 3

3 2 2 1

3 1 2

1



























c c L L (59)

Substituting Eq. (59) into Eq. (54), we get kinks solutions for Eq. (52) of the form

(60)

4. CONCLUSION

In this Letter, the cosine-function method has been successfully applied to find the solution for eight nonlinear partial differential equations. The cosine-function method is used to find a new exact solution. Thus, we can say that the proposed method can be extended to solve the problems of nonlinear partial differential equations which arising in the theory of solitons and other areas.

ACKNOWLEDGEMENT

The authors would like to thank the referees for their comments.

5. REFERENCES

[1]. A. R. Mitchell and D. F. Griffiths, The Finite Difference Method in Partial Differential Equations, John Wiley & Sons, 1980.

[2]. E. J. Parkes and B. R. Duffy, An automated tanh-function method for finding solitary wave solutions to nonlinear evolution equations, Comput. Phys. Commun. 98 (1998), 288-300.

[3]. A. H. Khater, W. Malfiet, D. K. Callebaut and E. S. Kamel, The tanh-function method, a simple transformation and exact analytical solutions for nonlinear reaction-diffusion equations, Chaos, Solitons &

Fractals 14 (2002), 513-522.

[4]. E. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A 277(4-5) (2000), 212-218.

[5]. E. Fan, Travelling wave solutions for generalized Hirota-Satsuma coupled KdV systems, Z Naturforsch A 56 (2001), 312-318.

[6]. A. M.Wazwaz, A sine-cosine method for handing nonlinear wave equations.,Math. Comput. Model., 40(2004), 499–508.

[7]. A. M. Wazwaz, The tanh and the sine-cosine methods for the complex modified KdV and the generalized KdV equations, Comput. Math. Appl. 49(7-8)(2005), 1101–1112.

[8]. H. A. Zedan, Blow-up the symmetry analysis for the Hirota-Satsuma equations. Izv. Nats. Akad. Nauk Armenii Mat. 43 (2008), no. 2, 53–65; translation in J. Contemp. Math. Anal., 43(2)(2008), 108–117.

[9]. S.A. EL-Wakil, M.A. Abdou, New exact traveling wave solution for two nonlinear physical models, Nonlinear Anal., in press.

[10]. H. Z. Zedan, S. J. Monaquel The Sine-Cosine Method For The Davey-Stewartson Equations Appl. Math.

E-Notes, 10(2010), 103-111

[11]. E. Yusufo˘glu, A. Bekir and M. Alp, Periodic and solitary wave solutions of Kawa- hara and modified Kawahara equations by using sine-cosine method. Chaos Soli-tons Fractals, 37(4)(2008), 1193–1197.

[12]. H-T. Chen, H-Q. Zhang, New double periodic and multiple soliton solutions of the generalized (2+1)- dimensional Boussinesq equation, Chaos. Soliton. Fract. 20 (2004) 765-769.

[13]. A.H.A. Ali, A.A. Soliman, K.R. Raslan, Soliton solution for nonlinear partial differential equations by cosine-function method, Physics Letters A 368 (2007) 299–304

[14]. A.A. Soliman, M.A. Abdou, Numerical solutions of nonlinear evolution equations using variational iteration method, J. Comput. Appl. Math. (2007), in press.

[15]. M.A. Abdou, On the variational iteration method, Phys. Lett. A (2007), in press.

 

^.

2 2 sec 1 3

2 , 2 sec 1 2 3

2 1 2



 



  





 



  





 



  



ct y L x

L c c w

ct y L x

c c v

ct y L x

c c u

EXACT SOLUTIONS FOR NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS BY USING COSINE-FUNCTION METHOD Mahmoud M. El-Borai