Solutions of some nonlinear equations by a direct algebraic method

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Selçuk J. Appl. Math. Selçuk Journal of Vol. 12. No. 1. pp. 19-30, 2011 Applied Mathematics

Solutions of Some Nonlinear Equations by a Direct Algebraic Method Ibrahim E.Inan1_{, Yavuz Ugurlu}2_{, Serbay Duran}3

1_{Firat University, Faculty of Education, 23119 Elazı˘}_{g, Turkiye}

e-mail: ieinan@ yaho o.com

2_{Firat University, Department of Mathematics, 23119 Elazı˘}_{g, Turkiye}

e-mail: yavuzugurlu@ mynet.com

3_{Adıyaman University, Faculty of Education, Adıyaman, Turkiye}

e-mail: serbayduran@ hotm ail.com

Received Date: December 7, 2009 Accepted Date: May 26, 2011

Abstract. In this paper, we present a direct algebraic method which is sug-gested by H. Zhang for the exact solutions of the Benjamin-Bona-Mahony (BBM), (3+1) dimensional Kadomtsev—Petviashvili (KP) and Caudrey-Dodd-Gibbon Kawada (CDGK) equation.

Key words: The Benjamin-Bona-Mahony; (3+1) Dimensional Kadomtsev-Petviashvili Equation; Caudrey-Dodd-Gibbon Kawada Equation; A Direct Al-gebraic Method.

2000 Mathematics Subject Classification: 35E05, 11Y35, 34A34, 83C15. 1.Introduction

Nonlinear phenomena play a crucial role in applied mathematics and physics. Calculating exact and numerical solutions, in particular, traveling wave solu-tions, of nonlinear equations in mathematical physics play an important role in soliton theory [1, 2]. Many explicit exact methods have been introduced in literature [3-16]. Some of them are: Bäcklund transformation, General-ized Miura Transformation, Darboux transformation, Cole—Hopf transforma-tion, tanh method, sine—cosine method, Painleve method, homogeneous balance method, similarity reduction method and so on.

In this study, we implemented a direct algebraic method [17] for finding the exact solutions of BBM equation, (3+1) dimensional KP equation and CDGK equa-tion. The BBM equation was first put forward as a model for small-amplitude long waves on the surface of water in a channel by Peregrine [18, 19] and later by Benjamin et al. [20]. In physical situations such as unidirectional waves

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propagating in a water channel, long-crested waves in near-shore zones, and many others, the BBM equation serves an alternative model to the Korteweg-de Vries (KdV) equation [21, 22]. In applied mathematics, the KP equation [23] is a nonlinear partial diﬀerential equation. It is also sometimes called the Kadomtsev-Petviashvili-Boussinesq equation. The KP equation is usually writ-ten as

(− 6+ )+ 3 2_

 = 0

where  = ∓1. This equation is implemented to describe slowly varying non-linear waves in a dispersive medium [24]. The above written form shows that the KP equation is a generalization of the KdV equation which is like the KdV equation; the KP equation is completely integrable.The KP equation was first discovered in 1970 by Kadomtsev and Petviashvili when they relaxed the restric-tion that the waves be strictly one-dimensional. In this study, we considered (3+1) dimensional KP equation as an Example 2 in section 2.

2. An Analysis of the Method and applications

Before starting to give a detail of the method, we will give a simple description of a direct algebraic method [17]. For doing this, one can consider in a two variables general form of nonlinear PDE

(1)  (    ) = 0 and transform Eq. (1) with

 ( ) =  ()   =  + 

where  and  are arbitrary constants, respectively. After the transformation, we get a nonlinear ODE for  ()

(2) 0³0 00 000 ´= 0

Then, the solution of the Eq. (2) we are looking for is expressed in the form as a (3)  ( ) =  () =  X =0 

 is a positive integer that can be determined by balancing the highest order derivative and with the highest nonlinear terms in equation, and   0 1   are parameters to be determined. Substituting solution (3) into Eq. (2) yields a set of algebraic equations for 0 _{or }_{, then, all coeﬃcients of }0

 _or  _{have to vanish.} _{After this separated algebraic equation, we can found}   0 1   constants.

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In this work, we aim to obtain solution of the Benjamin-Bona-Mahony (BBM), (3+1) dimensional Kadomtsev—Petviashvili (KP) and Caudrey-Dodd-Gibbon Kawada (CDGK) equation by using the direct algebraic method which is intro-duced by Zhang [17]. The author is using the following sub-equation

(4) ³0´

2

= 33+ 22

where 0 = _ and 2 3are constants. In his work, he has given several cases to get the solutions of Eq. (4).

Example 1. In the first example, we consider BBM equation [1, 2]

(5) + + − = 0

Let us consider the traveling wave solutions  ( ) =  ()   =  +  then Eq. (5) becomes

(6) ( + )  + 

2 2

− 200= 0

where integration constant is taken as zero. When balancing 2 with 00 then gives  = 1. Therefore, we may choose

(7)  () = 0+ 1 ()

Substituting (7) into Eq. (6) yields a set of algebraic equations for 0 1   2 3. These systems are finding as

0 + 2 0 2 + 0 = 0 (8) 1 + 01 + 1 − 122 = 0 2 1 2 − 3123 2 = 0

from the solutions of the system, we can found Case 1: (9) 0= 0 1= 32_ 3 −1 + 2_ 2   =  −1 + 2_ 2  _{− 1 + }226= 0

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with the aid of Mathematica. Substituting (7) and (9) into (4) we have obtained the following solution of Eq. (5). This solution is

Solution 1: (10)  ( ) = 32_ 3 −1 + 2_ 2 1 √_ 2_{+ } 2− √_ 2₋ 3 22 where 2 0 161222= 23  =  +_−1+2_ 2. Solution 2: (11)  ( ) = 32_ 3 −1 + 2_ 2 1 (√−2) + 2Si  (√−2) − 3 22 where 2 0 422 ¡ 2 1+ 22 ¢ = 2 3  =  +_−1+2_ 2. Solution 3: (12)  ( ) = 32_ 3 −1 + 2_ 2 3 4 2_{+ } 1 + 2 where 2 1= 32  =  + _−1+2_ 2 when 2= 0. (13)  ( 0) = −3 2_ 3 3 4 2 + 1 + 2 where 2 1= 32  =  when  = 0. Case 2: (14) 0= −2 2_ 2 1 + 2_ 2  1= −3 2_ 3 1 + 2_ 2   = − 1 + 2_ 2  6= 0 1 +  2_ 26= 0 If these values put into Eq. (7), following solutions of the Eq. (5) can be obtained;

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Solution 1: (15)  ( ) = −2 2_ 2 1 + 2_ 2− 32_ 3 1 + 2_ 2 1 √_ 2_{+ } 2− √_ 2₋ 3 22 where 2 0 161222= 23  =  −1+2_₂. Solution 2: (16)  ( ) = −2 2_ 2 1 + 2_ 2 − 323 1 + 2_ 2 1 (√−2) + 2Si  (√−2) − 3 22 where 2 0 422 ¡ 2 1+ 22 ¢ = 2 3  =  −1+2_₂. Solution 3: (17)  ( ) = 32_ 3 1 + 2_ 2 3 4 2 + 1 + 2 where 2 1= 32  =  − _1+2_₂ when 2= 0.

Example 2. Let’s consider as a second example, (3+1) dimensional KP equa-tion [1, 2]

(18) + 6 ()2+ 6− − −  = 0

Let us consider the traveling wave solutions  ( ) =  ()   = +++ then Eq. (18) becomes

(19) ¡_{ − }2_{− }2¢0+ 620_{− }4000 = 0

where integration constant is taken as zero. When balancing 0 with 000then gives  = 1. Therefore, we may choose

(20)  () = 0+ 1 ()

Substituting (20) into Eq. (19) yields a set of algebraic equations for 0 1      2 3. These systems are finding as

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−3134+ 6212= 0 from the solutions of the system, we can found

(22) 0=

24−  + 2+ 2 62  1=

32

2   6= 0

with the aid of Mathematica. Substituting (20) and (22) into (4) we have obtained the following solution of equation (18). This solution is

Solution 1: (23)  ( ) =2 4_{−  + }2 + 2 62 + 32 2 1 √_ 2_{+ } 2− √_ 2₋ 3 22 where 2 0 161222= 23  =  +  +  + . Solution 2: (24)  ( ) = 2 4_{−  + }2_{+ }2 62 + 32 2 1 (√−2) + 2Si  (√−2) − 3 22 where 2 0 422 ¡ 21+ 22 ¢ = 23  =  +  +  + . Solution 3: (25)  ( ) = − +  2 + 2 62 + 32 2 3 4 2 + 1 + 2 where 2 1= 32  =  +  +  +  when 2= 0. Example 3. We considered CDGK equation [25]

(26) + + 30+ 30+ 1802= 0

Let us consider the traveling wave solutions  ( ) =  ()   =  +  then Eq. (26) becomes

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where integration constant is taken as zero. When balancing 00 with (4)then gives  = 1. Therefore, we may choose

(28)  () = 0+ 1 ()

Substituting (28) into Eq. (27) yields a set of algebraic equations for 0 1   2 3. These systems are finding as

6030 + 0 = 0 1802₀1 + 300132+ 1522+ 1 = 0 (29) 180021 + 302132+ 450133+ 15 21 5_ 23= 0 603₁ + 452₁33+ 15 21 5_2 3

from the solutions of the system, we can found Case 1: (30) 0= 0 1= − 2_ 3 4   = − 5_2 2  6= 0

Solution 1: (31) _{ ( ) = −} 23 4 1 √_ 2_{+ } 2− √_ 2₋ 3 22 where 2 0 161222= 23  =  − 522. Solution 2: (32) _{ ( ) = −} 2_ 3 4 1 (√−2) + 2Si  (√−2) − 3 22 where 2 0 422 ¡ 2 1+ 22 ¢ = 2 3  =  − 522.

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Solution 3: (33) _{ ( ) = −} 2_ 3 4 3 4 2_{+ } 1 + 2 where 2 1= 32  =  − 522 when 2= 0. Case 2: (34) 0= 1 120 ¡ −152_ 2− √ 1052_ 2 ¢  1= − 2_ 3 4   = 1 8 ¡ −115_2 2− √ 1055_2 2 ¢   6= 0

Solution 1: (35)  ( ) = 1 120 ³ −1522− √ 10522 ´ − 2_ 3 4 1 √_ 2_{+ } 2− √_ 2₋ 3 22 where 2 0 161222= 23  =  +18 ¡ −115_2 2− √ 1055_2 2 ¢ . Solution 2: (36)  ( ) = 1 120 ³ −1522− √ 10522 ´ − 2_ 3 4 1 (√−2) + 2Si  (√−2) − 3 22 where 2 0 422 ¡ 2 1+ 22 ¢ = 2 3  =  +18 ¡ −115_2 2− √ 1055_2 2 ¢ . Solution 3: (37) _{ ( ) = −} 2_ 3 4 3 4 2 + 1 + 2 where 2 1= 32  =  + 1₈ ¡ −115_2 2− √ 1055_2 2 ¢  when 2= 0.

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3. Conclusion

In this paper, we present a direct algebraic method by using Eq. (4) and with aid of Mathematica [26], implement it in a computer algebraic system. An im-plementation of the method is given by applying it third order BBM equation, (3+1) dimensional Kadomtsev—Petviashvili (KP) and Caudrey-Dodd-Gibbon Kawada (CDGK) equation. The method can be used to many other nonlinear evolution equations or coupled ones. In addition, this method is also computeri-zable, which allows us to perform complicated and tedious algebraic calculation on a computer by the help of symbolic programs such as Mathematica, Maple, Matlab, and so on.

4. Appendix Example 1: 0_{[] =}q_₃_{∗  []}3 + 2∗  []2;  [] = 0+ 1∗  [] ; Expand ∙ ( + ) ∗  [] +₂∗  []2 −2_{∗  ∗  [ [ []  ]  ]}¤ 0+ 0+ 2 0 2 +  [] 1+  [] 1 + [] 01+ 1 2 [] 2 2 1 −2_{ [] } 12− 3 2 2_{ []}2 13 Reduce ∙½ 0+ 0+ 2 0 2 == 0  1+ 1+ 01− 212== 0 1 2 2 1− 3 2 2_ 13== 0 ¾  {0 1 } ¸ 0 = 0 1= 32_ 3 −1 + 2_ 2   =  −1 + 2_ 2  _{− 1 + }226= 0 0 = −2 2_ 2 1 + 2_ 2  1= −3 2_ 3 1 + 2_ 2   = − 1 + 2_ 2  6= 0 1 +  2_ 26= 0

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Example 2: Expand ∙_¡  ∗  − 2− 2¢∗ µ 1 q  []22+  []33 ¶ + 6 ∗ 2_{∗ (} 0+ 1∗  []) ∗ µ 1 q  []22+  []33 ¶ −4_∗ µ 12 q  []22+  []33 +3 [] 13 q  []22+  []33 ¶¸ −21 q  []22+  []33 −2_ 1 q  []22+  []33 +1 q  []22+  []33 +6201 q  []22+  []33 +62_{ [] }2 1 q  []22+  []33− 412 q  []22+  []33 −34_{ [] } 13 q  []22+  []33 Reduce £©_−21− 21+ 1 +62_ 01− 412== 0 62_2 1− 3413== 0ª {0 1}] 0= 24−  + 2+ 2 62  1= 32 2   6= 0 Example 3: Expand _{[ ∗ (}0+ 1∗  []) + 5_∗ µ  [] 122+ 15 2 [] 2 123 15 2 [] 3 123 ¶ + 30 ∗ 3_{∗ (} 0+ 1∗  []) ∗ µ  [] 12+ 3 2 [] 2 13 ¶ +60 ∗  ∗ (0+ 1∗  [])3 i

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0+ 6030+  [] 1+ 180 [] 201 +180 []2021+ 60 [] 3 3 1 +303_{ [] } 012+ 3030 []2212 +5_{ [] } 122+ 453 [] 2 013 +453_{ []}3 2 13+ 15 2 5_{ []}2 123 +15 2 5_{ []}3 123 Reduce £©0+ 6030== 0 1 +1802 01+ 303012 +5_ 122== 0 180021+ 303_2 12+ 453013 +15 2 5_ 123== 0 6031 +453_2 13+ 15 2 5_ 123== 0 ¾  {0 1 }] 0 = 0 1= − 2_ 3 4   = − 5_2 2  6= 0 0 = 1 120 ³ −1522− √ 10522 ´  1= − 2_ 3 4   = 1 8 ³ −11522− √ 105522 ´   6= 0 5. References

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