กรุณาใช้ไฟล์นี้ในการแก้ไข
- ปรับให้เป็น 1 คอลัมน์
- การอ้างอิงเอกสารในบทความใช้ระบบตัวอักษร ไม่ใช้ตัวเลข (ดูใน NOTE FOR CONTRIBUTORS)
- ระบุหมายเลขบรรทัด
26/01/11
Supaporn Suksern*
ABSTRACT
Some solutions for one class of nonlinear fourth-order partial differential equations u tt
=

  u
2

  uu xxxx
  u xxtt
  u u x xxx
  u
2 xx where , , , , xx
    
and
 are arbitrary constants are presented in the paper. This equation may be thought of as a fourth-order analogue of a generalization of the Camassa-Holm equation, about which there has been considerable recent interest. Furthermore, this equation is a
Boussinesq-type equation which arises as a model of vibrations of harmonic mass-spring chain. The idea of travelling wave solutions and linearization criteria for fourth-order ordinary differential equations by point transformations are applied to this problem.
Keywords : linearization problem, point transformation, nonlinear ordinary differential equation, travelling wave solution
INTRODUCTION
Almost all important governing equations in physics take the form of nonlinear differential equations, and, in general, are very difficult to solve explicitly. While solving problems related to nonlinear ordinary differential equations it is often expedient to simplify equations by a suitable change of variables.
Many methods of solving differential equations use a change of variables that transforms a given differential equation into another equation with known properties. Since the class of linear equations is considered to be the simplest class of equations, there is the problem of transforming a given differential equation into a linear equation. This problem, which is called a linearization problem. The reduction of an ordinary differential equation to a linear ordinary differential equation besides simplification allows constructing an exact solution of the original equation.
One of the most interested nonlinear problems but also difficultly in solving is the problem of nonlinear fourth-order partial differential equations where , , , , u
and tt

=

  u
2
 xx
  uu xxxx
  u xxtt
  u u x xxx
  u
2 xx
(1) are arbitrary constants. The main difficulty in solving this problem comes from the terms of nonlinear partial differential equations and the large number of order.
Because of this difficulty, there are only a few attempts to solve this problem.
In 2008, Ibragimov, Meleshko and Suksern (Ibragimov et al ., 2008)found the explicit form of the criteria for linearization of fourth-order ordinary differential equations by point transformations. Moreover, the procedure for the construction of the linearizing transformation
____________________________________________________________________
Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok, 65000, Thailand
E-mail: s_suksern@hotmail.com
1

กรุณาใช้ไฟล์นี้ในการแก้ไข
- ปรับให้เป็น 1 คอลัมน์
- การอ้างอิงเอกสารในบทความใช้ระบบตัวอักษร ไม่ใช้ตัวเลข (ดูใน NOTE FOR CONTRIBUTORS)
- ระบุหมายเลขบรรทัด
26/01/11
* Corresponding author are presented. By the virtue of (Ibragimov et al ., 2008) to bring about the idea for solving the problem of nonlinear fourth-order partial differential equations (1).
The way of solving the problem is organized as follows. Firstly, reducing the nonlinear fourth-order partial differential equations to the nonlinear fourth-order ordinary differential equations by substituting the form of travelling wave solutions. Secondly, reducing the nonlinear fourth-order ordinary differential equations to the linear fourth-order ordinary differential equations by applying the criteria for linearization in (Ibragimov et al ., 2008).
Finally, finding the exact solutions of linear equations and then substituting back to the exact solutions of the original problem.
LINEARIZATION CRITERIA FOR FOURTH-ORDER ORDINARY DIFFERENTIAL
EQUATIONS BY POINT TRANSFORMATIONS
The important tools for this research is the linearization criteria for fourth-order ordinary differential equations by point transformations. From (Ibragimov et al ., 2008) we have the following theorems.
Theorem 1 Any fourth-order ordinary differential equation y

F x y y y y
 can be reduced by a point transformation t
   
, u
   
, to the linear equation
(2) u
         
0, (3) where t and u are the independent and dependent variables, respectively, if it belongs to the class of equations
(4)
 
0
 
0

2

2 y

2

0
 y

+ D y +D y +D y +D y +D = 0,
4

4
3

3
2

2
1

0
(4) or
1 y +

2
2
1

0
1
+
( y' + r)
2

3
2

2
1

0

2
(5)
 
3

2
 
4 3 2 1 0

7

6

5
+ K y + K y + K y + K y
7 6 5 4

4 3 2
+ K y + K y + K y + K ]
3 2 1 0

0, where
A = A (x,y), B =B (x,y), C = C (x,y), D =D (x,y), r=r(x,y), F = F (x,y), H = H (x,y), j j j j j j j j j j j j j
and j
are arbitrary functions of , .
2

กรุณาใช้ไฟล์นี้ในการแก้ไข
- ปรับให้เป็น 1 คอลัมน์
- การอ้างอิงเอกสารในบทความใช้ระบบตัวอักษร ไม่ใช้ตัวเลข (ดูใน NOTE FOR CONTRIBUTORS)
- ระบุหมายเลขบรรทัด
26/01/11
Since this research deals with the first class, let us emphasize to the first class for other theorems that we need to use.
Theorem 2 Equation (4) is linearizable if and only if its coefficients obey the following conditions:
A - A = 0,
0y 1x
(6)
4B - 3A = 0,
0 1
(7)
(8)
1y 1 2
12A + 3A A - 4C = 0,
1x 0 1 1
0 0 1 0y 0x 1 1x
4C + A C - 24D = 0,
2y 1 2 4
1
4C + A C - 12D = 0,
1y 1 1 3
2
16C - 12A A - 3A A + 4A C + 8A C - 32D = 0,
1x 0x 1 0 1 0 1 1 0 2
3
192D + 36A A A - 48A C - 48C A - 288D + 9A A
2x 0x 0 1 0x 1 0x 1 1y 0 1
2
- 12A C - 36A A C + 48A D + 32C C = 0,
0 1 0 1 0 0 2 0 1
2
384D - [3((3A A - 4C )A + 16(2A D + C C ) - 16(A C - D )A )A
1xy 0 1 1 0 1 1 0 1 1 0 2 0 0
2
- 32(4(C D - 2C D + C D )+ (3A D - C )A ) - 96D A + 384D A
1 1 2 0 0 2 1 0 0 1 1y 0 0y 1
(9)
(10)
(11)
(12)
(13)
(14)
(15)
+ 1536D - 16(3A A - 4C )C + 12((3A A - 4C )A - 4(A C - 4D ))
0yy 0 1 1 0x 0 1 1 0 1 0 2
A ] = 0
0x
Theorem 3 Provided that the conditions (6)-(15) are satisfied, the linearizing transformation (2) is defined by a fourth-order ordinary differential equation for the function
  
, namely by the Riccati equation
40 d
 dx
- 20

2 = 8C - 3A -12A ,
0 0 0x
(16) for

=

 xx x
, (17) and by the following integrable system of partial differential equations for the function
  
4
 yy
=

A , y 1
4
 xy
=
 y 0

(18) and
2

กรุณาใช้ไฟล์นี้ในการแก้ไข
- ปรับให้เป็น 1 คอลัมน์
- การอ้างอิงเอกสารในบทความใช้ระบบตัวอักษร ไม่ใช้ตัวเลข (ดูใน NOTE FOR CONTRIBUTORS)
- ระบุหมายเลขบรรทัด
26/01/11
1600
 xxxx
= 9600
  xxx

2

2
+ 160 ( - 12A - 3A - 90 + 8C ) xx 0x 0 0
+ 40
 x
(12A A + 72A
0x

0x 0
3
0
2

- 12A C
0

3
  
2 2
+ 120 - 48 C + 24D - 8 ) + (144A + 72A A - 352A C
0 1 0x 0x 0 0x 0
- 160C - 80C A - 1600D +
0xx 0x 0 0y
 4 2
640D - 80 + 9A - 88A C
1x x 0 0 0
+ 160A D + 30A
0 1 0

- 400A D + 300
1 0
 
+ 144C ) + 1600
0

D , y 0
(19) where

is given by equation (17) and

is the following expression

0
3
0 0 1 0x 0x 0 0xx
.
Finally, the coefficients

and
 of the resulting linear equation (3) are

=

 x
3
,
 
4 -1 2 2
= (1600 ) ( - 144A - 72A A + 352A C + 160C + 80C A x 0x 0x 0 0x 0 0xx 0x 0
0y 1x
 x 0
4
0
2
0 0 1 0

+ 400A D - 300
1 0
 
- 144C ).
0
METHOD AND RESULT
Let us consider the nonlinear fourth-order partial differential equation (1) u tt
=

  u
2

  uu xxxx
 where , , , , xx
    
and
 are arbitrary constants.
 u xxtt
  u u x xxx
  u
2 xx
(20)
(21)
(22)
Of particular interest among solutions of equation (1) are traveling wave solutions: u(x,t)=H(x-Dt), where D is a constant phase velocity and the argument x-Dt is a phase of the wave.
Substituting the representation of a solution into equation (1), one finds

  2

H
       2

  2

   2
(23)
This is an equation of the form (4) with coefficients
A
1


H

 
D
2
, A
0

0,
C
2

C
1

D
4

D
3

0,
0,
D
2

B
0



H
 
D
2
C
0

2

H

H
 
D
2

D
2
2


H
 
D
2
,
,
D
1

D
0

,
0.
Substituting these coefficients into the linearization conditions (6)-(15), one obtains the following results.
1.1
Case 1 :
 
0
If
 
0 , then equation (23) is linearizable if and only if
3

กรุณาใช้ไฟล์นี้ในการแก้ไข
- ปรับให้เป็น 1 คอลัมน์
- การอ้างอิงเอกสารในบทความใช้ระบบตัวอักษร ไม่ใช้ตัวเลข (ดูใน NOTE FOR CONTRIBUTORS)
- ระบุหมายเลขบรรทัด
26/01/11
0.
The solution of equation (23) is
  
C
1 sin
 
D
2

D
2
 x

Dt
 
C
2 cos
 
D
2

D
2
 x

Dt

, (24) where C
1
and C
2
are arbitrary constants.
1.2

Case 2 :
If
 

0,


0

0 and
 
0 , then equation (23) is linearizable if and only if
 
0,
 
D
2
.
Next, to find the linear form of equation (23) and it's solutions. From theorem 3, we found that equation (23) is mapped by the transformations
  
Dt ,
 
H to the linear equation
 
0.
Hence, the solution of equation (1) is
  
C
0

1
 
Dt
 
C
2
 x

Dt

2 
C
3
 x

Dt

3
, (25) where C C C
0
,
1
,
2
and C
3
are arbitrary constants.

If
 
0,
 
0 and
 
3

, then equation (23) is linearizable if and only if
    
D
2
.
Next, to find the linear form of equation (23) and it's solutions. By using theorem 3, we obtained that equation (23) is mapped by the transformations
  
Dt ,
 
H
2
2

D

2

H to the linear equation
 
0.
So that we obtain the implicit solution of equation (1) in the form u
2

D
2

2
 u

C
0

1
 
Dt
 
C
2
 x

Dt

2 
C
3
 x

Dt

3
, (26) where C C C
0
,
1
,
2
and C
3
are arbitrary constants.
DISCUSSION AND CONCLUSION
In the present work, we found some solutions for nonlinear fourth-order partial differential equations (1), that are in the form of equations (24), (25) and (26), respectively.
An interesting aspect of the results in this paper is that the class of exact solutions of the original nonlinear problems, which can not find by the classical methods.
ACKNOWLEDGEMENTS
4

กรุณาใช้ไฟล์นี้ในการแก้ไข
- ปรับให้เป็น 1 คอลัมน์
- การอ้างอิงเอกสารในบทความใช้ระบบตัวอักษร ไม่ใช้ตัวเลข (ดูใน NOTE FOR CONTRIBUTORS)
- ระบุหมายเลขบรรทัด
26/01/11
This work was financially supported by Faculty of Science, Naresuan University. The author wishes to express thanks to Prof.Dr. Sergey V. Meleshko, Suranaree University of
Technology for his guidance during the work.
REFERENCES
Ibragimov, N. H., Meleshko, S. V., Suksern, S. (2008). Linearization of fourth-order ordinary differential equations by point transformations, J. Phys. A: Math. Theor. 41 (235206) 19pp.
2