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Solutions for One Class of Nonlinear Fourth-Order Partial Differential
Equations
Supaporn Suksern*
ABSTRACT
Some solutions for one class of nonlinear fourth-order partial differential equations
utt =  u + u 2  uuxxxx  uxxtt   uxuxxx   u 2 xx


xx
where  ,  ,  ,  , 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, in
which there has been considerable interest recently. 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 is 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 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 interesting nonlinear problems but also difficult to solve is the
problem of nonlinear fourth-order partial differential equations
(1)
utt =  u + u 2  uuxxxx  uxxtt   ux uxxx   u 2 xx


xx
where  ,  ,  ,  , and  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 have been 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
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* Corresponding
author
was 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)(This is not a complete
sentence! Please rewrite!)
The way to solve this 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.
(You can see that this highlighted part does not contain a single complete sentences, but a
series of clauses. Please rewrite these clauses into sentences.)
LINEARIZATION CRITERIA FOR FOURTH-ORDER ORDINARY DIFFERENTIAL
EQUATIONS BY POINT TRANSFORMATIONS
The important tool for this research is the linearization criteria for fourth-order
ordinary differential equations by point transformations. Based on Ibragimov et al (2008), we
have the following theorems.
Theorem 1
Any fourth-order ordinary differential equation
y  4  F  x, y, y, y, y 
can be reduced by a point transformation
t    x, y  , u    x, y  ,
to the linear equation
u  4    t  u    t  u  0,
(2)
(3)
where t and u are the independent and dependent variables, respectively, if it What does this
‘it’ refer to? t or u? or something else? belongs to the class of equations
y(4) +  A1y+A 0  y+B0 y2 +  C2 y2 +C1y+C0  y
(4)
+ D4 y4 +D3 y3 +D2 y2 +D1y+D0 = 0,
or
1
4
y  +
( -10 y + F2 y'2 + F1y + F0 )y'''
y + r
1
+
[ 15y3 + ( H 2 y2 + H1y+ H 0 )y2
(5)
2
( y' + r)
+ (J 4 y4 + J 3 y3 + J 2 y2 + J1y +J 0 )y
+ K 7 y7 + K 6 y6 + K 5 y5 + K 4 y4 + K 3 y3 + K 2 y2 + K1y+ K 0 ]  0,
2
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where
A j = A j (x,y), B j =B j (x,y), C j = C j (x,y), D j =D j (x,y), r=r(x,y), Fj = Fj (x,y), H j = H j (x,y),
J j = J j (x,y) and K j =K j (x,y) are arbitrary functions of x, y .
Since this research deals with the first class, let us emphasize 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 0y - A1x = 0,
(6)
(7)
4B0 - 3A1 = 0,
2
1
12A1y + 3A - 8C2 = 0,
(8)
12A1x + 3A0 A1 - 4C1 = 0,
(9)
32C0y + 12A0x A1 - 16C1x + 3A02A1 - 4A0C1 = 0,
(10)
4C2y + A1C2 - 24D 4 = 0,
(11)
4C1y + A1C1 - 12D3 = 0,
(12)
2
16C1x - 12A 0x A1 - 3A 0 A1 + 4A 0C1 + 8A1C0 - 32D 2 = 0,
192D 2x + 36A 0x A 0 A1 - 48A 0x C1 - 48C0x A1 - 288D1y + 9A 03A1
- 12A 0 2 C1 - 36A 0 A1C0 + 48A 0 D 2 + 32C0 C1 = 0,
(13)
(14)
384D1xy - [3((3A 0 A1 - 4C1 )A 0 2 + 16(2A1D1 + C 0C1 ) - 16(A1C0 - D 2 )A 0 )A 0
- 32(4(C1D1 - 2C 2 D 0 + C0 D 2 )+ (3A1D 0 - C0 2 )A1 ) - 96D1y A 0 + 384D0y A1
(15)
+ 1536D0yy - 16(3A 0 A1 - 4C1 )C 0x + 12((3A 0A1 - 4C1 )A 0 - 4(A1C0 - 4D 2 ))A 0x ] = 0
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
  x  , namely by the Riccati equation
40
d
- 20 2 = 8C0 - 3A 0 2 -12A 0x ,
dx
(16)
for
 xx
,
(17)
x
and by the following integrable system of partial differential equations for the function
  x, y 
=
4 yy =  y A1 , 4 xy =  y (A 0 + 6),
and
2
(18)
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1600 xxxx = 9600 xxx  + 160 xx ( - 12A 0x - 3A 0 2 - 90 2 + 8C0 )
+ 40 x (12A 0x A 0 + 72A 0x  - 16C0x + 3A 03 +18A 0 2  - 12A 0C0
+ 120 3 - 48 C0 + 24D1 - 8) +  (144A 0x 2 + 72A 0x A 0 2 - 352A 0x C0
(19)
- 160C0xx - 80C0x A 0 - 1600D0y + 640D1x - 80x + 9A 0 4 - 88A 0 2C0
+ 160A 0 D1 + 30A 0  - 400A1D0 + 300 + 144C0 2 ) + 1600 y D0 ,
where  is given by equation (17) and  is the following expression
 = A 03 - 4A 0 C0 + 8D1 - 8C0x + 6A 0x A 0 + 4A 0xx .
Finally, the coefficients  and  of the resulting linear equation (3) are

= 3 ,
x
(20)
(21)
 = (1600x 4 )-1 ( - 144A 0x 2 - 72A 0x A 0 2 + 352A 0x C0 + 160C0xx + 80C0x A 0
+ 1600D0y - 640D1x + 80 x - 9A 0 4 + 88A 0 2 C0 - 160A 0 D1 - 30A 0 
(22)
+ 400A1D0 - 300 - 144C0 2 ).
METHOD AND RESULT
Let us consider the nonlinear fourth-order partial differential equation (1)
utt =  u + u 2   uuxxxx  uxxtt   ux uxxx   u 2 xx
xx
where  ,  ,  ,  , and  are arbitrary constants.
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
(23)
 H+ D2  H4 +  HH+ H2 +  2 H+ -D2  H + 2 H2 =0.
This is an equation of the form (4) with coefficients
A1 

 H  D
,
A0  0, B0 

,
 H   D2
2 H    D2
C2  C1  0, C0 
,
 H   D2
2
D4  D3  0, D2 
, D1  D0  0.
 H   D2
2
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
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      0.
The solution of equation (23) is
u  x, t   C1 sin
  D2
  D2
x

Dt

C
cos


 x  Dt  ,
2
 D2
 D2
(24)
where C1 and C2 are arbitrary constants.
1.2 Case 2 :   0
 If   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 its solutions (Now it is not a
complete sentence. Please rewrite this). From theorem 3, we found that equation
(23) is mapped by the transformations
  x  Dt ,   H
to the linear equation
  4  0.
Hence, the solution of equation (1) is
2
3
u  x, t   C0  C1  x  Dt   C2  x  Dt   C3  x  Dt  ,
(25)
where C0 , C1 , C2 and C3 are arbitrary constants.
If   0,   0 and   3 , then equation (23) is linearizable if and only if
  4 ,   D 2 .
Next, to find the linear form of equation (23) and its solutions (Now it is not a
complete sentence. Please rewrite this). By using theorem 3, we obtained that
equation (23) is mapped by the transformations
H 2 D2 
  x  Dt ,  

H
2

to the linear equation
  4  0.
So that we obtain the implicit solution of equation (1) in the form
u 2 D2 
2
3
(26)

u  C0  C1  x  Dt   C2  x  Dt   C3  x  Dt  ,
2

where C0 , C1 , C2 and C3 are arbitrary constants.

DISCUSSION AND CONCLUSION
In the present work, we found some solutions for nonlinear fourth-order partial
differential equations (1), which 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 (Change this to passive voice) by the classical
methods. (This clause should be rewritten as a complete sentence.
4
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ACKNOWLEDGEMENTS
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