58 
 

 American Scientific Research Journal for Engineering, Technology,  and Sciences  (ASRJETS) 
ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 

© Global Society of Scientific Research and Researchers  
http://asrjetsjournal.org/  

 

A New Approach of Bernoulli Sub-ODE Method to Solve 

Nonlinear PDEs 

Md. Abdus Salama*, Md. Shafiqul Islamb, Md. Hamidul Islamc, Md. Abdul 

Azizd 

a,bDept. of Mathematics, Mawlana Bhashani Science and Technology University, Tangail-1902, Bangladesh 
cDept. of Mathematics and Physics, North South University, Dhaka, Bangladesh 

dDept. of Electronics and Telecommunication Engineering, Prime University, Dhaka-1216, Bangladesh 
aEmail: salam.a.math03@gmail.com,    bEmail: shafiquemath31@gmail.com 

cEmail: hamidul.islam@northsouth.edu,   dEmail: maaziz17@gmail.com 

 

Abstract 

In this paper, a new approach of the Bernoulli Sub-ODE method is proposed and this method is applied to solve 

the modified Liouville equation and the regularized long wave equation. As a result some new traveling wave 

solutions for them are successfully established. When the parameters are taken as special values, the solitary 

wave solutions are originated from these traveling wave solutions. Further, graphical representation of some 

solutions are given to visualize the dynamics of the equation. The results reveal that this method may be useful 

for solving higher order nonlinear partial differential equations. 

Keywords:  Modified Liouville equation; regularized long wave equation; traveling wave solutions. 

1. Introduction  

The investigation of traveling wave solutions(exact solutions) of nonlinear partial differential equations(PDEs) 

plays an important role not only in theoretic research but also in the applications. They describe different types 

of physical systems, ranging from gravitation to fluid dynamics. The interest of finding travelling wave 

solutions of nonlinear PDEs is increasing day by day and has now become a hot topic to researchers. In recent 

years, many researchers who are interested in the nonlinear physical phenomena have investigated exact 

solutions of nonlinear PDEs.  

------------------------------------------------------------------------ 

* Corresponding author.  

http://asrjetsjournal.org/
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59 
 

With the development of soliton theory and the application of computer symbolic system such as Maple and 

Mathematica, many powerful methods for obtaining exact solutions of nonlinear evolution equations are 

presented, such as the tanh-method [1-3], the extended tanh method [4-5], the Jacobi elliptic function expansion 

[6-8], the Bucland transformation[9-12], the homogeneous balance method[13], the inverse scattering method 

[14], the variational iteration method [15], the exp-function method [16], (G'/G)-expansion method [17], 

modified simple equation method [18], F-expansion method[19-20]  and so on. In 2011, Ben Jing proposed 

Bernoulli Sub-ODE method for finding exact solutions of nonlinear PDEs. After reducing the nonliner PDEs 

𝑃𝑃(𝑢𝑢,𝑢𝑢𝑡𝑡 ,𝑢𝑢𝑥𝑥,𝑢𝑢𝑡𝑡𝑡𝑡 ,𝑢𝑢𝑥𝑥𝑡𝑡 ,𝑢𝑢𝑥𝑥𝑥𝑥 , … … ) = 0  to nonlinear ODEs 0),,,,,,( 2 =′′′′−′′′′− LLLuucucuucuP   by 

choosing  ),(),( ξutxu =  where tcx −=ξ , he assumed the solution of  nonlinear ODEs in the form 𝑢𝑢 =

∑ 𝑎𝑎𝑖𝑖𝐻𝐻𝑖𝑖 ,𝑚𝑚
𝑖𝑖=0  where 𝐻𝐻(𝜉𝜉) can be determined from the first order ODE  𝐻𝐻′ = 𝐻𝐻2 − 𝐻𝐻 [21-22]. But in our proposed 

method, we consider a set of first order ODEs in the form: 0))(()( =±′ nHH ξξ (where 2≥n ) instead of 

𝐻𝐻′ = (𝐻𝐻2 − 𝐻𝐻) for getting a set of exact solutions and then a set of solitary wave solutions in a sequential 

manner. 

Here 0))(()( =±′ nHH ξξ give the following 

)1.1(
oddis,)1/(1])1([

1,)1/(1])1([

1

evenis,)1/(1])1([

1

)(

11

1













−−±

−
−−±

−−±
=

nnncnnc

nnnc
H

ξξ

ξ
ξ

In this method, we will get the solutions in terms of  ctx
nc

−=
−±

ξ
ξ

,
])1([

1

1

, where the singularities 

occur in the case of .0))(1(1 =−−± ctxnc  

2. Methodology 

Suppose that a nonlinear partial differential equation in two independent variables x and t, is given by 

𝑃𝑃(𝑢𝑢,𝑢𝑢𝑡𝑡 ,𝑢𝑢𝑥𝑥,𝑢𝑢𝑡𝑡𝑡𝑡 ,𝑢𝑢𝑥𝑥𝑡𝑡 ,𝑢𝑢𝑥𝑥𝑥𝑥 , … … … ) = 0                                             (2.1) 

where ),( txuu = is an unknown function, P is a polynomial in ),( txuu = and its various partial derivatives, 

the highest order derivatives and nonlinear terms are involved. The outline of the method is given below: 

Step-1: Combine the independent variables x and t into one variable ξ , by choosing  

),(),( ξutxu =  where tcx −=ξ                                                                                        (2.2) 

The traveling wave transformation (2.2) permits us to transform equation  (2. 1) to the following ODE: 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2018) Volume 44, No  1, pp 58-67 

60 
 

,0),,,,,,( 2 =′′′′−′′′′− LLLuucucuucuP                                         (2.3) 

where the prime denotes the differential with respect to 𝜉𝜉. 

Step-2: We suppose that equation (2. 3) has the solution of the form in the finite series: 

)(,)(
0

ξξ HHHau i
m

i
i == ∑

=

                                                   (2.4)
 

where 0, ≠mi aa are constants to be determined, the positive integer m can be determined by considering the 

homogeneous balance between the highest order derivatives and the nonlinear terms appearing in equation (2.3), 

and ( )ξHH =  satisfies the equation : 

,0))(()( =±′ nHH ξξ  where 2≥n                                              (2.5) 

Step-3: We substitute equation (2.4) into equation (2.3) and use equation (2.5) and then we account the function 

H(ξ). As a result of this substitution, we get a polynomial of H(ξ). We equate all the coefficients of same power 

of H(ξ) to zero. This procedure yields a system of algebraic equations whichever can be solved to find ia .  

Step-4: Substituting the values ia  into equation (2.4) along with general solutions of equation (2.5) complete 

the determination of the solution of equation (2.1).Finally particular choice of unknown parameters in exact 

solutions gives the desired solitary wave solutions. 

3. Applications of the method 

To illustrate the idea of the proposed method, we have selected two nonlinear PDEs, such as the modified 

Liouville equation and the regularized long wave equation which arise in mathematical physics.  

Example-1: Solution of modified Liouville equation 

𝑤𝑤𝑡𝑡𝑡𝑡 = 𝑎𝑎2𝑤𝑤𝑥𝑥𝑥𝑥 + 𝑏𝑏𝑒𝑒𝛽𝛽𝛽𝛽                                                        (3.1) 

that arises in hydrodynamics, where w(x, t) is the stream function and a, b,β are nonzero constants[23-24].                                                                                           

We first use the Painleve transformation ( ) wetxu β=, , so that 

𝑤𝑤 = 1
𝛽𝛽
𝑙𝑙𝑙𝑙𝑢𝑢.                                   (3.2) 

Now the wave transformation equations 𝑢𝑢(𝑥𝑥, 𝑡𝑡) = 𝑢𝑢(𝜉𝜉), 𝜉𝜉 = 𝑥𝑥 − 𝑐𝑐𝑡𝑡 and equation (3.2) reduces equation (3.1) 

into the following ODE: 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2018) Volume 44, No  1, pp 58-67 

61 
 

032 =+′−′′ kuuuu  where 22 ca
bk
−

=
β

 and ac ±≠   (3.3) 

Let (3.3) has the solution of the form:    )(,)(
0

ξξ HHHau i
m

i
i ==∑

=      
                                                       (3.4) 

Hence for different for values of n  in (2.5), the corresponding exact solutions of (3.1) are obtained below: 

 (a) 2=n : By considering the homogeneous balance between 𝑢𝑢′′′ and 𝑢𝑢𝑢𝑢′appearing in eq.(3.3), we get 𝑚𝑚 =

2. As a result, (3.4)  takes the form: 

2
210)( HaHaau ++=ξ                                                      (3.5) 

where 𝑎𝑎0, 𝑎𝑎1, 𝑎𝑎2 are unknown constants to be determined and )(ξH satisfies the eq.(2.5) and this function is 

determined from eq. (1.1) by setting 𝑙𝑙 = 2. Substituting (3.5) in the reduced ODE (3.3) and collecting the 

coefficients of various power of 𝐻𝐻(𝜉𝜉) yields the following system of algebraic equations. 

02: 2
2

3
2

6 =+ akaH  

034: 2
2121

5 =+ akaaaH  

0363: 2
20202

2
1

2
1

4 =+++ akaaaakaaH  

062: 3
121010

3 =++ kaaakaaaH  

033: 2
2

0
2

10
2 =+ akaakaH  

03: 1
2

0
1 =akaH  

0: 3
0

0 =kaH  

Solving the above equations, we get   .2,0,0 210 k
aaa −

===
                                                                  

(3.6)
 

Hence the solution of (3.4) takes the form 

2
1 )(

2)(
ξ

ξ
±
−

=
ck

u
                                                              

(3.7) 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2018) Volume 44, No  1, pp 58-67 

62 
 

Finally putting 𝜉𝜉 = 𝑥𝑥 − 𝑐𝑐𝑡𝑡,  and using equation (1.1), we get the following desired exact solution of (3.1)  









−±
−−

= 2
1

22

1 )]([
)(2ln1),(

ctxcb
catxw

ββ                                                     
(3.8) 

Similarly, for ,....7,6,5,4,3=n the corresponding exact solutions of eq. (3.1) are  

(b) 3=n ,  








−±
−−

= 2
1

22

2 )](2[
)(8ln1),(
ctxcb

catxw
ββ

 

(c) 4=n   








−±
−−

= 2
1

22

3 )](3[
)(18ln1),(

ctxcb
catxw

ββ
 

(d) 5=n    








−±
−−

= 2
1

22

4 )](4[
)(32ln1),(

ctxcb
catxw

ββ
 

(e) 6=n  








−±
−−

= 2
1

22

5 )](5[
)(50ln1),(

ctxcb
catxw

ββ
 

(f) 7=n :   








−±
−−

= 2
1

22

6 )](6[
)(72ln1),(

ctxcb
catxw

ββ
 

In general, the solution of (3.1) is  









−−±
−−−

=− 2
1

222

1 )])(1([
)()1(2ln1)(

ctxncb
canwn ββ

ξ  , where 
0

)])(1([
)()1(2

2
1

222

>
−−±
−−−

ctxncb
can

β                                    (3.9) 

Singularity: From the above obtained solutions we observe that the solutions have singularities for the values 

x  and t  satisfying 2,0))(1(1 ≥=−−± nctxnc or .0
)])(1([

)()1(2
2

1

222

>
−−±

−−
ctxncb

can
β  

Justification: Here, 𝑤𝑤𝑖𝑖(𝑥𝑥, 𝑡𝑡), 𝑖𝑖 = 1,2,3,4,5,6  obtained in different cases are exact solutions of the modified 

Liouville equation because they fully satisfy the equation (3.1). This justification is checked by MAPLE-13 and 

the corresponding MAPLE code is given below: 

 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2018) Volume 44, No  1, pp 58-67 

63 
 

* Justification of the solutions of the modified Liouville equation* 

 

>  

 

>  

 

>  








−−±
−−−

=− 2
1

222

1 )])(1([
)()1(2ln1)(

ctxncb
canwn ββ

ξ  

 

>  

                                                            0 

Figure 3 

Velocity profile of 𝑤𝑤1(𝑥𝑥, 𝑡𝑡) with wave speed, c=2 

 

Figure 3.1: (3D Plot): Profile of (3.9), 

when 𝑙𝑙 = 2, 𝑎𝑎 = 1, 𝑐𝑐1 = 1, 𝑏𝑏 = 1,𝛽𝛽 = 1 

 

Figure 3.2: (2D Plot): Profile of (3.9), when 

𝑙𝑙 = 2, 𝑎𝑎 = 1, 𝑐𝑐1 = 1, 𝑏𝑏 = 1,𝛽𝛽 = 1 and time t=1. 

  

Velocity profile of 𝑤𝑤1(𝑥𝑥, 𝑡𝑡) with wave speed, c=3 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2018) Volume 44, No  1, pp 58-67 

64 
 

 

Figure 3.3: (3D Plot): Profile of (3.9), 

when 𝑙𝑙 = 2, 𝑎𝑎 = 1, 𝑐𝑐1 = 1, 𝑏𝑏 = 1,𝛽𝛽 = 1 

 

Figure 3.4: (2D Plot): Profile of (3.9), when 

𝑙𝑙 = 2, 𝑎𝑎 = 1, 𝑐𝑐1 = 1, 𝑏𝑏 = 1,𝛽𝛽 = 1 and time t=1. 

 

Example-2: Solution of regularized long-wave equation 

The regularized long-wave equation is  

0,,06 >=−−+ babuuuauu xxtxxt                                             (3.10) 

where a,b are real constants[25]. In the above procedure, we can also find the solutions of this equation and the 

general form of the solution of (3.10) is: 

0))(1(,
)])(1([

)1(2
6

),( 12
1

2

1 ≠−−±
−−±

−
+

−
=− ctxnc

ctxnc
bcncatxun

                                      
(3.11) 

Velocity profile of 𝑢𝑢1(𝑥𝑥, 𝑡𝑡) with wave speed, c=2 

 

Figure 3.5: (3D Plot): Soliton profile of (3.11), when 𝑙𝑙 = 2, 𝑎𝑎 = 1, 𝑏𝑏 = 1, 𝑐𝑐1 = 1 

Velocity profile of 𝑢𝑢1(𝑥𝑥, 𝑡𝑡) with wave speed, c=3 

 

Figure 3.6: (3D Plot): Soliton profile of (3.11), when 𝑙𝑙 = 2, 𝑎𝑎 = 1, 𝑏𝑏 = 1, 𝑐𝑐1 = 1 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2018) Volume 44, No  1, pp 58-67 

65 
 

Singularity: From the above obtained solutions we observe that the solutions (3.11) have singularities for the 

values x  and t  satisfying 2,0))(1(1 ≥=−−± nctxnc . For example, the solution (3.11) has a singularity at 

𝑥𝑥 = −6, when .1,2,2,81 ==== tcnc  

 

Figure 3.7: Singularity of  𝑢𝑢1(𝑥𝑥, 𝑡𝑡) at  𝑥𝑥 = −6 

4. Results and Discussions 

In case of our proposed method, we have got many exact solutions for differetnt values of 𝑙𝑙. So we will discuss 

some solutions only. 

Equation (3.8) represents the soliton type solutions (shown in Figure 3.1(3D plot)  and Figure 3.4(3D plot)) of 

the modified Liouville equation. The corresponding two-dimensional plot are given in Figure 3.2(2D plot)  and 

Figure 3.4(2D plot)  respectively.  

And finally Figure 3.5(3D plot) and Figure 3.6(3D plot) give also the soliton profile for the solutions of the 

equation regularized long wave equation where the waves move with 2 and 3 respectively. A singularity has 

been shown in Figure 3.7. 

5. Conclusion  

The main achievement of this work is to explore a new approach of the Bernoulli Sub-ODE method, which is 

capable of producing more fruitful and new solitary wave solutions of several nonlinear evolution equations. 

Using this method, we have successfully obtained exact solutions and then solitary wave solutions in terms of 

fractional functions.  Besides, there are several avenues for further investigation:  

 Seeking exact solutions to NEEs having (2+1) or (3+1) dimension through the proposed method is 

another possible study. 

 Applications of the obtained solutions can be discussed. 

References 

[1]. Abdou M. A., The extended tanh-method and its applications for solving nonlinear physical models, 

Appl. Math. Comput., 190( 2007), 988-996.  

[2]. Malfliet W., Solitory wave solutions of nonlinear wave equations, Am. J. Phys.,60(1992), 650–654. 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2018) Volume 44, No  1, pp 58-67 

66 
 

[3]. Hereman W. and Malfliet W., The tanh method I: exact solutions of nonlinear evolution and wave 

equations, Phys. Scripta, 54(1996), 563–568. 

[4]. Wazwaz A. M., The tanh method: solitons and periodic solutions for the Dodd-Bullough-Mikhailov 

and the Tzitzeica-Dodd-Bullough equations, Chaos Soliton Fractal, 25(2005), 55–63. 

[5]. Wazwaz A. M., The extended tanh method for new soliton solutions for many forms of the fifth-order 

KdV equations, Appli. Math. Comput. 184(2007), 1002–1014. 

[6]. Chen Y., Wang Q., Extended Jacobi elliptic function rational expansion method and abundant families 

of Jacobi elliptic functions solutions to the (1+1)- dimensional nonlinear dispersive long wave 

equation. Chaos, Solitons & Fractals. 24(2005): 745-757 . 

[7]. Li Z., Wang M. L., A sub-ODE method for finding exact solutions of a generalized KdV-mKdV 

equation with higher order nonlinear terms. Phys. Lett. A. 361(2007): 115-118 . 

[8]. Yan Z., Abundant families of Jacobi elliptic functions of the (2+1)-dimensional integrable Davey- 

Stawartson-type equation via a new method. Chaos, Solitons & Fractals. 18(2003): 299-309.  

[9]. Boiti M., Leon J., Pempinelli P.,  Spectral transform for a two spatial dimension extension of the 

dispersive long wave equation. Inverse Probl. 3(1987): 371-387. 

[10]. Miura M. R, Backlund Transformation. Springer, Berlin, (1978) 

[11]. Ren Y., Zhang H., New generalized hyperbolic functions and auto-Backlund transformation to find 

exact solutions of the (2+1)-dimensional NNY equation. Phys. Lett. A. 357(2006): 438-448. 

[12]. Rogers C., Shadwick W. F., Backlund Transformation. Academic Press, New York, (1982) 

[13]. Moussa M. H. M., ElShikh R. M., Two applications of the homogeneous balance method for solving 

the generalized Hirota-Satsuma coupled KdV system with variable coefficients, Inter. J. Nonlinear Sci. 

7(2009): 29-38. 

[14]. Vakhmenko V. O., Parkers E. J., The calculation of multi-soliton solutions of the Vakhnenko equation 

by the inverse scattering method. Chaos, Solutions & Fractals. 13(2003): 1819-1826. 

[15]. Abdou M. A., Soliman A. A., New applications of variational iteration method. Physica D, 211 (1–2), 

2005, 1–8.  

[16]. Bekir A. and Boz A., Exact solutions for nonlinear evolution equation using Exp-function method, 

Physics Letters A 372(2008), 1619–1625.  

[17]. Islam M. E., Khan K., Akbar M. A. and Islam R., Traveling Wave Solutions of Nonlinear Evolution 

Equation via Enhanced (G'/G)-expansion Method. GANIT J. Bangladesh Math. Soc., (ISSN 1606-

3694) Vol. 33, 2013, 83-92. 

[18]. Jawad A. J. M., Petkovi´c M. D. and Biswas A., Modified simple equation method for nonlinear 

evolution equations, App. Math. and Compu., vol. 217, no. 2, (2010), pp. 869–877.  

[19]. Wang M., Li X., Extended F-expansion and periodic wave solutions for the generalized Zakharov 

equations. Phys. Lett. A. 343(2005): 48-54.  

[20]. Wang M., Li X.,  Applications of F-expansion to periodic wave solutions for a new Hamiltonian 

amplitude equation. Chaos, Solutions & Fractals. 24(2005): 1257-1268. 

[21]. Ben Jing. A new Bernoulli Sub ODE method for constructing travelling wave solutions for two 

nonlinear equations with any order. U.P.B science bulletin, series A, vol.73, issue 3, page 85-93, 2011. 

[22]. Xu F., and Feng Q. A.  Generalized Sub-ODE Method and Applications for Nonlinear Evolution 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2018) Volume 44, No  1, pp 58-67 

67 
 

Equation s. J. Sci. Res. Report. 2(2): 571-581, 2013. 

[23]. Mahmoud A.E. Abdelrahman.  Exact Traveling Wave Solutions for Fitzhugh-Nagumo(FN) Equation 

and Modified Liouville Equation.  International Journal of Computer Applications (0975 8887), 

Volume  113 - No. 3, March 2015. 

[24]. Salam M. A., Traveling-Wave Solution of Modified Liouville Equation by Means of Modified Simple 

Equation Method, ISRN Applied Mathematics, Article ID 565247,4 pages doi:10.5402/2012/565247, 

2012. 

[25]. Wazwaz A.M.  A. sine-cosine method for handling nonlinear wave equations. Math. And Comput. 

Modelling. 40, 499-508, 2004.