Series Solution and Wave Solution of Time Fractional Generalized 

Korteweg-de Vries Equation 

Mitu Nagpal1 and Rajeev Kumar2 

1,2Department of Mathematics, MMEC, Maharishi Markandeshwar (Deemed to 

be University), Mullana, Ambala-133207(Haryana), India  

Corresponding should be addressed to Mitu Nagpal; gakharmitu@gmail.com 

                                        

 

 

Keywords: Korteweg-de Vries Equation, Lie Symmetry, Conformable Derivative, 

Series Solution, Wave Solution,  (
𝑮′

𝑮
)– Expansion Method. 

1. Introduction

1.1 Scope 

In applied mathematics and physics, nonlinear phenomena are commonly present. In 

applied mathematics and mathematical physics Non-Linear Partial Differential 

Equations (NLPDEs) are applied for modeling several scientific processes and issues. 

In obtaining the Non-Linear Fractional Partial Differential Equations NLFPDEs [1] 

approximate solutions are vital in all of these fields of study. We mainly concentrate on 

estimating the precise solutions because we don’t have a mechanism for finding the 

precise solutions of these kinds of Fractional Partial Differential Equations (FPDEs). 

Abstract: In the present article, exact solutions of nonlinear fractional 𝑝𝑡ℎ order

Korteweg-de Vries equation time fractional derivatives are deduced and analyzed. The 

Lie symmetry approach has been used to identify the fractional KdV equation's 

infinitesimal generators and symmetry reductions. Some new exact solutions are 

obtained with using (
𝑮′

𝑮
)–expansion method. The wave solutions and series solutions 

of KdV nonlinear fractional partial differential equation has been evaluated and in the 

form of rational and exponential. All the calculations have done in Maple. 

Communications on Applied Nonlinear Analysis 
ISSN: 1074-133X 
Vol 32 No. 10s (2025) 

https://internationalpubls.com 102

Article History: Received: 12-01-2025, Revised: 15-02-2025, Accepted: 01-03-2025

mailto:gakharmitu@gmail.com


To solve FPDEs, a number of techniques have been used, e.g., Elzaki transform 

decomposition method [2], Laplace transforms approach [3], and Adomian's 

decomposition technique [4] furthermore. There are various techniques to analyze the 

NLPDEs but lie group of symmetry is one of the approaches for deducing the solution 

of NLPDEs. Lie Symmetries studies differential equation invariance under a one-

parameter group transformations that turn into a new solution and minimize the order 

of differential equation. Lie developed the theory of one parameter group of 

transformation in 17th century which applied for findings the solution of Partial 

Differential Equations (PDEs). Later on, many contributors like Bluman [6], Birkoff 

[5] and Olver [7] et. al. Many researchers developed different methods [8] for finding

exact solution of NLPDEs, such as tanh method [9], the hyperbolic B-spline 

differential quadrature method [10] and new extended (
𝐺′

𝐺
) method [11], etc. 

1.2 Review of Related Work:  

Korteweg-de Vries (KdV) equation is the generalized form of fractional PDEs. KdV 

equations can be utilized for predicting solitary waves, long waves, tides and wave 

propagation in a shallow canal [12-16]. Fluid mechanics [17], viscoelasticity, signal 

processing, fractional kinetics and hydrology are among some of the disciplines that 

use the KdV equations.  

J. S. Russell's (1834) provided the concept of KdV equations. However, the equation is 

developed by Lord Rayleigh, Joseph Boussinesq (1870) and KdV (1895). The 

numerical and exact solitary wave solutions of the generalized long wave and KdV 

equation has obtained by D. Kaya and S. M. EL Sayad [18] for the initial conditions. 

D. D. Bhatta and M. I. Bhatti [19] presented an algorithm for numerical solutions of

KdV equation on reformed Bernstein polynomials. 

Communications on Applied Nonlinear Analysis 
ISSN: 1074-133X 
Vol 32 No. 10s (2025) 

https://internationalpubls.com 103



A. R. Seadawy [20] studied the exact solution for the KdV equation of higher order 

non-linearity by using the variational technique. Exact solutions of KdV equation of 

higher order non-linearity are determined for utilizing the variational principle method 

without requiring for significant calculations. A. R. Seadawy, D. Lu and C. Yue 

studied fifth-order generalized KdV equations [21] on water wave equations. 

G. Wang obtained series solution and invariant solutions of KdV–Burgers–Kuramoto

generalized equation and deduced invariants and invariant solutions based on the Lie 

point symmetries [22]. A. A. Alderremy, A. Shaban, et.al. [23] investigated third order 

KdV equations and coupled Burgers equations with the help of two methods. 

H. Rezazadeh, A. G. Davodi, et.al. [24] applied the (
𝑮′

𝑮
)  expansion technique for

traveling wave solutions of the Schrödinger-KdV equations by the conformable 

derivative. 

1.3 Motivation of study 

In the above related work we observed that the solution of KdV equation solved by the 

theory of Riemann – Liouville. This theory has limitation that cannot obtain the 

traveling wave solutions. Fractional derivatives have changed as a result of the use of 

conformable fractional derivatives [26] by Khalil et.al. to identify Lie symmetries in 

differential equations. In this paper, by conformable fractional derivatives [32-37] we 

will find Lie symmetry of KdV equation. 

The generalized pth order KdV nonlinear fractional partial differential equation: 

𝑢𝑡
𝛼 − 𝑢𝑥𝑥 − 𝐴𝑢

𝑝𝑢𝑥 = 0 , 0 < 𝛼 ≤ 1, 𝑝 > 0  ………………. (1)

and 𝑢𝑡
𝛼 is the conformable fractional derivative [38-39].

Communications on Applied Nonlinear Analysis 
ISSN: 1074-133X 
Vol 32 No. 10s (2025) 

https://internationalpubls.com 104



Objective of the study: 

a. To find the lie symmetry of KdV NLFPDEs.

b. To find series solution of the problem.

c. To apply the (
𝑮′

𝑮
) - expansion approach to determine the wave solution. 

This paper divides in following sections: 

Section 2, Preliminaries are described. 

Section3, Analysis of NLFPDEs by Lie Symmetry. 

Section 4, Includes the description of (
𝐺′

𝐺
)- Expansion Method. 

Section 5, Symmetry analysis of KdV NLFPDEs. 

Section 6, Findings of series solution of KdV equation  

Section 7, Exact traveling wave solution of fractional KdV equation is provided in 

trigonometric form. 

At last, the brief of the research work is recorded in the section 8.  

2. Preliminaries:

2.1 Conformable Derivative and Conformable Fractional Derivatives: 

If function ƭ:  [0,∞) → 𝑅 then conformable fractional derivative [7] is defined by: 

𝑇𝛼 (ƭ)(ŧ) = lim
𝜖→0

ƭ(ŧ + 𝜖ŧ1−𝛼 ) − ƭ(ŧ)

𝜖

For all ŧ ˃ 0,∈ (0,1), if  ƭ is 𝛼 -differentiable in some (0, 𝛼), a >0 and lim
𝑡→∞

ƭ𝛼(𝑡) occurs,

then define ƭ𝛼( 0) =lim
𝑡→0

ƭ𝛼 (ŧ). We generally use ƭ𝛼(ŧ)   for   𝑇𝛼 (ƭ)(ŧ) to represent

conformable fractional derivatives and consideration that  𝑇𝛼 (ŧ
𝑝  ) = 𝑝ŧ𝑝−𝛼 .

Communications on Applied Nonlinear Analysis 
ISSN: 1074-133X 
Vol 32 No. 10s (2025) 

https://internationalpubls.com 105



2.2 Conformable Fractional Derivative: 

A conformable fractional derivative, ƭ of  order 𝛼 with respect to function ƭ: [0,∞) →

𝑅  defined as 𝐷𝛼ƭ(𝑥) = lim
ℎ→0

ƭ(𝑥+ℎ𝑒(𝛼−1)𝑥)−ƭ(𝑥)

ℎ
Where 𝑥 > 0 , 𝛼 ∈ (0,1), f is 𝛼 –

differentiable in (0, 𝛼) , and lim
𝑥→0+

𝐷𝛼ƭ (𝑥) exists and (𝐷𝛼ƭ)(0) = lim
𝑥→0+

(𝐷𝛼ƭ)(𝑥) .

Based on the product rule and quotient rule, conformable derivative [7] yields 

conclusion which is similar to the Mean Value Theorem and Rolle's Theorem. 

3. Analysis of NLFPDEs by Lie Symmetry:

The time-fractional partial differential equation is    

𝜕𝛼𝑤

𝜕𝑡𝛼
=  H [𝑤],   0 < 𝛼 ≤ 1  …………., (2) 

where 𝑤 = 𝑤 (x, t), non-linear differential operator is H [𝑤] and conformable fractional 

derivative is (
𝜕𝛼

𝜕𝑡𝛼
). Now, we analyze symmetry transformation of equation (2), take 

invertible point transformations 

𝑥 = X (x, t, 𝑤,𝜀) ,  𝑡 ̌ = T (x, t, 𝑤,𝜀),   �̌�= W (x, t, 𝑤,𝜀) ……….. ,            (3) 

based on a continuous parameter 𝜀 and Eq. (1) that have the same form in the new 

variable  𝑥,̌ 𝑡,̌ �̌�  are said to be symmetry transformation. The symmetry group is a 

continuous group made up of   H* transformations. The Lie group is another name of 

the symmetry group H*. The key point in Lie group of transformation [8] is to deduced 

infinitesimal generator [9] and determining equations.  

Infinitesimal transformation of (3) be 

𝑥  =  𝑥 + 𝜀𝜉(𝑥, 𝑡, 𝑤) + 𝑜(𝜀2),

𝑡 ̌  = 𝑡 + 𝜀𝜏(𝑥, 𝑡, 𝑤) + 𝑜(𝜀2) ,                                                                   (4)

�̌�  = 𝑤 + 𝜀𝜂(𝑥, 𝑡, 𝑤) + 𝑜 (𝜀2),

W = 𝜉(𝑥, 𝑡, 𝑤)
𝜕

𝜕𝑥
 + 𝜏(𝑥, 𝑡, 𝑤)

𝜕

𝜕𝑡
 +  𝜂(𝑥, 𝑡, 𝑤)

𝜕

𝜕𝑤
 ………,                                 (5)

Communications on Applied Nonlinear Analysis 
ISSN: 1074-133X 
Vol 32 No. 10s (2025) 

https://internationalpubls.com 106



Here, W is the infinitesimal operator. 

𝑑𝑥 ̌ 

𝑑𝜀
  =  𝜉(𝑥, 𝑡,̌ �̌�),   

𝑑𝑡,̌

𝑑𝜀
= 𝜏(𝑥, 𝑡,̌ �̌�),  

𝑑�̌�

𝑑𝜀
  =  𝜂(𝑥, 𝑡,̌ �̌�) 

can be computed to obtain the group transformation (3) relating to operator (5) and 

subject to initial conditions 

𝑥|𝜀=0 = 𝑥   ,    𝑡 ̌|𝜀=0 = 𝑡     ,   �̌�|𝜀=0 = 𝑤 …….,        (6)  

A surface  𝑤 =  𝑤 (x, t) is mapped as the group transformation, generated by W if 

W (𝑤 - 𝑤 (x, t)) = 0 when 𝑤 = 𝑤 (x, t) …………,                                             (7) 

Here,  �̌�(𝑥 ̌, 𝑡 ̌) satisfies to  
𝜕𝛼𝑤 ̌

𝜕�̃�𝛼
  = H [�̌�] ,    0<𝛼 ≤ 1……….,      (8) 

As the function  𝑤 =  𝑤 (𝑥, t)  satisfies Eq. (2), then the transformation (3) forms a 

symmetry group H of Eq. (2). Extended transformation (4) of fractional differentiation 

𝜕𝛼𝑤

𝜕𝑡𝛼
and operator of 𝑥 differentiation of several order  

𝜕𝛼𝑤

𝜕𝑥𝑘
 , k=1, 2, 3, …. , we can find 

𝜕𝛼�̌�

𝜕�̃�𝛼
= 
𝜕𝛼𝑤

𝜕𝑡+𝛼
+ 𝜀𝜂𝛼

𝑡 (𝑥, 𝑡, 𝑤) +o (𝜀2),

𝜕�̌�

𝜕�̃�
=  

𝜕𝑤

𝜕𝑥
+𝜀𝜂𝑥(𝑥, t,𝑤) + o (𝜀2),

𝜕2�̌�

𝜕�̃�2
=  

𝜕2𝑤

𝜕𝑥2
+ 𝜀𝜂𝑥𝑥(𝑥, t,𝑤) + o (𝜀2),

𝜕3�̌�

𝜕�̃�3
=  

𝜕3𝑤

𝜕𝑥3
+ 𝜀𝜂𝑥𝑥𝑥(𝑥, t,𝑤) + o (𝜀2),

   : 

    : 

where     

𝜂𝑥  =  𝐷𝑥(𝜂) - 𝑤𝑡𝐷𝑥(𝜏) - 𝑤𝑥𝐷𝑥(𝜉),

𝜂𝑥𝑥  =  𝐷𝑥(𝜂
𝑥) - 𝑤𝑥𝑡𝐷𝑥(𝜏) - 𝑤𝑥𝑥𝐷𝑥(𝜉), (9) 

𝜂𝑥𝑥  =  𝐷𝑥(𝜂
𝑥𝑥) - 𝑤𝑥𝑥𝑡𝐷𝑥(𝜏) - 𝑤𝑥𝑥𝑥𝐷𝑥(𝜉).

      :     

Here, derivative operator 𝐷𝑥 is defined as

Communications on Applied Nonlinear Analysis 
ISSN: 1074-133X 
Vol 32 No. 10s (2025) 

https://internationalpubls.com 107



𝐷𝑥  =  
𝜕

𝜕𝑥
 +𝑤𝑥  

𝜕

𝜕𝑤
 + 𝑤𝑥𝑥

𝜕

𝜕𝑤𝑥
 + 𝑤𝑡𝑥

𝜕

𝜕𝑤𝑡
 +………,                                                       (10) 

and 
𝜕𝛼�̌�

𝜕𝑡 ̌𝛼
 = 

𝜕𝛼𝑤

𝜕𝑡𝛼
 + 𝜀𝜂𝛼

𝑡  + o (𝜀2), where  𝜂𝛼
𝑡  extended infinitesimal associated to 

conformable fractional time derivative of  𝑣 𝑎𝑛𝑑 �̃�  differentiable functions. The 

prolongation of the point transformation (3) to the αth derivative for some α ∈ (0, 1]. 

4. Description of (
𝑮′

𝑮
)-Expansion Method [40]: 

(
𝐺′

𝐺
)- Expansion Method (Zayed, 2011) has been discussed in this section as a way to 

obtain traveling wave solutions of NLPDEs. Assume that non-linear equation,  

F (𝑢, 𝑢𝑥 , 𝑢𝑥𝑥, 𝑢𝑡 , 𝑢𝑡𝑡...) = 0.……….,           (11) 

Where F is higher order derivatives and non-linear terms of polynomial with unknown 

variable 𝑢 = 𝑢(𝑥, 𝑡) and its derivatives 𝑢, 𝑢𝑥, 𝑢𝑥𝑥, 𝑢𝑡 , 𝑢𝑡𝑡. 

A transformation 𝑢 = 𝑢(𝑥, 𝑡) = Q (ξ) and ξ = 𝑔𝑥 +  ℎ𝑦 − 𝑐𝑡 , 𝑄 (𝜉) represents the 

traveling wave solutions at speed c and g and h define the wave numbers. 

The method can be applied utilizing the following methods to find the wave solutions: 

Step 1:  Firstly, change NLPDEs equation into nonlinear ordinary differential equation 

(ODE) using transformation (Guner and Ozkan 2016) and the system reduce into ODE. 

𝐹 (𝑄, 𝑄′, 𝑄′′ … . . )  =  0 ……………….,                         (12) 

Step 2: Assume that the solution to equation (12) may be defined as : 

Q (ξ)  = 𝑎𝑚 (
𝐺′

𝐺
)
𝑚
+ 𝑎𝑚−1 (

𝐺′

𝐺
)
𝑚−1

+…………,          (13) 

the following form, 𝐺 =  𝐺 (𝜉) of second order linear differential equation is satisfied: 

𝐺′′ +  𝜆𝐺′ +  µ𝐺 =  0………..,             (14) 

Whereas  𝑎𝑚 , 𝑎𝑚−1, . . . . . 𝑎0, 𝜆 𝑎𝑛𝑑 µ are constants, 𝑎𝑚  ǂ 0. The non-linear terms and 

highest order derivatives in Eq. (11) are balanced by using the homogeneous balance 

method for find the positive integer m. 

Communications on Applied Nonlinear Analysis 
ISSN: 1074-133X 
Vol 32 No. 10s (2025) 

https://internationalpubls.com 108



Step 3: Using Eq. (14) and putting in Eq. (13) into Eq. (12). Collecting the same order 

terms of (
𝐺′

𝐺
), and then equating each coefficient of polynomial to zero. A set of 

equations for am, am−1, ....... a0  are obtained. 

Step 4: As a result of Eq. (14) we able to obtain travelling wave solutions for NLPDEs 

(2). The general solutions to Eq. (14) are given as   

 

 
𝐺′

𝐺
=

{
 
 
 
 

 
 
 
 
−
𝜆

2
+√

𝜆2−4𝜇

2
 (

𝑐1 sinh
√𝜆2−4𝜇

2
𝜉+𝑐2 cosh

√𝜆2−4𝜇

2
𝜉

𝑐1 cosh
√𝜆2−4𝜇

2
𝜉+𝑐2 sinh

√𝜆2−4𝜇

2
𝜉

) , 𝜆2 − 4𝜇 > 0

−
𝜆

2
+√

4𝜇−𝜆2

2
 (

−𝑐1 sin
√4𝜇−𝜆2

2
𝜉+𝑐2 cos

√4𝜇−𝜆2

2
𝜉

𝑐1 cos
√4𝜇−𝜆2

2
𝜉+𝑐2 sin

√4𝜇−𝜆2

2
𝜉

) , 𝜆2 − 4𝜇 < 0

−
𝜆

2
+

𝑐1

𝑐1+𝑐1𝜉
  , 𝜆2 − 4𝜇 = 0

 

 

 

5. Symmetry Analysis of KdV NLFPDEs:  

The following  𝑝 th order KdV equation in fractional form is  

ũƫ
ᾱ − ũ𝜘𝜘 − 𝐴ũ

𝑝ũ𝜘 = 0 , 0 ≤ 𝛼 ≤ 1  

and 𝛼 (parameter) expresses the order of conformal fractional derivative.  

Here ũ(𝑥, ƫ) = ũ(𝜉), 𝜉 = 𝑘𝜘 −
𝜆𝑡ᾱ

𝜏(1+ᾱ)
. 

 

 

Construct an ODE for the expression ũ(𝑥, ƫ)= 𝑄 (𝜉) using the Eq. (1), 

𝑐2𝑄′(𝜉) − 𝑄′′(𝜉) 𝑔2 − 𝑎2𝑄(𝜉) + 𝑏2𝑄3(𝜉) = 0………..,                                  (15)  

Applying lie group of transformation in Eq. (1), we get Infinitesimal equation    

Communications on Applied Nonlinear Analysis 
ISSN: 1074-133X 
Vol 32 No. 10s (2025) 

https://internationalpubls.com 109



ɳƫ
ᾱ − ɳ𝜘𝜘 − 𝐴(ũ𝑝ɳ𝜘 + 𝑝ũ𝑝−1ɳũ𝜘)=0                                                  (16) 

Substituting the values of ɳ𝜘𝜘 and ɳƫ
ᾱ into (16). In partial derivatives of where 0 <

 ᾱ ≤ 1 and the parameter ᾱ represents order of conformable fractional derivatives, 

equate the coefficients of the various equations. The infinitesimal equation is given by 

ũ and   determining equations are obtained for the symmetry of Eq. (16) by the Lie 

theory. 

 

 𝐷1(𝜏) = 0 

𝐷3,3(𝜉) = 0, 𝐷3(𝜉) = 0, 𝐷3,3(ɳ) = 0 

𝐷1,1(𝜉) − ƫ
1−ᾱ𝐷2(𝜉) − 𝐴ũ

𝑝𝐷1(𝜉) − 2𝐷1,3(ɳ) − 𝐴𝑝ũ
𝑝−1ɳ = 0 

                                    − 𝜏ᾱƫ−ᾱ + ƫ1−ᾱɳũ − ƫ
1−ᾱ𝜏ƫ + 𝜏ƫ

−ᾱ = 0                              (17) 

 ƫ1−ᾱɳƫ − 𝑎ũ
𝑝ɳ𝜘 − ɳ𝜘𝜘 = 0 

 2𝜉𝜘 − ɳũ = 0 

These determining equations can be solved to obtain 

ɳ =  ũƈ1 

                                                           𝜉 = −𝑝ƈ1𝜘 + ƈ2                                  (18)         

𝜏 =
−2𝑝ƈ1ƫ

ᾱ
+ ƈ3ƫ

1−ᾱ 

Vector fields span the associated symmetry group are 

𝑉1 = ũ
𝜕

𝜕ũ
− 𝑝𝜘

𝜕

𝜕𝜘
− 2𝑝

ƫ

ᾱ

𝜕

𝜕ƫ
 

                                                                 𝑉2 =
𝜕

𝜕𝜘
                                                (19)          

𝑉3 = ƫ
1−ᾱ

𝜕

𝜕ƫ
 

The equivalent solution is 𝜉 = 𝑥 −  ƈ
ƫᾱ

ᾱ
 for the symmetry 𝑉2 + 𝑉3. 

 

Communications on Applied Nonlinear Analysis 
ISSN: 1074-133X 
Vol 32 No. 10s (2025) 

https://internationalpubls.com 110



6. Series solution of KdV NLFPDEs: 

𝑉1 = ũ
𝜕

𝜕ũ
− 𝑝𝜘

𝜕

𝜕𝜘
− 2𝑝

ƫ

ᾱ

𝜕

𝜕ƫ
 

                                                           
𝑑𝑥

−𝑝𝜘
=

ᾱ𝑑ƫ

−2𝑝ƫ
=

𝑑ũ

ũ
    ………….,                    (20) 

Now solving Eq. (20), we get the similarity variable 

                                                      ũ = ƫ
−
ᾱ

2𝑝𝐹(𝜉),     𝜉 = 𝑥ƫ
ᾱ

2  …………,            (21)  

Inserting these values in Eq. (1), which yields 

−
ᾱ

2𝑝
ƫ
−
ᾱ

2𝑝
−ᾱ
𝐹(𝜉) −

ᾱ

2
ƫ
−
ᾱ

2𝑝
−ᾱ
𝜉𝐹′(𝜉) − ƫ

−
ᾱ

2𝑝𝐹′′(𝜉)ƫ−ᾱ − 𝐴ƫ−
ᾱ

2𝐹𝑝(𝜉)ƫ
−
ᾱ

2𝑝𝐹′(𝜉)ƫ−
ᾱ

2 =

0              ………..,        (22)  

To obtain a solution of Eq. (22)                                                                                                                               

We take  𝐹(𝜉) = 𝐴(𝜉)𝑝    …….,    (23) 

Where A and p are constant. Equating the similar exponent of  𝜉, we get 𝑝 = −
1

𝑞
  . 

Substituting (23) in to (22), the following solution is 

ũ = −𝑥𝑝ƫ
−ᾱ(1+𝑝2)

2𝑝 (
𝑝 + 1

𝑝
)

1
𝑝+1

 

 

 

7. Wave Solution of KdV Fractional Equation: 

The (
𝑮′

𝑮
) - expansion method [23] is used to solve the NLFPDEs defined as the KdV as 

given   

                                           ũƫ
ᾱ − ũ𝜘𝜘 − 𝐴ũ

𝑝ũ𝜘 = 0  …………..,  (24) 

And wave transformation is 

 ũ = 𝐹(𝜉), 𝜉 = 𝑥 −  ƈ
ƫᾱ

ᾱ
 ……………….,     (25) 

Now modify the Eq. (1) into ODE as 

Communications on Applied Nonlinear Analysis 
ISSN: 1074-133X 
Vol 32 No. 10s (2025) 

https://internationalpubls.com 111



                                    −ƈ𝐹′(𝜉) − 𝐹′′(𝜉) − 𝐴𝐹′(𝜉)𝐹𝑝(𝜉) = 0 ………..,                (26) 

integrate Eq. (27), we get 

                                        −ƈ𝐹(𝜉) − 𝐹′(𝜉) − 𝐴
𝐹𝑝+1(𝜉)

𝑝+1
= 0 ……..,                           (27) 

The non-linear terms and highest order derivatives identified in Eq. (26) that is not 

positive integers then using the homogeneous balance approach, we obtain the positive 

integer 𝑚 =
1 

𝑝
 . Considering the solution in the form as 

                                                     𝐹(𝜉) = 𝐴 (
𝐺′

𝐺
)

1

𝑝
, 𝑝 > 0 ………,            (28) 

Where A is constant to be find out.  

Now Eq. (28) becomes 

−ƈ𝐴𝑋(𝜉)
1

𝑝 +
𝐴𝑢𝑋(𝜉)

1
𝑝

𝑃𝑋(𝜉)
+
𝐴𝜆

𝑝
𝑋(𝜉)

1

𝑝 +
𝐴𝜆

𝑝
𝑋(𝜉)

1

𝑝
+1
−
𝐴𝑝+2

𝑝+1
𝑋(𝜉)

1

𝑝
+1
= 0………,          (29)    

Where 𝑋(𝜉) =
𝐺′

𝐺
 , After collecting the coefficient of  𝑋(𝜉)

1

𝑝, 𝑋(𝜉)
1

𝑝
+1

and 𝑋(𝜉)
1

𝑝
−1

 ,we 

get 𝐴 = (
𝑝+1

𝑝
)

1

𝑝+1
, 𝑢 = 0 𝑎𝑛𝑑 𝜆 = 𝑝ƈ ……….,             (30) 

 

Determined wave solutions of Eq. (30) as 

𝐹(𝜉) = (
𝑝+1

𝑝
)

1

𝑝+1
(

𝜆𝑒−𝜆𝜉

𝑐1𝑒
−𝜆𝜉+𝑐2

)

1

𝑝+1
where 𝜉 = 𝑥 −  ƈ

ƫᾱ

ᾱ
 

The exact solution of Eq. (1) is determined as 

𝐹(𝜉) = (
𝑝 + 1

𝑝
)

1
𝑝+1

(

 
𝜆𝑒

−𝜆(𝑥− ƈ
ƫᾱ

ᾱ)

𝑐1𝑒
−𝜆(𝑥− ƈ

ƫᾱ

ᾱ
)
+ 𝑐2)

 

1
𝑝+1

 

 

 

Communications on Applied Nonlinear Analysis 
ISSN: 1074-133X 
Vol 32 No. 10s (2025) 

https://internationalpubls.com 112



8. Conclusion: 

The conformable derivatives have been utilized to interpret the NLFPDEs generalized 

KdV equation. The symmetry aspects of KdV equation are studied by Lie group 

analysis methodology. Afterwards, vector fields of KdV equation are discussed on the 

basis of the point symmetry. Also, the symmetry reductions are constructed. 

Additionally, the paper shows exact traveling wave solutions of nonlinear fractional pth 

order Korteweg-de Vries equation that have been obtained by using the (
𝐺′

𝐺
)-expansion 

method.  At last, the generalized fractional KdV equation series solution and several 

explicit and exact solutions are established. The results of this paper provide practical 

and significant insights for future research. Additionally, the exact solutions we have 

derived may be useful across different areas of applied mathematics, particularly for 

analyzing specific physical phenomena. 

 

 

References: 

1. Y.Keskin and G.Oturanc, “Reduced differential transform method for partial 

differential equations,” International Journal of Nonlinear Sciences and 

Numerical Stimulation, vol. 10, pp. 741–750, 2009. 

2. H.Khan, A.Khan, P.Kumam, D.Baleanu, and M.Arif, “An approximate 

analytical solution of the Navier-Stokes equations within Caputo operator and 

Elzaki transform decomposition method,” Advances in Difference Equations, 

vol. 622, pp. 1–23, 2020. 

3. H.Khan, A.Khan, M.Al-Qurashi, R.Shah and D.Baleanu, “Modified modelling 

for heat like equations within Caputo operator,” Energies, vol. 13, pp. 2002, 

2020. 

4. D.J.Evans and K.R.Raslan, “The Adomian decomposition method for solving 

delay differential equation,” International Journal of Computer Mathematics, 

vol. 82, pp. 49–54, 2005. 

Communications on Applied Nonlinear Analysis 
ISSN: 1074-133X 
Vol 32 No. 10s (2025) 

https://internationalpubls.com 113



5. G.W.Bluman and J.D.Cole, “Similarity Methods for Differential Equations,” 

In Applied Mathematical Science, Springer-Verlag, New York, 1974. 

6.  G.W. Bluman and J.D. Cole, “The General Similarity Solution of the Heat 

Equation,” Journal of Mathematics and Mechanics, vol. 18, pp. 1025–1042, 

1969. 

7. P.J. Olver, “Applications of Lie groups to Differential Equations,” Graduate 

Texts in Mathematics, Springer-Verlag, New York, vol. 107, 1986. 

8. J. B. Bell, P. Colella, and H. M. Glaz, “A second-order projection method for 

the incompressible Navier-stokes equations,” Journal of Computational 

Physics, vol. 85, no. 2, pp. 257–283, 1989.  

9. W. Malfliet, “The tanh method: a tool for solving certain classes of nonlinear 

evolution and wave equations,” Journal of Computational and Applied 

Mathematics, vol. 164-165, pp. 529–541, 2004.  

10.  M. Kapoor and V. Joshi, “Solution of Non-linear fisher’s Reaction-Diffusion 

Equation by Using Hyperbolic B-Spline Based Differential Quadrature 

Method,” Journal of Physics: Conference Series, vol. 1531, no. 1, p. 012064, 

2020.  

11.  A. T. Khan and H. Naher, “Solitons and periodic solutions of the fisher 

equation with nonlinear ordinary differential 

equationasauxiliaryequation,”AmericanJournalofAppliedMathematics and 

Statistics, vol. 6, no. 6, pp. 244–252, 2018. 

12.  L. Jin, “Application of variational iteration method to the fifth-order KdV 

equation,” International Journal of Contemporary Mathematical Sciences, vol. 

3, pp. 213-221, 2008. 

13.  D. Baldwin, U. Goktas, W. Hereman, L. Hong, R.S. Martino, J.C. Miller, 

“Symbolic computation of exact solutions in hyperbolic and elliptic functions 

for nonlinear PDEs,” Journal of Symbolic Computation. Vol. 37, pp. 669-705, 

2004. 

14.  B.A. Kupershmidt, “A super KdV equation: an integrable system,” 

PhysicsLetters A, vol.102, pp. 213-215, 1984. 

15.  R.R. Long, “Solitary waves in one- and two-fluids systems,” Tellus, vol. 8, 

pp. 460-471, 1956. 

16.  D.J. Benney, “Long nonlinear waves in fluid flows,” Journal of Mathematics 

and Physics,vol. 45, pp. 52-63, 1966. 

Communications on Applied Nonlinear Analysis 
ISSN: 1074-133X 
Vol 32 No. 10s (2025) 

https://internationalpubls.com 114



17. T. Kawahara, “Oscillatory solitary waves in dispersive media,” Journal of 

Physical Society of Japan, vol. 33, pp. 260-264, 1972. 

18. D. Kaya and S. M. EL Sayad, “An application of the decomposition method 

for the generalized KdV and RLW equations,” Chaos, Solitons & Fractals , 

vol. 17, pp. 869-877, 2003. 

19.  D.D. Bhatta and M.I. Bhatti, “Numerical solution of Kdv equation using 

modified Bernstein polynomials,” Computers & Mathematics with 

Applications, vol. 174, pp. 1255-1268, 2006. 

20.  A.R. Seadawy , “New exact solutions for the Kdv equation with higher order 

nonlinearity by using the variational method,” Computers & Mathematics with 

Applications, vol. 62, pp. 3741-3755, 2011. 

21.  Aly R. Seadawy, Dianchen Lu & Chen Yue , “Travelling wave solutions of 

the generalized nonlinear fifth-order KdV water wave equations and its 

stability,” Journal of Taibah University for Science ,vol.11, pp. 623-633, 2017. 

22.  G. Wang, “Symmetry analysis, analytical solutions and conservation laws of a 

generalized KdV–burgers–kuramoto equation and its fractional version,” world 

scientific, vol. 29, 2021. 

23.  A.A. Alderremy, Shaban Aly, Rabia Fayyaz, Adnan Khan,Rasool Shah, 

and  Noorolhuda Wyal , “The Analysis of Fractional-Order Nonlinear Systems 

of Third Order KdV and Burgers Equations via a Novel Transform,” 

Applications of Delay Differential Equations in Biological Systems 

2021,vol.22, pp.1-24, 2022. 

24.  H. Rezazadeh, A.G. Davodi and D. Gholami, “Combined formal periodic 

wave-like and soliton-like solutions of the conformable Schrödinger-KdV 

equation using the G′/G-expansion technique,” Results in Physics, vol. 47, 

2023. 

25.  R.Khalil, M.A.Horani, A.Yousef, and M.Sababheh, “A new definition of 

fractional derivative,” Journal of Computational and Applied Mathematics, 

vol. 264, pp. 65–70, 2014.  

26.  A. Atangana, D. Baleanu, and A. Alsaedi, “New properties of conformable 

derivative,” Open Mathematics, vol. 13, pp. 889–898, 2015.  

27.  R.Kumar, A.Bansal and S.Saini,“Traveling wave solutions and bifurcation 

analysis of Chaffee–infante equation,” In Proceedings of International 

Communications on Applied Nonlinear Analysis 
ISSN: 1074-133X 
Vol 32 No. 10s (2025) 

https://internationalpubls.com 115



Conference on Trends In Computational and Cognitive Engineering, TCCE 

2019, pp. 153–161, 2020.  

28. R.Kumar, R.K.Gupta and S.S.Bhatia, “The New Generalized (
𝐺′

𝐺
)-Expansion

Method For Solving (2+1)-Dimensional PKP Equation,” International Journal

of Nonlinear Science, vol. 14, pp. 48-52, 2012.

29. A.A.Tayyan and A.H.Sakka, “Symmetries and exact solutions of conformable

fractional partial differential equations,” Palestine Journal of Mathematics, vol.

9, pp. 300–311, 2020.

30. O.P. Bhutani, G. Chandrasekaran and P. Mittal, “On the Exact Solutions of

Nonlinear Differential Equations of Nonlinear Engineering System via

Invariant Variational Principles,” International Journal of Engineering Science,

vol. 26, pp. 243–248, 1988.

31. R.Kumar, R.Kumar, A.Bansal, A.Biswas, Y.Yildirim, S.P.Moshokoa and

A.Asiri, “Optical Solitons and Group Invariants for Chen-Lee-Liu Equation

with Time-Dependent Chromatic Dispersion and Nonlinearity by Lie

Symmetry,” Ukrainian Journal of Physics Optics, vol. 24, pp. 04021-04021,

2023.

32. O. S. Iyiola and E. R. Nwaeze, “Some new results on the new conformable

fractional calculus with application using D’Alambert approach,” Progress In

Fractional Differentiations And Applications, vol. 2, pp. 115–122, 2016.

33. M. Z. Sarikaya, “Gronwall Type Inequality for Confromable Fractional

Integrals, RGMITA Research Report Collection,” 2016.

34. F. Usta and M. Z. Sarikaya, “On Generalization Conformable Fractional

Integral Inequalities, RGMITA Research Report Collection,” 2016.

35. P. Michal and P. Skripkova, “Sturm’s theorems for conformable fractional

differential equations,” Mathematical Communications, vol. 21, pp. 273–281,

2016.

36. U.Fuat and Z.S.Mehmet, “Explicit bounds on certain integral inequalities via

conformable fractional calculus,” Coget Mathematics, vol. 4, no. 1, article

1277505, 2017.

37. B. A. Tayyan and A. H. Sakka, “Symmetries and exact solutions of

conformable fractional partial differential equations,” Palestine Journal of

Mathematics, vol. 9, pp. 300–311, 2020.

38. M.A.Hammad and R.Khalil,“Abel’s formula for conformable fractional

differential equations,” International Journal of Differential Equations and

Applications, vol. 13, pp. 177–183, 2014.

39. T. Abdeljawad, M. A. Horani, and R. Khalil, “Conformable fractional

semigroups of operators,” Journal of Semigroup and Applications, 2015.

40. R.Kumar, R.K.Gupta and S.S.Bhatia, “The New Generalized (
𝐺′
)-Expansion

𝐺

Method For Solving (2+1)-Dimensional PKP Equation,” International Journal

of Nonlinear Science, vol. 14, pp. 48-52, 2012.

Communications on Applied Nonlinear Analysis 
ISSN: 1074-133X 
Vol 32 No. 10s (2025) 

https://internationalpubls.com 116