Research ArticleLie Symmetry Analysis, Exact Solutions, and ConservationLaws for the Generalized Time-Fractional KdV-Like Equation

Maria Ihsane El Bahi and Khalid Hilal

Sultan Moulay Sliman University, Bp 523, 23000 Beni Mellal, Morocco

Correspondence should be addressed to Maria Ihsane El Bahi; m.bahi@usms.ma

Received 9 December 2020; Revised 21 December 2020; Accepted 26 December 2020; Published 7 January 2021

Academic Editor: Ismat Beg

Copyright © 2021 Maria Ihsane El Bahi and Khalid Hilal. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

In this paper, the problem of constructing the Lie point symmetries group of the nonlinear partial differential equation appeared inmathematical physics known as the generalized KdV-Like equation is discussed. By using the Lie symmetry method for thegeneralized KdV-Like equation, the point symmetry operators are constructed and are used to reduce the equation to anotherfractional ordinary differential equation based on Erdélyi-Kober differential operator. The symmetries of this equation are alsoused to construct the conservation Laws by applying the new conservation theorem introduced by Ibragimov. Furthermore,another type of solutions is given by means of power series method and the convergence of the solutions is provided; also, somegraphics of solutions are plotted in 3D.

1. Introduction

Fractional calculus is the generalization of the ordinary dif-ferentiation and integration to noninteger (arbitrary) order;the subject is as old as calculus of differentiation and integra-tion and goes back to the time when Leibniz, Newton, andGauss invented this kind of calculation. Additionally, it isconsidered as one of the most interesting topics in severalfields, especially mathematics and physics, due to its applica-tions in modelization of physical process related to its histor-ical states (nonlocal property), which can be effectivelydescribed by using the theory of derivatives and integrals offractional order, that make the models described by frac-tional order more realistic than those described by integerorder. The most important part of fractional calculus isdevoted to the fractional differential equations (FDEs); inthe literature, there are diverse definitions for fractionalderivative including Riemann-Liouville derivative, Caputoderivative, and Conformable derivative, but the most popularone is Riemann-Liouville derivative. The fractional derivativehas attracted the attention of many researchers in differentareas such as viscoelasticity, vibration, economic, biology,and fluid mechanics (see [1–11]). Unfortunately, it is almost

difficult to solving and detecting all solutions of nonlinearpartial differential equations (PDEs) which renders it a chal-lenging problem, because of this, an interesting advance hasbeen made, and some methods for solving this type of equa-tions have been discussed; among them are subequationmethod, homotopy perturbation method, the first integralmethod, and Lie group method (see [12–18]). Lie symmetryanalysis was introduced by Sophus Lie (1842, 1899), a Nor-wegian mathematician who made significant contributionsto the theories of algebraic invariants and differential equa-tions. It can be said that the Lie symmetry method is the mostimportant approach for constructing analytical solutions ofnonlinear PDEs. It is based to study the invariance of differ-ential equations (DEs) under a one-parameter group oftransformations which transforms a solution to anothernew solution and is also used to reduce the order such asthe number of variables of DEs; moreover, the conservationlaws (CLs) can be constructed by using the symmetries ofthe DEs (see [19–24]). A short time ago, the Lie symmetryanalysis is also used for FDEs; in [25], Gazizov et al. showedus how the prolongation formulae for fractional derivatives isformulated; by this work, the Lie group method becomesValid for FDEs; after that, many researches are devoted for

HindawiJournal of Function SpacesVolume 2021, Article ID 6628130, 10 pageshttps://doi.org/10.1155/2021/6628130

studying the FDEs by using Lie symmetry analysis method,for more details see ([26–29]). The CLs are very important,are a mathematical description of the statement that the totalamount of a certain physical quantity including such energy,linear momentum, angular momentum, and charge, andremains unchanged during the evolution of a physical sys-tem. It is also the first step towards finding a solution; fur-thermore, the concept of integrability will be possible if theequation has conservation laws, and the strategy of con-structing the CLs of FPDEs is given by the combination oftwo works of Ibragimov [30] and Lukashchuk et al. [31]and has shown how the CLs of FDEs can be constructed eventhose equations without fractional Lagrangians. For the firsttime in 1895, the Korteweg-de Vries (in short words KdV)equation

∂u∂t

+ 6uux + uxxx = 0, ð1Þ

emerged as an evolution equation representing the waves ofsurface gravity propagation in a water shallow canal (see,[32]) and largely used by the mathematicians and physiciststo model a variety of different physical phenomena as hydro-magnetic collision-free waves, ion-acoustic waves, acousticsolitons in plasmas, lattice dynamics, stratified waves inte-rior, internal gravity waves, and so on (see, e.g., [33–36]).Such equations have been studied extensively, especially, forthe soliton solutions, solitary wave solutions, and periodicwave solutions. In order to find other properties that can bedifficult for the standard type or for finding some solutionsin common, a different equation is founded known as KdV-Like equations, for more details (see [37–42]). In [43], aninquiry was undertaken to increase the reliability and preci-sion of a genetic programming-based method to deducemodel equations from a proven analytical solution, especiallyby using the solitary wave solution; the program, instead ofgiving the Eq.(1), surprisingly gave the following generalizedKdV-Like equation

∂u∂t

+ 3 1 − δð Þu + δ + 1ð Þ uxxu

� �ux − δuxxx = 0: ð2Þ

The benefit of using fractional derivatives in Eq.(2) whilemodeling the real-world problems is the nonlocal property,and this implies that the next state of the system relies notonly upon its present state but also upon all of its historicalstates (see [44, 45]). Because of that, we will study in thispaper the generalized KdV-Like equation with the fractionalorder derivative presented as follows:

∂αu∂tα

+ 3 1 − δð Þu + δ + 1ð Þ uxxu

� �ux − δuxxx = 0, 0 < α ≤ 1,

ð3Þ

where ∂αt u is the Riemann-Liouville (R-L) fractional deriva-tive of order α with respect to t and δ is an arbitrary constant.For α = 1, we have two cases; the first is δ = −1; here, the Eq.(3) is reduced to the classical KdV equation which are con-sidered by many authors (see [46–49]), and the second case

is δ = 1, so the Eq. (3) is reduced to the classical KdV-Likeequation which are studied by using Lie symmetry methodand other approaches in [37, 38, 43]. The paper is structuredas follows. In Sec. 2, we present some main results of Lie sym-metries analysis for FPDEs in the general form. In Sec. 3, weconstruct the Lie point symmetries and similarity reductionof generalized fractional KdV-Like equation. By means ofIbragimov’s theorem, the conservation laws of Eq. (3) aregiven in Sec. 4. In Sec. 5, we suggest an extra type of solutionsin the form of power series by using the power series method.Finally, some conclusions are given in Section 6.

2. Some Main Results in Lie Symmetry Analysis

In this section, we present a brief introduction of Lie symme-try analysis for fractional partial differential equations(FPDEs), and we will give some results which will be usedthroughout this study, so let us consider a general form ofthe FPDE introduced by

∂αu x, tð Þ∂tα

= F x, t, u, ux, uxx, uxxxð Þ, ð4Þ

where the subscripts indicate the partial derivatives and ∂α/∂tα is R-L fractional derivative operator given by

∂α

∂tαf x, tð Þ =

∂n

∂tnf x, tð Þ, if α = n,

1Γ n − αð Þ

∂n

∂tn

ðt0t − sð Þn−α−1 f x, sð Þds, 0 ≤ n − 1 < α < n,

8>>><>>>:

n ∈ℕ∗:

ð5Þ

Now, assuming that Eq. (4) is invariant under the follow-ing one-parameter Lie group of transformations expressed as

t = t + ετ x, t, uð Þ +O εð Þ, ð6Þ

x = x + εξ x, t, uð Þ +O εð Þ, ð7Þu = u + εη x, t, uð Þ +O εð Þ, ð8Þ

uαt = uαt + εηαt x, t, uð Þ +O εð Þ, ð9Þux = ux + εηx x, t, uð Þ +O εð Þ, ð10Þuxx = uxx + εηxx x, t, uð Þ +O εð Þ, ð11Þuxxx = uxxx + εηxxx x, t, uð Þ +O εð Þ, ð12Þ

where ε is the group parameter and ξ, τ, η are the infinitesi-mals and their corresponding extended infinitesimals of order1, 2, and 3 are the functions ηx, ηxx, ηxxx, and α presented by

ηx =Dx ηð Þ − uxDx ξð Þ − utDx τð Þ,ηxx =Dx ηxð Þ − uxxDx ξð Þ − uxtDx τð Þ,ηxxx =Dx ηxxð Þ − uxxxDx ξð Þ − uxxtDx τð Þ,ηαt =Dα

t ηð Þ + ξDαt uxð Þ −Dα

t ξuxð Þ +Dαt Dt τð Þuð Þ

−Dα−1t τuð Þ + τDα+1

t uð Þ,

ð13Þ

2 Journal of Function Spaces

whereDx is the total derivative operator with respect to x writ-ten as

Dx =∂∂x

+ ux∂∂u

+ uxx∂∂ux

+ utx∂∂ut

+⋯, ð14Þ

the explicit form of the extended infinitesimal ηαt of order α isgiven by

ηαt = ∂αt ηð Þ + ηu − αDt τð Þð Þ∂αt u − u∂αt ηuð Þ + μ

+ 〠∞

n=1

α

n

!∂nt ηu −

α

n + 1

!Dn+1t τð Þ

" #∂α−nt u

− 〠∞

n=1

α

n

!Dnt ξð Þ∂α−nt uxð Þ,

ð15Þ

where Dnt is the operator of differentiation of integer order n

and

μ = 〠∞

n=2〠n

m=2〠m

k=2〠k−1

r=0

α

n

!n

m

!r

k

!1k!

tn−α

Γ n + 1 − αð Þ

× −u½ �r dm

dtmuk−rh i ∂n−m+kηu

∂tn−m∂uk:

ð16Þ

The expression of μ vanishes when the infinitesimal ηðx,t, uÞ is linear in the variable u, which means

η x, t, uð Þ = u x, tð Þf x, tð Þ + h x, tð Þ⇒ μ = 0: ð17Þ

The generator of the one-parameter Lie group (6) or theinfinitesimal operator is the differential operator defined as

X = τ x, t, uð Þ ∂∂t + ξ x, t, uð Þ ∂∂x

+ η x, t, uð Þ ∂∂u

: ð18Þ

The corresponding prolonged generator Xðα,3Þ of order(α, 3) is

X α,3ð Þ = X + ηαt∂∂αt u

+ ηx∂∂ux

+ ηxx∂

∂uxx+ ηxxx

∂∂uxxx

: ð19Þ

Theorem 1 (Infinitesimal criterion of invariance). The vectorfield X = τðx, t, uÞð∂/∂tÞ + ξðx, t, uÞð∂/∂xÞ + ηðx, t, uÞð∂/∂uÞ,is a point symmetry of Eq. (4) if

Xα,3 Δð Þ��Δ=0ð Þ = 0, Δ = ∂αt u − F: ð20Þ

Remark 2 (Invariance condition). In the Eq. (4), the lowerlimit of the integral must be invariant under (6), which means

X tð Þj t=0ð Þ = 0⇒ τ x, t, uð Þjt=0 = 0: ð21Þ

Definition 3. A solution u = f ðx, tÞ is an invariant solution of(4) if satisfies the following conditions

(i) u = f ðx, tÞ is an invariant surface of (18), which isequivalent to

Xf = 0⇒ τ x, t, uð Þ ∂∂t + ξ x, t, uð Þ ∂∂x

+ η x, t, uð Þ ∂∂u

� �f = 0,

ð22Þ

(ii) u = f ðx, tÞ satisfies Eq. (4)

3. Lie Symmetry Analysis for the GeneralizedKdV-Like Equation

In this section, the Lie symmetry group of the generalizedKdV-Like equation is performed.

The symmetry group of Eq. (3) is generated by the vectorfield (18), so applying the third prolongation Xðα,3Þ to the Eq.(3), we obtain the infinitesimal criterion of invariance corre-sponding Eq. (3), expressed as

ηαt + η 3 1 − δð Þ − δ + 1ð Þ uxxu2

h iux + ηx 3 1 − δð Þu + δ + 1ð Þ uxx

u

h i+ ηxx δ + 1ð Þ ux

u− ηxxxδ = 0:

ð23Þ

Substituting the explicit expressions ηx, ηxx, ηxxx , and ηαtinto (23) and equating powers of derivatives up to zero, weget an overdetermined system of linear partial differentialequations; after resolving this system, the infinitesimals func-tions are given by

τ t, x, uð Þ = 3C1t,ξ t, x, uð Þ = C1αx + C2,η t, x, uð Þ = −2C1αu,

ð24Þ

where C1, C2 are arbitrary constants. The corresponding Liealgebra is given by

X = C1αx − C2ð Þ ∂∂x

+ 3C1t∂∂t

− 2C1αu∂∂u

: ð25Þ

If we set

X1 =∂∂x

, X2 = 3t ∂∂t

+ αx∂∂x

− 2αu ∂∂u

, ð26Þ

we can see that the vector fields X1, X2 is closed under Liebracket defined by ½Xi, Xj� = XiXj − XjXi; therefore, the Liealgebra X is generated by the vectors fields Xi (1,2), whichmeans

X = C1X1 + C2X2: ð27Þ

3Journal of Function Spaces

Now, by solving the following characteristic equation

dtτ x, t, uð Þ = dx

ξ x, t, uð Þ = duη x, t, uð Þ , ð28Þ

corresponding X1 and X2, we can obtain the reduced formsof generalized KdV-Like equation in the form of FODE.

Case 1. Reduction with X1 = ∂/∂x :solving the characteristic equation

dt0 = dx

1 = du0 : ð29Þ

Leads to the following similarity variable and similaritytransformation

ζ = t, u = f tð Þ, ð30Þ

with f ðtÞ satisfies

Dαt f tð Þ = 0: ð31Þ

Therefore, the group invariant solution corresponding to X1,is presented as

u1 = k1tα−1, ð32Þ

where k1 is an arbitrary constant. Figure 1 presents the graphof solution u1ðx, tÞ for some different values of α,

Case 2. Reduction with X2 = 3tð∂/∂tÞ + αxð∂/∂xÞ − 2αuð∂/∂uÞ:

The similarity variable ζ and similarity transformationf ðζÞ corresponding to the infinitesimal generator X2 isobtained by solving the associated characteristic equationgiven by

dt3t =

dxαx

= du−2αu : ð33Þ

Thus,

ζ = xt−α3 , f ζð Þ = t

2α3 u: ð34Þ

Therefore,

u = t−2α3 f xt

−α3

� �: ð35Þ

Theorem 3. By using the similarity transformation (34) in(3), the generalized KdV-Like equation is transformed intoa non-linear FODE given by

P−5α3 +1,α

3α

f� �

ζð Þ + 3 1 − δð Þf + δ + 1ð Þ f ζζf

� �f ζ − δf ζζζ = 0,

ð36Þ

with ðPδ,αλ f ÞðζÞ is the Erdélyi-Kober differential operator

given by

Pδ,αλ f

� �ζð Þ =

Yn−1i=0

δ + i −1λζddζ

� �Kδ+α,n−α f� �

ζð Þ,

m =α½ � + 1, α ∉ℕ,α, α ∈ℕ,

( ð37Þ

where Kδ,αλ is Erdélyi-Kober fractional integral operator

introduced by

Kδ,αλ f

� �ζð Þ = 1

Γ αð Þð∞1

z − 1ð Þα−1z− δ+αð Þ f ζz1λ

� �dz, α > 0,

f ζð Þ, α = 0:

8><>:

ð38Þ

Proof. By using the Riemann-Liouville fractional derivativedefinition for the similarity transformation

u = t−2α3 f xt

−α3

� �, ð39Þ

we have

∂αu∂tα

= ∂n

∂tn1

Γ n − αð Þðt0t − sð Þn−α−1s−2α3 f xs

−α3

� �ds

� �: ð40Þ

Let v = t/s, we have ds = −t/v2, so the above expressioncan be expressed as

2

4

6

8

10

12

14

0 0.2

𝛼 = 0.9𝛼 = 0.75

𝛼 = 0.5𝛼 = 0.25

0.4 0.6t

0.8 1

Figure 1: The solution u1ðx, tÞ for Eq. (1) for K1 = 1 and (α = 0:9,α = 0:75, α = 0:5, α = 0:25).

4 Journal of Function Spaces

∂αu∂tα

= ∂n

∂tntn−

5α3

1Γ n − αð Þ

ð∞1

v − 1ð Þn−α−1v− n+1−5α3ð Þ f ζv

α3

� �dv

� �,

= ∂n

∂tntn−

5α3 K

1−2α3 ,n−α

3α

f� �

ζð Þh i

:

ð41Þ

We have

t∂∂t

ϕ ζð Þ = −txα

3 t−α3 −1ϕ′ ζð Þ = −

α

3 ζϕ′ ζð Þ: ð42Þ

Then,

∂nu∂tn

= ∂n−1

∂tn−1∂∂t

tn−5α3 K

1−2α3 ,n−α

3α

f� �

ζð Þ� �� �

,

= ∂n−1

∂tn−1tn−

5α2 −1 n −

5α3 −

α

3 ζ∂∂ζ

� �K

1−2α3 ,n−α

3α

f� �

ζð ÞÞ� �

:

ð43Þ

By repeating this procedure n − 1 times, we get

∂nu∂tn

= ∂n−1

∂tn−1tn−

5α2 −1 n −

5α3 −

α

3 ζ∂∂ζ

� �K

1−2α3 ,n−α

3α

f� �

ζð ÞÞ� �

,

⋮

= t−5α3Yn−1j=0

1 − 5α3 + j −

α

3 ζ∂∂z

� �K

1−2α3 ,n−α

3α

f� �

ζð Þ,

= t−5α3 P

−5α3 +1,α

3α

f� �

ζð Þ:ð44Þ

Continuing further by calculating ux , uxx, and uxxx for(35) and replacing in Eq. (3), we find that time-fractional generalized KdV-Like equation is reduced intoa FODE given by

P−5α3 +1,α

3α

f� �

ζð Þ + 3 1 − δð Þf + δ + 1ð Þ f ζζf

� �f ζ − δf ζζζ = 0:

ð45Þ

So the proof becomes complete.

4. Conservation Laws

In this section, based on its Lie point symmetry and by usingthe new conservation theorem [30], the conservation laws ofthe generalized KdV-Like equation are constructed.

All vectors C = ðCt , CxÞ verify the following conservationequation

Dt Ctð Þ +Dx Cxð Þ = 0: ð46Þ

In all solutions of Eq. (3), it is called conservation law of Eq.(3), with Dt and Dx are the total derivative operators withrespect to t and x, respectively.

The formal Lagrangian of Eq. (3) is given by

L = θ∂αu∂tα

+ 3 1 − δð Þu + δ + 1ð Þ uxxu

� �ux − δuxxx

� �, ð47Þ

where θðx, tÞ is a new dependent variable. The adjoint equa-tion of generalized KdV-Like equation is written as

δL

δu= 0, ð48Þ

where δ/δu is the Euler-Lagrange operator defined as

δ

δu= ∂∂u

+ Dαtð Þ∗ ∂

∂ Dαt uð Þ −Dx

∂∂ uxð Þ +D2

x∂

∂ uxxð Þ −D3x

∂∂ uxxxð Þ+⋯,

ð49Þ

ðDαt Þ∗ is the adjoint operator of Dα

t

Dαtð Þ∗ = −1ð ÞnPn−α

T DnTð Þ= t

cDαt , ð50Þ

with

Pn−αt f x, tð Þ = 1

Γ n − αð ÞðSt

f x, sð Þs − tð Þ1+α−n ds, n = α½ � + 1: ð51Þ

and ðDαt Þ∗ is the right-side Caputo operator. Now, the con-

struction of CLs for FPDEs is similar to PDEs. So, the funda-mental Neoter identity is given by

X α,3ð Þ +Dt τð Þ +Dx ξð Þ =Wiδ

δu+Dt Ntð Þ +Dx Nxð Þ, ð52Þ

where Nx,Nt are Neoter operators, Xðα,3Þ is given by (19),and Wi are the characteristic functions written as

Wi = ηi − τiut − ξiux: ð53Þ

The x-component of Neoter operator is introduced byNx

Nx = ξ +Wi∂∂ux

−Dx∂

∂uxx

� �+Dxx

∂∂uxxx

� �� �

+Dx Wið Þ ∂∂uxx

−Dx∂

∂uxxx

� �� �+Dxx Wið Þ ∂

∂uxxx:

ð54Þ

For R-L time-fractional derivative, Nt is determined by

Nt = τ + 〠m−1

k=0−1ð ÞkDα−1−k Wið ÞDk

t∂

∂Dαt u

− −1ð ÞmI Wi,Dmt

∂∂ Dα

tð Þ� �

,

ð55Þ

where I is defined as

I f , hð Þ = 1Γ m − αð Þ

ðt0

ðTt

f τ, xð Þh ϕ, xð Þϕ − τð Þα+1−m dϕdτ: ð56Þ

5Journal of Function Spaces

By applying (52) on (47) for Xið1, 2Þ and for all solutions,we deduce that Xðα,3Þ +DtðτÞL +DxðξÞL = 0, and δL/δu = 0;therefore,

Dt NtLð Þ +Dx NxLð Þ = 0, ð57Þ

Observing that (57) satisfies (46), so the components ofthe conserved vector rewritten as

Ct =NtL = τL + 〠m−1

k=0−1ð ÞkDα−1−k Wið ÞDk

t∂L∂Dα

t u

− −1ð ÞmI Wi,Dmt

∂L∂ Dα

tð Þ� �

,

= 〠m−1

k=0−1ð ÞkDα−1−k Wið ÞDk

t∂L∂Dα

t u

− −1ð ÞmI Wi,Dmt

∂L∂ Dα

tð Þ� �

,

Cx =NxL = ξL +Wi∂L∂ux

−Dx∂L∂uxx

� �+Dxx

∂L∂uxxx

� �� �

+Dx Wið Þ ∂L∂uxx

−Dx∂L∂uxxx

� �� �+Dxx Wið Þ ∂L

∂uxxx,

=Wi∂L∂ux

−Dx∂L∂uxx

� �+Dxx

∂L∂uxxx

� �� �

+Dx Wið Þ ∂L∂uxx

−Dx∂L∂uxxx

� �� �+Dxx Wið Þ ∂L

∂uxxx:

ð58Þ

Now, by using the definitions and the results describedabove, we can construct the conservation laws of Eq. (3).According to the previous section, the time-fractional gener-alized KdV-Like equation accepts two infinitesimal genera-tors defined in Section 3 by

X1 =∂∂x

, X2 = 3t ∂∂t

+ αx∂∂x

− 2αu ∂∂u

: ð59Þ

The characteristic functions corresponding of each gen-erator are presented by

W1 = −ux,W2 = −2αu − 3tut − αxux:

C1t = θDα−1

t −uxð Þ + I −ux, θtð Þ,

C1x = −ux 3 1 − δð Þu + uxx

u

� �θ −Dx

uxuθ

� �+Dxx −δθð Þ

h i−Dx uxð Þ ux

uθ −Dx −δθð Þ

h i,

+Dxx uxð Þ δθð Þ,

= δθxxux − 3 1 − δð Þθuux + θxu2xu

− θu3xu2

− δθxuxx

− θuxuxxu

+ δθuxxx:

C2t = θDα−1

t −2αu − 3tut − αxuxð Þ + I −2αu − 3tut − αxux, θtð Þ,

C2x = − 2αu + 3tut + αxuxð Þ

� 3 1 − δð Þu + uxxu

� �θ −Dx

uxuθ

� �+Dxx −δθð Þ

h i−Dx 2αu + 3tut + αxuxð Þ ux

uθ −Dx −δθð Þ

h i+Dxx 2αu + 3tut + αxuxð Þ δθð Þ,

= − 2αu + 3tut + αxuxð Þ 3 1 − δð Þθu − θxuxu

+ θu2xu2

− δθxx

� �

− 3αux + αxuxx + 3tuxtð Þ θuxu

+ δθx� �

+ 4αuxx + αxuxxx + 3tuxxtð Þ δθð Þ:ð60Þ

5. Power Series Solution

In this section, an additional exact solution is extracted byapplying the power series method; also, the convergence ofpower series solutions is proved.

Firstly, let us use the special fractional complex transfor-mation introduced by

u x, tð Þ = u ζð Þ, ζ = ax −btα

Γ 1 + αð Þ , ð61Þ

where a and b are two arbitrary constants, so substituting(61) into Eq. (3), we get

−buζ + 3 1 − δð Þu + δ + 1ð Þa2 uζζu

� �auζ − a3δuζζζ = 0: ð62Þ

Hence, (62) leads to

−buζu + 3 1 − δð Þu2 + δ + 1ð Þa2uζζ

auζ − a3δuuζζζ = 0:ð63Þ

Now, we integrate (63) with respect to ζ; we obtain

−12 bu

2 + a 1 − δð Þu3 + δ + 12

� �a3u2ζ − a3δuuζζ + c = 0,

ð64Þ

where c is an integration constant. Now, we seek a solution of(64) in power series form

u ζð Þ = 〠∞

n=0anζ

n = 〠∞

n=0an ax −

btα

Γ 1 + αð Þ� �n

: ð65Þ

6 Journal of Function Spaces

We have

uζ = 〠∞

n=0n + 1ð Þan+1ζn, ð66Þ

uζζ = 〠∞

n=0n + 2ð Þ n + 1ð Þan+2ζn: ð67Þ

Substituting (65), (66), and (67) into (64), we obtain

−12 b 〠

∞

n=0anζ

n

!2

+ a 1 − δð Þ 〠∞

n=0anζ

n

!3

+ δ + 12

� �a3

� 〠∞

n=0n + 1ð Þan+1ζn

!2

− δa3 〠∞

n=0anζ

n

!

� 〠∞

n=0n + 2ð Þ n + 1ð Þan+2ζn

!+ c = 0:

ð68Þ

Therefore,

−12 b〠

∞

n=0〠n

i=0an−iaiζ

n + a 1 − δð Þ〠∞

n=0〠n

i=0〠i

j=0an−iai−jajζ

n

+ δ + 12

� �a3 〠

∞

n=0〠n

i=0n − i + 1ð Þ i + 1ð Þan−i+1ai+1ζn

− δa3 〠∞

n=0〠n

i=0n − i + 2ð Þ n − i + 1ð Þan−i+2aiζn + c = 0:

ð69Þ

Comparing the coefficient of ζ, for n = 0, we obtain

a2 =−ba20 + 2a 1 − δð Þa30 + 2δ + 1ð Þa3a21 + 2c

4δa3a0: ð70Þ

The second case is when n ≥ 0

an+2 =1

δa3a0 n + 2ð Þ n + 1ð Þ −12 b〠

n

i=0an−iai

"

+ a 1 − δð Þ〠n

i=0〠i

j=0an−iai−jaj + δ + 1

2

� �a3 〠

n

i=0

� n − i + 1ð Þ i + 1ð Þan−i+1ai+1�, n = 1, 2:⋯

ð71Þ

Hence, the power series solution of Eq. (3) becomes

u ζð Þ = a0 + a1ζ + a2ζ2 + 〠

∞

n=1an+2ζ

n+2, = a0 + a1 ax −btα

Γ 1 + αð Þ� �

+ −ba20 + 2a 1 − δð Þa30 + 2δ + 1ð Þa3a21 + 2c4δa3a0

� �

� ax −btα

Γ 1 + αð Þ� �2

+ 〠∞

n=1

1δa3a0 n + 2ð Þ n + 1ð Þ�

� −12 b

�〠n

i=0an−iai + a 1 − δð Þ〠

n

i=0〠i

j=0an−iai−jaj

+ δ + 12

� �a3 〠

n

i=0n − i + 1ð Þ i + 1ð Þan−i+1ai+1

�

× ax −btα

Γ 1 + αð Þ� �n+2

,

ð72Þ

where a0 and a1 are the arbitrary constants, and by using (70)and (71), all the coefficients of sequence fang∞3 can be calcu-lated. Now, we can prove the convergence of the power seriessolution.

Observing that

an+2j j ≤ C 〠n

i=0an−iai + 〠

n

i=0〠i

j=0an−iai−jaj + 〠

n

i=0n − i + 1ð Þ i + 1ð Þan−i+1ai+1

" #,

ð73Þ

where C =max fj−b/2δa3a0j, jað1 − δÞ/δa3a0j, jð2δ + 1Þa3/2δa3a0jg. Now, by using another power series introduced by

M =M ζð Þ = 〠∞

n=0mnζ

n, ð74Þ

with

m0 = a0j j,m1 = a1j j,m2 = a2j j

mn+1 = C 〠n

i=0an−ij j aij j + 〠

n

i=0〠i

j=0an−ij j ai−j

�� ��aj��"

+ 〠n

i=0an−i+1j j ai+1j j

�, n = 1, 2,⋯,

ð75Þ

we can see easily that janj ≤mn, for n = 0, 1, 2,⋯; then,MðζÞ =∑∞

n=0mnζn is a majorant series of (65). It remains

to show that MðζÞ has a positive radius of convergence.

M ζð Þ =m0 +m1ζ +m2ζ2 + C 〠

∞

n=1〠n

i=1mn−imiζ

n+2"

+ 〠∞

n=1〠n

i=0〠i

j=0mn−imi−jmjζ

n+2 + 〠∞

n=1〠n

i=1mn−i+1mi+1ζ

n+2#,

7Journal of Function Spaces

=m0 +m1ζ +m2ζ2 + C M2 −m2

0

ζ2 + M3 −m30

ζ2

h+ M −m0ð Þ2 −m2

1ζ2i:

ð76Þ

After that, we define an implicit functional equation withrespect to the independent variable ζ by the followingform

R ζ, rð Þ = r −m0 −m1ζ −m2ζ2 − C M2 −m2

0

ζ2h

+ M3 −m30

ζ2 + M −m0ð Þ2 −m2

1ζ2i:

ð77Þ

Then, from (77), it is clear that Rðζ, rÞ is analytic in theneighborhood of ð0,m0Þ, with

R 0,m0ð Þ = 0, Rm′ 0,m0ð Þ = 1 ≠ 0: ð78Þ

2. × 10104. × 10106. × 10108. × 10101. × 1011

1.2 × 10111.4 × 10111.6 × 1011

0

0.040

–10

t x

(a) n = 6

4. × 1031

3. × 1031

2. × 1031

1. × 1031

0

0.040

–10

t x

(b) n = 20

Figure 3: The solution u2ðx, tÞ for Eq.(3) when a0 = 2, a1 = 3, δ = 3, c = 1, a = 2, b = 5.

3. × 1011

2. × 1011

1. × 1011

0

0

–10

0.04xt

(a) n = 6

3. × 1032

4. × 1032

2. × 1032

1. × 1032

0

0.040

–10

t x

(b) n = 20

Figure 2: The solution u2ðx, tÞ for Eq. (3) when a0 = 2, a1 = 3, δ = 3, c = 1, a = 2, b = 5:

8 Journal of Function Spaces

With the aid of the implicit function theorem given in [50,51], MðζÞ is analytic in a neighborhood of ð0,m0Þ with apositive radius; therefore, the power series M =MðζÞ =∑∞

n=0mnζn converges in a neighborhood of ð0,m0Þ; conse-

quently, (65) is convergent in a neighborhood of ð0,m0Þ,so the proof is complete.

Finally, the graphics of the exact power series solutions(72) are plotted in Figures 2 and 3 by choosing the appropri-ate parameters for different values of α.

Case 1: α = 0:25:Case 2: α = 0:75.

6. Conclusion

In this paper, Lie point symmetry and similarity reduction ofgeneralized KdV-Like equation are investigated by using theLie symmetry analysis. With the aid of similarity transforma-tion, the equation is reduced into a FODE with Erdélyi-Koberthe fractional differential operator. The CLs are also con-structed by using Ibragimov’s approach. Finally, by meansof the power series method, other kinds of solutions are pre-sented; moreover, we investigated the convergence analysisfor the obtained explicit solution, and we presented some ofthe 3D graphics of power series solutions. Our outcomesshow that the fractional-Lie symmetry analysis approachand the power series method provide the useful and powerfulmathematical tools to study other FDEs in mathematicalphysic and engineering. No more of that, the Lie analysis ofFODEs which are related to the Erdélyi-Kober is not yetexplored in the area of fractional calculus. Hence, it will beeligible as a future subject works.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

[1] R. L. Bagley and P. J. Torvik, “A theoretical basis for the appli-cation of fractional calculus to viscoelasticity,” Journal of Rhe-ology, vol. 27, no. 3, pp. 201–210, 1983.

[2] V. V. Kulish and J. L. Lage, “Application of fractional calculusto fluid mechanics,” Journal of Fluids Engineering, vol. 124,no. 3, pp. 803–806, 2002.

[3] K. B. Oldhan and F. Spanier, The Fractional Calculus, Aca-demic Press, New York, NY, USA, 1974.

[4] I. Podlubny, Fractional Differential Equationq, an Introductionto Fractional Derivatives, Fractional Differantial Equations,some Methods of Thier Applications, Academic Press, SanDiego, CA, USA, 1999.

[5] S. Samko, A. A. Kilbas, and O. Marichev, Fractional Integralsand Derivatives :Theory and Applications, Gordon and BreachScience, Yverdon, Switzerland, 1993.

[6] K. Diethemk, “Springer,” in The analysis of fractional differen-tial equations, Berlin, 2012.

[7] D. Baleanu, J. A. T. Machado, and A. C. J. Luo, FractionalDynamics and Control, Springer, New York, NY, USA, 2011.

[8] L. Debnath, “Recent applications of fractional calculus to sci-ence and engineering,” International Journal of Mathematicsand Mathematical Sciences, vol. 2003, no. 54, Article ID3301486, 3442 pages, 2003.

[9] K. Guida, K. Hilal, and L. Ibnelazyz, “Existence results for aclass of coupled Hilfer fractional pantograph differential equa-tions with nonlocal integral boundary value conditions,”Advances in Mathematical Physics, vol. 2020, Article ID8898292, 8 pages, 2020.

[10] K. Hilal, L. Ibnelazyz, K. Guida, and S. Melliani, “FractionalLangevin equations with nonseparated integral boundary con-ditions,” Advances in Mathematical Physics, vol. 2020, ArticleID 3173764, 8 pages, 2020.

[11] M. Bouaouid, K. Hilal, and S. Melliani, “Nonlocal telegraphequation in frame of the conformable time-fractional deriva-tive,” Advances in Mathematical Physics, vol. 2019, Article ID7528937, 7 pages, 2019.

[12] B. Ghazanfari and A. Sepahvandzadeh, “Adomian decomposi-tion method for solving fractional Bratutype equations,” Jour-nal of Mathematics and Computer Science, vol. 8, pp. 236–244,2014.

[13] V. S. Ertürk and S. Momani, “Solving systems of fractional dif-ferential equations using differential transform method,” Jour-nal of Computational and AppliedMathematics, vol. 215, no. 1,pp. 142–151, 2008.

[14] A. Arikoglu and I. Ozkol, “Solution of fractional differentialequations by using differential transform method,” Chaos Sol-itons Fractals, vol. 34, no. 5, pp. 1473–1481, 2007.

[15] X. Zhang, J. Zhao, J. Liu, and B. Tang, “Homotopy perturba-tion method for two dimensional time-fractional wave equa-tion,” Applied Mathematical Modelling, vol. 38, no. 23,pp. 5545–5552, 2014.

[16] S. Zhang, “Application of Exp-function method to a KdVequation with variable coefficients,” Physics Letters A,vol. 365, no. 5-6, pp. 448–453, 2007.

[17] S. Zhang and H. Q. Zhang, “Fractional sub-equation methodand its applications to nonlinear fractional PDEs,” Physics Let-ters A, vol. 375, no. 7, pp. 1069–1073, 2011.

[18] S. Guo, L. Mei, Y. Li, and Y. Sun, “The improved fractionalsub-equation method and its applications to the space-timefractional differential equations in fluid mechanics,” PhysicsLetters A, vol. 376, no. 4, pp. 407–411, 2012.

[19] P. J. Olver,Applications of Lie Groups to Differential Equations,Grad. Texts in Math., vol. 107, Springer, New York, NY, USA,1993.

[20] P. E. Hydon, Symmetry Methode for differential Aquations,Compridge University Press, UK, 2002.

[21] M. S. Osman, D. Baleanu, A. R. Adem, K. Hosseini,M. Mirzazadeh, and M. Eslami, “Double-wave solutions andLie symmetry analysis to the (2 + 1)-dimensional coupledBurg-ers equations,” Chinese Journal of Physics, vol. 63,pp. 122–129, 2020.

[22] Z. Zhao, “Conservation laws and nonlocally related systems ofthe Hunter-Saxton equation for liquid crystal,” Analysis andMathematical Physics, vol. 9, pp. 2311–2327, 2019.

[23] Z. Zhao, “Backlund transformations, rational solutions andsoliton-cnoidal wave solutions of the modified Kadomtsev-Petviashvili equation,” Applied Mathematics Letters, vol. 89,pp. 103–110, 2019.

9Journal of Function Spaces

[24] Z. Zhao and B. Han, “Residual symmetry, Bäcklund transfor-mation and CRE solvability of a (2++1)-dimensional nonlin-ear system,” Nonlinear Dynamics, vol. 94, pp. 461–474, 2018.

[25] R. K. Gazizov, A. A. Kasatkin, and S. Y. Lukashchuk, “Contin-uous transformation groups of fractional differential equa-tion,” Vestnik USATU, vol. 9, pp. 125–135, 2007.

[26] A. Hussain, S. Bano, I. Khan, D. Baleanu, and K. Sooppy Nisar,“Lie symmetry analysis, explicit solutions and conservationlaws of a spatially two-dimensional Burgers-Huxley equation,”Symmetry, vol. 12, no. 1, p. 170, 2020.

[27] A. Naderifard, S. R. Hejazi, and E. Dastranj, “Symmetry prop-erties, conservation Laws and exact solutions of time-fractional irrigation,” Wave in Rondom and Complex Media,vol. 29, no. 1, pp. 178–194, 2019.

[28] S. Rashidi and S. Reza Hejazi, “Lie symmetry approach for theVlasov-Maxwell system of equations,” Journal of Geometryand Physics, vol. 132, pp. 1–12, 2018.

[29] J.-G. Liu, X.-J. Yang, Y.-Y. Feng, and H.-Y. Zhang, “On thegeneralized time fractional diffusion equation: symmetry anal-ysis, conservation laws, optical system and exact solutions,”International Journal of Geometric Methods inModern Physics,vol. 17, no. 1, p. 2050013, 2019.

[30] N. H. Ibragimov, “A new conservation theorem,” Journal ofMathematical Analysis and Applications, vol. 333, no. 1,pp. 311–328, 2007.

[31] S. Y. Lukashchuk, “Conservation laws for time-fractional sub-diffusion and diffusion-wave equations,” Nonlinear Dynamics,vol. 80, no. 1-2, pp. 791–802, 2015.

[32] D. J. Korteweg and G. de Vries, “On the change of form of longwaves advancing in a rectangular canal and on a new type oflong stationary waves,” Philosophical Magazine, vol. 39,pp. 422–443, 1895.

[33] M. K. Fung, “KdV equation as an Euler-Poincaré equation,”Chinese Journal of Physics, vol. 35, pp. 789–796, 1997.

[34] D. J. Benney, “Long nonlinear waves in fluid flows,” Journal ofMathematics and Physics, vol. 45, no. 1-4, pp. 52–63, 1966.

[35] N. J. Zabusky and M. D. Kruskal, “Interaction of solitons in acollisionless plasma and the recurrence of initial states,” Phys-ical Review Letters, vol. 15, no. 6, pp. 240–243, 1965.

[36] H. Washimi and T. Taniuti, “Propagation of Ion-Acoustic sol-itary waves of small amplitude,” Physical Review Letters,vol. 17, no. 19, pp. 996–998, 1966.

[37] A. H. Abdel Kader, M. S. Abdel Latif, F. el Biaty, and A. Elsaid,“Symmetry analysis and some new exact solutions of somenonlinear KdV-Like equations,” Asian-European Journal ofMathematics, vol. 11, no. 3, article 1850040, 2018.

[38] A. M. Wazwaz, “Peakon and solitonic solutions for KdV-likeequations,” Physica Scripta, vol. 90, no. 4, p. 045203, 2015.

[39] N. Bildik, “On the solution of KdV-like equations by the opti-mal perturbation iteration technique,” International Journal ofApplied Physics and Mathematics, vol. 9, no. 1, pp. 29–37,2019.

[40] Y. Zhang and W.-X. Ma, “Rational solutions to a KdV-likeequation,” Applied Mathematics and Computation, vol. 256,pp. 252–256, 2015.

[41] D. Kaya, “On the solution of a korteweg-de vries like equationby the decomposition method,” International Journal of Com-puter Mathematics, vol. 72, no. 4, pp. 531–539, 1999.

[42] Z. Wang and X. Liu, “Bifurcations and exact traveling wavesolutions for the KdV-like equation,” Nonlinear Dynamics,vol. 95, no. 1, pp. 465–477, 2019.

[43] A. Sen, D. P. Ahalpara, A. Thyagaraja, and G. S. Krishnas-wami, “A KdV-like advection-dispersion equation with someremarkable properties,” Communications in Nonlinear Scienceand Numerical Simulation, vol. 17, no. 11, pp. 4115–4124,2012.

[44] X.-J. Yang, J. Hristov, H. M. Srivastava, and B. Ahmad,“Modelling fractal waves on shallow water surfaces via localfractional korteweg-de vries equation,” Abstract and AppliedAnalysis, vol. 2014, Article ID 278672, 10 pages, 2014.

[45] D. Kumar, J. Singh, S. Kumar, Sushila, and B. P. Singh,“Numerical computation of nonlinear shock wave equationof fractional order,” Ain Shams Engineering Journal, vol. 6,no. 2, pp. 605–611, 2015.

[46] D. Y. Chen, Introduction to Solitons, Science Press, Beijing,2006.

[47] A. Nakamura, “A direct method of calculating periodic wavesolutions to nonlinear evolution equations. II. Exact one- andtwo-periodic wave solution of the coupled bilinear equations,”Journal of the Physical Society of Japan, vol. 48, pp. 1365–1370,1980.

[48] Z. Zhu, “The soliton solutions of generalized KdV equation,”Acta Physica Sinica, vol. 41, pp. 1057–1062, 1992.

[49] W. Ma and D. Zhou, “On solitary wave solutions to a general-ized KdV equation,” Acta Physica Sinica, vol. 42, pp. 1731–1734, 1993.

[50] W. Rudin, Principles of Mathematical Analysis, ChinaMachine Press, Beijing, 3 edition, 2004.

[51] N. H. Asmar, Partial Differential Equations with Fourier Seriesand Boundary Value Problems, Machine Press, Beijing, China,2nd edition, 2005.

10 Journal of Function Spaces