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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; [email protected]
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.
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