# Lie Symmetry Analysis ofC1m,a,b Partial Differential Equations

Authors: Hengtai Wang; Aminu Ma’aruf Nass; Zhiwei Zou
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

In this article, we discussed the Lie symmetry analysis ofC1m,a,b fractional and integer order differential equations. The symmetry algebra of both differential equations is obtained and utilized to find the similarity reductions, invariant solutions, and conservation laws. In both cases, the symmetry algebra is of low dimensions.


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## Body

## 1. Introduction

In the last century, fractional partial differential equations (FPDEs) have played important rules in the fields of science and engineering, for instance, physics, chemistry, biology, andcontrol theory. Recently, those class of differential equations has also attracted much more interest of mathematicians and physicists [1–6].Finding the best methods of obtaining the exact solutions of differential equations remains one of the unanswered questions in the field. Many approaches have been developed by mathematicians to study the solutions of PFDEs, such as Adomian decomposition method, the fractional subequation method, numerical method, the first integral method, and Lie symmetry method [7–14]. In this article, we consider one of the powerful techniques of solving and analyzing differential equations, i.e., the Lie symmetry method. The Lie symmetry method is widely used to transformed partial differential equations (PDEs) into ordinary differential equations (ODEs), and the ODE is later solve numerically or analytically using similarity invariant [7, 9, 10, 12, 14–22]. Lie symmetry is also utilized in obtaining the conservation laws (Cls) [23]. The method developed by Noether theorem [24] and Ibraginov’s [25] is one of the best and simplest methods of evaluating Cls of differential equations.Consider general forms of fractional differential equations:(1)∂αu∂tα=Fu,where u=ux,t denotes the unknown function, and Fu=Fx,u,ux,uxx,⋯ is a known function. The fractional order α is a real number and ∂αu/∂tα denotes Riemann-Liouville(R-L) derivative defined in [1, 3, 26] as
(2)Dtαu=∂nu∂tn,α=n∈ℕ,1Γn−α∂n∂tn∫0tuτ,xt−τα+1−ndτ,n−1<a<n,n∈ℕ,where Γx denotes the standard gamma function defined by
(3)Γx=∫0∞e−ttx−1dt.TheCNm,a,b partial differential equations
(4)ut+umx+1bua∇2ubx=0,m>1,1<n=a+b,where N denotes the spatial dimension , have been introduced in [27]. In particular, if N=1, it becomes
(5)ut+umx+1buaubxxx=0,which is equivalent to
(6)ut+mum−1ux+b−1a+b−2ua+b−3ux3+a+3b−3ua+b−2uxuxx+ua+b−1uxxx=0.Therefore, the fractional form ofC1m,a,b differential equation is define as
(7)∂αu∂ta=−mum−1ux−b−1a+b−2ua+b−3ux3−a+3b−3ua+b−2uxuxx−ua+b−1uxxx,where ∂au/∂ta is given by (2).


## 2. Lie Symmetries of Eq. (6)

In this section, we first consider the Lie symmetry analysis of Eq.(6). To obtained the Lie symmetry analysis, we first consider a one-parameter Lie group of transformations
(8)x¯=x+εξx,t,u+Oε2,t¯=t+ετx,t,u+Oε2,u¯=u+εφx,t,u+Oε2,with a small parameter ε≪1. The vector field associated with the one-parameter group of transformation is
(9)V=ξx,t,u∂∂x+τx,t,u∂∂t+ϕx,t,u∂∂u.Thus, expanding the infinitesimals generator to include the transformation of the derivatives, we obtained the following third prolongationpr3V(10)pr3V=V+ϕx∂∂ux+ϕt∂∂ut+ϕxx∂∂uxx+ϕxxx∂∂uxxx.In (10), ϕx, ϕt, ϕxx, and ϕxxx are all undetermined functions, which are given by the following formulae
(11)ϕx=Dxϕ−ξux−τut+ξuxx+τuxt,(12)ϕt=Dtϕ−ξux−τut+ξuxt+τutt,(13)ϕxx=Dx2ϕ−ξux−τut+ξuxxx+τuxxt,(14)ϕxxx=Dx2ϕ−ξux−τut+ξuxxx+τuxxxt,where Dx and Dt represent the total derivatives with respect to x and t, respectively. e refers the reader(s) to [28] for details of how to evaluate the prolongation formulae.If the vector field (9) forms a symmetry of Eq.(6), the infinitesimal generator V must satisfy the following invariance criterion for Eq.(6), given as
(15)pr3V∆∆=0=0,where
(16)Δ=ut+mum−1ux+b−1a+b−2ua+b−3ux3+a+3b−3ua+b−2uxuxx+ua+b−1uxxx.Substituting Eqs. (11)–(14) into Eq. (15) and equating the coefficients of various powers of partial derivatives of u to zero, an overdetermined system of equations known as determining equations is obtained. Solving the determining equations the following infinitesimals for C1m,a,b has been derived
(17)ξx,t,u=pc1x+c2,τx,t,u=qc1t+c3,ϕx,t,u=c1u,where c1, c2, and c3 are arbitrary constants, and throughout this paper, we denote
(18)p=a+b−m2,qa+b−3m+22.Therefore, we have the following conclusion.Theorem 1.
For the arbitrary parametersm,a,b, if m>1, a+b>1, the vector field admitted by the differential equation (6) is
(19)V1=∂x,V2=∂t,V3=px∂x+qt∂t+u∂u.The sketch of the proof has been stated above. In details, we should divide it into the following four cases. Obviously, the vector fields of each case are the special cases of (19)
(i)
Ifa+b=2, a+3b≠3, the vector field is
(20)V1=∂x,V2=∂t,V3=1−m2x∂x+2−3m2t∂t+u∂u(ii)
Ifa+b=2, a+3b=3, the vector field is the same as (i)(iii)
Ifa+b≠2, a+3b=3, the vector field is
(21)V1=∂x,V2=∂t,V3=px∂x+qt∂t+u∂u(iv)
Ifa+b≠2, a+3b≠3, b≠1, the vector field is the same as (iii).The vector fieldsV1, V2, and V3 form a Lie algebra under the following Lie bracket
(22)f∂i,g∂j=f∂ig∂j−g∂jf∂i,where i and j stand for x,t,u.That is to say(23)V1,V2=0,V1,V3=pV3,V2,V3=qV3.By solving the following ordinary differential equations with the initial conditions:(24)dx∗dε=ξx∗,t∗,u∗,x∗ε=0=x,dt∗dε=τx∗,t∗,u∗,t∗ε=0=t,du∗dε=ϕx∗,t∗,u∗,u∗ε=0=u.We therefore obtain the group transformation which is generated by infinitesimal generatorsV1,V2,V3, respectively
(25)G1:x,t,u↦x+ε,t,u,(26)G2:x,t,u↦x,t+ε,u,(27)G3:x,t,u↦epεx,eqεt,eεu.Remark 2.
In (25)–(27), an arbitrary element in Gii=1,2,3 can transfer one solution of Eq. (6) to another one, so do the products of the elements from G1, G2, and G3.Remark 3.
The Lie groupG1×G2 is a normal Lie subgroup of G1G2G3. The Lie algebra generated by V1 and V2 is an ideal of L.Theorem 4.
Ifu=fx,t is a solution of Eq. (1.3), then u1, u2, and u3 as follows are solutions of Eq. (6) as well.
(28)u1=fx−ε,t,u2=fx,t−ε,u3=eεfe−pεx,e−qεt.


## 3. Similarity Reductions to Eq. (6)

In the preceding section, we obtained the group symmetry analysis of Eq. (6). In this section, the characteristic equations of vector fields are obtained by making use of (19) and utilized to perform the symmetry reduction.ForV=V1, the characteristic equation can be presented as follows:
(29)dx1=dt0=du0.Solving the characteristic equation (29), we have u=fξ, where ξ=t which yields a trivial solution.ForV=V2, the characteristic equation can be expressed as follows:
(30)dx0=dt1=du0,from which we have u=fξ, where ξ=x which transformed (6) to an ODE
(31)mfm−1f′+b−1a+b−2fa+b−3f′3+a+3b−3fa+b−2f′f″+fa+b−1f‴=0.ForV=V3, the characteristic equation can be written as follows:
(32)dxpx=dtqt=duu.Solving the characteristic equation, we have the following invariantsξ=xt−p/q, u=t1/qfξ, which transformed (6) to an ODE
(33)1qt1q−1f−pqxt−pqt1q−1−f′+mtpq−1fm−1f′+b−1a+b−2ta+b−2mqfa+b−3f′3+a+3b−3ta+b−3pqf′f″+ta+b−3pqf′f″+ta+b−3pqfa+b−1f‴=0.


## 4. Nonlinear Self-Adjointness of Eq. (6)

In this section, we shall show that Eq. (6) is nonlinearly self-adjoint. Let it start by presenting the definitions of nonlinear self-adjointness of differential equations according to Ibragimov’s [13, 25].Definition 5.
Given a differential functionF and the new dependent variable v=vx known as the adjoint variable or local variable [13, 25], the formal Lagrangian for the differential equation F=0 is the differential function given by
(34)F∗≔vF.Definition 6. [25].
The differential equation (6) is said to be nonlinearly selfadjoint if there exists a substitution
(35)v=ϕx,t,u≠0,such that
(36)F∗v=φx,t,u=λF,for some undermine function λ.Theorem 7.
The differential equation (6) is self-adjoint.Proof.
The formal Lagrangian is(37)L=vut+mum−1ux+b−1a+b−2ua+b−3ux3+a+3b−3ua+b−2uxuxx+ua+b−1uxxx.
Substituting into36=0, we have the adjoint equation to Eq. (6)
(38)−mum−1vx−aa+b−2ua+b−3vxux2−2aua+b−2vxuxx−vt−2aua+b−2vxxux−ua+b−1vxxx=0.
Letv=ϕx,t,u and the left hand of (38) be λ·∆, we shall get λ=−φu, and
(39)φ=c1u+c2,b=a+1,φ=c3ub−a−1,b≠a+1.
Then, we prove that Eq. (6) is self-adjoint.☐Generally speaking, we are now to calculate the conserved vectors. However, by using the method of Ibragimov [25], so we here omit the process.


## 5. Lie Symmetry and Reductions of Eq. (7)

In this section, we deal with all of the point symmetries of Eq. (7). We now assume that n−1<α<n in this section. Thus, Eq. (1.4) is an FPDE with the infinitesimal generator given by (2.2). If the vector field (9) generates a symmetry of Eq. (7), then V must satisfy the following Lie symmetry condition:
(40)pr3VΔ1Δ1=0=0,where
(41)Δ1=∂αu∂tα+mum−1ux+b−1a+b−2ua+b−3ux3+a+3b−3ua+b−2uxuxx+ua+b−1uxxx.Also, the invariant condition yields [29]
(42)τx,t,ut=0=0,and the αth extended infinitesimal related to Riemann-Liouville fractional time derivative with (42) is given by [30, 31].
(43)ϕα0=∂αϕ∂tφ+ϕu−αDtτ∂αu∂tα−u∂αϕu∂tα+μ−∑n−1∞αnDtnξDtα−nux+∑n−1∞αn∂αϕu∂tα−αn+1Dtn+1τDtα−nu,where
(44)μ=∑n=2∞∑m=2n∑k=2m∑r=0k−1αnnmkr1k!tn−αΓn+1−α−un−α∂∂tm×uk−r∂n−m+kϕ∂tn−m∂uk.The expression ofμ is complicated; however, it should converges to zero when the infinitesimal φ is linear in u, because of the existence of the derivatives ∂n−m+k/∂tn−m∂uk,k≥2 in the above expression (44).Thus, the Lie group classification method for the FPDE leads to the following result.Theorem 8.
The infinitesimal symmetry group of the equation (7) is spanned by the two vector fields
(45)V1=∂xV2=px∂x+1αqt∂t+u∂u.Proof.
Considering the invariance criterion (40), we have
(46)ϕα0+mm−1um−2ϕux+mum−1ϕx+b−1a+b−2a+b−3ua+b−4ϕux3+3b−1a+b−2ua+b−3ϕxux2+a+3b−3a+b−2ua+b−3ϕuxuxx+a+3b−3ϕuxxx+a+3b−3ϕxxux+a+b−1ϕua+b−2uxxx+ua+b−1ϕxxx=0solving the determining equation (46), and we obtained the following infinitesimals
(47)ξx,t,u=pc1x+c2,τx,t,u=1αqc1t,ϕx,t,u=c1u,which follows that the fractional differential equation (7) admitted two dimensional symmetries and is spanned by (45).☐Next, we utilized the admitted Lie symmetry and perform similarity reductions, present the reduced nonlinear fractional ordinary differential equations (FODEs), and classify the corresponding group-invariant solutions of the fractionalC1m,a,b.Case 9.
ForV1=∂/∂x, the characteristic equation is
(48)dx1=dt0=du0,and the following similarity variables are obtained by solving the characteristic equation u=fζ, ζ=t, which yields the following reduced ODE
(49)∂tαft=0.
The fractional ODE has a polynomial general group-invariant solution(50)u=a1tα−1,where a1 is an arbitrary constant of integration [6, 15, 16].Case 10.
ForV2=px∂x+1/2qt∂t+u∂u, the characteristic equation is
(51)dxpx=αdtqt=duu,and by solving the above equation, we get the group-invariant solution
(52)u=tαqgζ,ζ=xt−pαq,where g is an arbitrary function of ζ. Using these invariants, Eq. (7) transforms to a special nonlinear ODE of fractional order. Thus, we have the following theorem corresponding to this case.Theorem 11.
The transformation (52) reduces (7) to the following nonlinear ordinary differential equation of fractional order with the Erd’elyi-Kober fractional differential operator Pβτ,α of order [5]
(53)Pβτ+α,n−αg≔∏j=0n−1τ+j−1βζddζKβτ,αgζ,n=α+1,α≠ℕα,α∈ℕ,where
(54)Kβτ,αgζ≔1Γα∫1∞u−1α−1u−uτ+αgζu1βdu,α>0,gζ,α=0.is the Erd’elyi-Kober fractional integral operator.Proof.
Letn−1<α<n, n=1,2,3,⋯, according to the Riemann-Liouville fractional derivative, and one can obtain
(55)∂αu∂tα=∂n∂tn1Γn−α∫0tt−sn−α−1sαqgxs−pαqds.
Letv=t/s, and then ds=−t/v2dv;so, (55) can be written as
(56)∂αu∂tα=∂n∂tntn−α+αq1Γn−α∫1∞v−1n−α−1v−n−α+αq+1gζvpαqdv.
In view of the Erd’elyi-Kober fractional integral operator (54), one can get
(57)∂αu∂tα=∂n∂tntn−α+αqKqpα1+αq,n−αgζ.
Therefore, the right hand side of (57) becomes
(58)∂n∂tntn−α+αqKqpα1+αq,n−αgζ=∂n−1∂tn−1∂∂ttn−α+αqKqpα1+αq,n−αgζ=∂n−1∂tn−1tn−α+αq−1n−α+αq−1−pαqζdfζKqpα1+αq,n−αgζ.
Repeating the same proceduren−1 times, one can obtain
(59)∂n∂tntn−α+αqKqpα1+αq,n−αgζ=t−α+αq∏j=0n−11−α+αq+j−pαqζddζKqpα1+αq,n−αgζ.
Using the definition of the Erd’elyi-Kober fractional differential operator (53), we get
(60)∂n∂tntn−α+αqKqpα1+αq,n−αgζ=t−α+αqPqpα1−α+αq,αgζ.
Substituting (60) into (57), we obtain an expression for the time fractional derivative
(61)∂αu∂tα=t−α+αqPqpα1−α+αq,αgζ.
Thus, the fractionalC1m,a,b equation (7) can be reduced into a fractional order.
ODE:(62)Pqpα1−α+αq,αgζ=−mgm−1gζ−b−1a+b−2ga+b−3gζ3−a+3b−3ga+b−2gζ3−a+3b−3ga+b−2gζuζζ−ga+b−1gζζζ.☐


## 6. Conservation Law of Eq. (7)

In this section, we will construct the conservation laws of the fractionalC1m,a,b differential equation (7).LetL be the formal Lagrangian of Eq. (7) written as L=vΔ1, where v=vx,t is the new introduced dependent variable; so, the adjoint equation of (1.4) is written as δL/δu=0, where δ/δu is the Euler-Lagrange operator with respect to u and defined by
(63)δδu=∂∂u+Dtα∗∂∂Dtαu+∑k=1∞−1kDi1Di2⋯Dik∂∂u1i2⋯ik,and Dtα∗ is the adjoint operator of Dtα. For Riemann-Liouville fractional differential operators, we have
(64)Dtα∗=−1nIrn−αDtn=∑tCDrα,where
(65)Irn−αDtn=1Γn−α∫trτ−tn−α−1fx,τdτ,n=α+1is the right-sided fractional integral operators of order α.In this case, the formal Lagrangian of Eq. (7) is written as
(66)L=vx,t∂αu∂tα+mum−1ux+b−1a+b−2ua+b−3ux3+a+3b−3ua+b−2uxuxx+ua+b−1uxxx.Every Lie point symmetry (45) admitted by the Eq. (7) leads to a conservation law DiCi=0 where the components Ci constructed by the following formula [5, 32]:
(67)Ct=∑k=0n−1−1kDtα−1−kWDk∂L∂Dtαu−JW,Dtn∂L∂Dtαu,Cx=W∂L∂ux−Dx∂L∂uxx+Dx2∂L∂uxxx+DxW∂L∂uxx−Dx∂L∂uxxx+Dx2W∂L∂uxxx,where W=ϕ−ξux−τut and L are the formal Lagrangian which are written in the symmetric form, and the mixed derivatives J are given by the integral:
(68)Jf,g=1Γn−α∫0t∫trfx,ξgx,μμ−ξn−α−1dμdξ,for 0<α<1. For other cases, the calculations of conserved vector components are similar. In this case,
(69)Ct=Dtα−1W∂L∂Dtαu−JW,Dt∂L∂Dtαu.The conserved vector componentsCt,Cx are as follows:
(70)Ct=Dtα−1Wiv−JWi,vt,Cx=Wimvum−1+3b−1a+b−2ua+b−3vux2+a+3b−3ua+b−2vuxx−a+3b−3a+b−2.ua+b−3vux2−a+3b−3ua+b−2vxux−a+3b−3ua+b−2vuxx+a+b−1ua+b−2vux+ua+b−1vx+a+b−1a+b−2ua+b−3vux2+2a+b−1ua+b−2vxux+a+b−1ua+b−2vuxx+ua+b−1vxx+DxWia+3b−3ua+b−2vux−a+b−1ua+b−2vux−ua+b−1vx+Dx2Wiua+b−1v.where W1=−ux, W2=u−pxux−1/αqtut.



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