# Lie Symmetry Analysis and Invariant Solutions of a Nonlinear Fokker-Planck Equation Describing Cell Population Growth

Authors: Jia Zheng
Journal: Advances in Mathematical Physics
 (2020)

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## Abstract

In this paper, a complete Lie symmetry analysis is performed for a nonlinear Fokker-Planck equation for growing cell populations. Moreover, an optimal system of one-dimensional subalgebras is constructed and used to find similarity reductions and invariant solutions. A new power series solution is constructed via the reduced equation, and its convergence is proved.


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## Body

## 1. Introduction

During the last few decades, Lie symmetry group theory has been developed considerably and plays an increasingly important role in many scientific fields such as constructing similarity solutions, conservation laws, and symmetry-preserving difference schemes [1–5]. For the partial differential equations (PDEs), the Lie symmetry analysis method provides similar variables which are used to construct new differential equations with lower dimension, then group-invariant solutions of the studied PDEs is constructed via the reduced differential equations. With the benefit of the Lie symmetry analysis method, many differential equations were studied successfully [6–15].To construct inequivalent invariant solutions which means that it is impossible to connect them with some group transformation, one needs to seek a minimal list of group generators in the simplest form that span these inequivalent group-invariant solutions. Such a scenario motivates emergence of the definition of an optimal system of subalgebra. From an algorithmic perspective of an optimal system, we need to simplify a general element of the infinitesimal operators to several simple and inequivalent forms by using adjoint transformations; refer to [1, 3] for details.In this paper, we use the Lie symmetry method to study a nonlinear diffusion-type PDE describing cell population growth:(1)∂f∂t+v∂f∂μ=epvfn∂2f∂v2,where p and n are nonzero constants. This model describes the changes of cell population density f=fμ,v,t with the maturation of the cell populations μ∈0,1, the maturation velocity v∈0,∞, and time t∈0,∞. Equation (1) incorporates an exponential function in v and power function in f in the diffusion coefficient and extends the model proposed by Rotenberg [16]:
(2)∂f∂t+v∂f∂μ=D∂2f∂v2,where D is a diffusion constant. This equation is analyzed numerically in [17] and closed-form solutions under different reproduction rules of it are constructed. Equation (2) without a diffusion term is considered in [18]. A stationary modified version of Rotenberg’s model with a nonlinear transition rate is studied in [19].The remainder of this article is organized as follows. In Section2, we first determine the symmetry group of equation (1) and derive an optimal system of one-dimensional subalgebras. Consequently, similarity reductions for equation (1) are performed and an explicit power series solution of equation (1) is presented. Finally, the last section summarizes our work.


## 2. Main Results

### 2.1. Determination of Lie Symmetry

Consider a local one-parameter Lie group of point transformation:(3)μ∗=μ+εξμ,v,t,f+Oε2,v∗=v+εζμ,v,t,f+Oε2,t∗=t+ετμ,v,t,f+Oε2,f∗=f+εημ,v,t,f+Oε2,where ε∈ℝ is the group parameter.In Lie’s framework, the following infinitesimal operator characterises the one-parameter Lie group (3) completely [2]:
(4)X=ξμ,v,t,f∂∂μ+ζμ,v,t,f∂∂v+τμ,v,t,f∂∂t+ημ,v,t,f∂∂f.Thus, if Lie group (3) leaves equation (1) invariant, then on the solution space of equation (1), operator (4) must satisfy the infinitesimal invariance criterion below:
(5)Pr2X∂f∂t+v∂f∂μ−epvfn∂2f∂v2=0,where the second prolongation Pr2X is given by
(6)Pr2X=X+ηt1∂∂ft+ημ1∂∂fμ+ηvv2∂∂fvv,in which ημ1 and ηvv2 are determined by classical formulae [1–3].For the sake of determining Lie group (3) admitted by equation (1), inserting (6) into condition (5) and making the coefficients of different order derivatives of f equal to zero, we obtain a linear overdetermined system of PDEs about ξ=ξμ,v,t,f, ζ=ζμ,v,t,f, τ=τμ,v,t,f, and η=ημ,v,t,f:
(7)∂ζ∂μ=∂ζ∂v=∂ζ∂t=∂ζ∂f=0,∂η∂μ=∂η∂v=∂η∂t=0,∂ξ∂f=∂ξ∂v=0,∂τ∂μ=∂τ∂v=∂τ∂f=0,η−f∂η∂f=0,ζ−∂ξ∂t=0,pfζ+nη+f∂ξ∂μ=0,pfζ+nη+f∂τ∂t=0.By solving equation (7), we have
(8)ξ=c2−pμ+t−c1nμ+c4,ζ=c2,τ=−c1nt−c2pt+c3,η=c1f,where c1, c2, c3, and c4 are arbitrary constants.Consequently, the infinitesimal generators admitted by equation (1) are given by
(9)X1=−pμ+t∂∂μ+∂∂v−pt∂∂t,X2=−nμ∂∂μ−nt∂∂t+f∂∂f,X3=∂∂t,X4=∂∂μ.In the wake of these infinitesimal generatorsXii=1,2,3,4, we obtain four Lie groups of point transformation admitted by equation (1):
(10)G1:μ,v,t,f⟶e−pεμ+tε,v+ε,e−pεt,f,G2:μ,v,t,f⟶e−nεμ,v,e−nεt,eεf,G3:μ,v,t,f⟶μ,v,t+ε,f,G4:μ,v,t,f⟶μ+ε,v,t,f.where ε∈ℝ is the group parameter.That is to say, iff=θμ,v,t satisfies equation (1), then fii=1,2,3,4 are also solutions of equation (1):
(11)f1=θepεμ−tε,v−ε,epεt,f2=eεθe−nεμ,v,e−nεt,f3=θμ,v,t−ε,f4=θμ−ε,v,t.


### 2.2. Optimal System of One-Dimensional Subalgebras

In this subsection, we will find a one-dimensional optimal system of Lie subalgebras admitted by equation (1) up to adjoint representation. First of all, the commutator table of Xii=1,⋯,4 is shown in Table 1 where the i,j entry means Xi,Xj=XiXj−XjXi.Table 1
Commutator table.
X1
X2
X3
X4
X1
0
0
pX3−X4
pX4
X2
0
0
nX3
nX4
X3
−pX3+X4
−nX3
0
0
X4
−pX4
−nX4
0
0EachXi generates an adjoint representation AdeεXiXj defined by [3]
(12)AdeεXiXj=Xj−εXi,Xj+ε22!Xi,Xi,Xj−⋯.Using (12) in conjunction with Table 1, we get Table 2 where the i,j-th entry represents AdeεXiXj.Proposition 1.
An optimal system of one-dimensional subalgebras spanned byX1, X2, X3, and X4 admitted by equation (1) is
(13)X1+aX2,X2,X3+X4,X3,X4,where a is an arbitrary constant.Proof.
Consider an arbitrary element spanned byX1~X4:
(14)X=a1X1+a2X2+a3X3+a4X4,where, hereinafter, aii=1,⋯,4 are arbitrary constants. Our target is to simplify as many of the coefficient ai as possible through the adjoint maps to X. We start with the coefficient of X1 and consider two cases about a1.
Case 1.a1≠0.
Without loss of generality, we assume thata1=1. According to Table 2, we act X by Adea3/a2n+pX3:
(15)X′=Ade−a3/a2n+pX3X=X1+a2X2+a4′X3,where a4′=a4+a3/a2n+p. With the adopted adjoint action, the coefficient of X4 disappears.
In order to cancel coefficienta4′, we further act on X′ by Ade−a4′/a2′n+pX4, then we get X′′=Ade−a4′/a2′n+pX4X′=X1+a2X2,
To summarize,X with a1≠0 is equivalent to X1+aX2, where a is an arbitrary constant.
Case 2.a1=0.
Considera2≠0. Following the above procedure, we act on X by Adea3/nX3 and Adea4′/nX4 successively to make the coefficient a3 and a4 zero. Thus, every one-dimensional subalgebra generated by X with a1=0 and a2≠0 is equivalent to the subalgebra spanned by X2.
Fora1=0, a2=0, and a3≠0, acting AdeεX1 on X, we obtain
(16)X′=AdeεX1X=e−nεX3+a4X4,which depends on the sign of a4. In fact, we can simplify the coefficient of X4 to either +1, −1, or 0. Moreover, we adopt the discrete symmetry μ,v,t,f⟶−μ,v,t,f, which maps X3−X4 to X3+X4. Thus, the one-dimensional subalgebra spanned by X3+a4X4 is equivalent to one spanned by either X3+X4 or X3.
Therefore, an optimal system of one-dimensional subalgebras admitted by equation (1) is determined by
(17)X1+aX2,X2,X3+X4,X3,X4,where a is an arbitrary constant. It completes the proof.Table 2
The adjoint representation ofX1, X2, X3, and X4.
Ad
X1
X2
X3
X4
X1
X1
X2
e−pεX3+εX4
e−pεX4
X2
X1
X2
e−nεX3
e−nεX4
X3
X1−εX4−pX3
X2+nX3
X3
X4
X4
X1+pεX4
X2+nX4
X3
X4


### 2.3. Similarity Reductions

In this subsection, we perform similarity reductions and construct invariant solutions for equation (1) based on the optimal system calculated in the preceding subsection.Case 3.
Reduction byX1+aX2.
Fora≠0, the characteristic equation for the generator aX1+X2, is
(18)dμ−anμ−pμ+t=dv1=dt−an−pt=dfaf,which gives the similarity variables x=ean+pvt, y=v−μ/t, and Fx,y=eavfμ,v,t. Substituting them into equation (1), we obtain
(19)an+p2x3Fn∂2F∂x2+an+px2Fnan+2a+p∂F∂x+2∂2F∂x∂y+xFna2F+2a∂F∂y+∂2F∂y2−x∂F∂x+y∂F∂y=0.
Fora=0, repeating the above procedure, the reduced equation corresponds to equation (19) with a=0.Case 4.
Reduction byX2.
Similarly, the similarity variables forX2 are y=μ/t and Fv,y=μ1/nfμ,v,t. Then, the corresponding reduced equation is
(20)vnF+y2−vy∂F∂y+epvFn∂2F∂v2=0.
Next, we use its symmetry to perform further symmetry reductions for equation (20). The infinitesimal operator of symmetry admitted by equation (20) is
(21)Y=∂∂v+∂∂y+Fny1−py∂∂F,whose similarity variables are x=y−v and Fv,y=gxy1/nepy/n. In fact, we can transform X1 to Y via y=μ/t and Fv,y=μ1/nfμ,v,t, that is to say, Y inherited symmetries of equation (1). Substituting the similarity variables into equation (20) leads to g=gx, satisfying the following reduced ordinary differential equation (ODE):
(22)2npgndgdx+n2gnd2gdx2+p2gn+1+n2xdgdx+ng=0.
Obviously, it is difficult to directly solve equation (22) by integration; thus, in the next subsection, we will construct power series solutions.Case 5.
Reduction byX3+X4.
Via the symmetryX3+X4, we get the group-invariant solution of the form fμ,v,t=Fv,y in which y=μ−t and F=Fv,y satisfy
(23)v−1∂F∂y=epvFn∂2F∂v2.
Again, using the symmetryY=−ny∂/∂y+F∂/∂F for equation (23), which actually is the inherited symmetry of equation (1) (X2⟶Y), we obtain the group-invariant solution Fv,y=y−1/ngv, where g=gv satisfies
(24)epvgn−1d2gdv2=1n1−v.
In particular, forn=1, we obtain a solution of equation (1):
(25)fμ,v,t=p−pv−2p3e−pv+c1v+c2μ−t−1,where c1 and c2 are arbitrary constants.
The evolutionary procedure of solution (25) is shown in Figure 1 by choosing appropriate parameters from different perspectives.Case 6.
Reduction byX3.
For the generatorX3, we have fμ,v,t=Fμ,v, where F=Fμ,v satisfies a 1+1-dimensional PDE:
(26)v∂F∂μ=epvFn∂2F∂v2.
Similarly, by means of the corresponding infinitesimal operatorY=−nμ∂/∂μ+F∂/∂F of equation (26), we reduce equation (26) into an ODE:
(27)v=epvgvn−1d2gvdv2,where Fμ,v=μ−1/ngv.
In particular, forn=1, we get a solution of equation (1):
(28)fμ,v,t=μ−1/n−pv+2p3e−pv+c1v+c2,where c1 and c2 are arbitrary constants.Case 7.
Reduction byX4.
ForX4, the group-invariant solution is fμ,v,t=Fμ,v, where F=Fμ,v satisfies
(29)∂F∂t=epvFn∂2F∂v2.
Equation (29) admits two Lie symmetries with infinitesimal operators Y1=−nt∂/∂t+F∂/∂F and Y2=∂/∂v−pt∂/∂t. The symmetry Y1 produces the group-invariant solution with the form Fv,t=t−1/ngv, where g=gv satisfies
(30)−1ng=epvgnd2gdv2.
In particular, forn=1, equation (1) has a solution:
(31)fμ,v,t=t1/n−e−pvp2+c1v+c2.
On the other hand, the similarity variables ofY2 are x=texppv and g=F. Then, the group-invariant solution is Fv,t=gx, where g=gx satisfies
(32)dgdx=p2xgnd2gdx.Figure 1
Solution (25) of Equation (1) by choosing suitable parameters: p=1, n=1, and μ=0.8. (a) Perspective view. (b) Overhead view. (c) Propagation pattern of the wave along the t axis.
(a)
(b)
(c)


### 2.4. Power Series Solutions via the Reduced Equation byX2

In this subsection, we will seek a power series solution of equation (22). Suppose that equation (22) has a power series solution of the form
(33)Fx=∑i=0∞Cixi,where ci are undetermined constants.Observe that(34)F′x=∑i=0∞i+1ci+1xi,F″x=∑i=0∞i+2i+1ci+2xi.Substituting expressions (33) and (34) into equation (22), we have
(35)∑i=0∞cixin2np∑i=0∞i+1ci+1xi+n2∑i=0∞i+2i+1ci+2xi+p2∑i=0∞cixi+n2x∑i=0∞i+1ci+1xi+n∑i=0∞cixi=0.Equating the coefficients of different powers ofx gives rise to the explicit expressions of ci. For i=0, one has
(36)c0n2npc1+2n2c2+p2c0+nc0=0,which leads to
(37)c2=−pnc1−p22n2c0−12nc0n−1.Generally fori≥1, we have
(38)ci+2=−1n2i+2i+1c0n∑k=0i−1∑j1+j2+⋯+jn=i−kcj1cj2⋯cjn×2npk+1ck+1+n2k+2k+1ck+2+p2ck−1n2i+2i+1c0nn2ici+nci+c0n2npi+1ci+1+p2cic0n.Therefore, for the chosen constantsc0 and c1, the sequence cii=0∞ can be determined by equations (37) and (38) successively.Now, we show that the power series solution (33) with ci given by (37) and (38) is convergent. As a matter of fact,
(39)∣c2∣≤M2∣c0∣+∣c1∣,and for i≥1,
(40)ci+2≤M∑k=0i−1∑j1+⋯jn=i−kcj1cj2⋯cjnck+1+ck+2+ck+3ci+ci+1,where M=max2p/nc0n,1/nc0n,p2/n2c0n,2p/n,p2/n2.Consider another power seriesQz=∑i=0∞Qizi, where
(41)Q0=c0,Q1=c1,Q2=M2c0+c1=MQ0+2Q1,Qi+2=M∑k=0i−1∑j1+⋯+jn=i−kQj1Qj2⋯QjnQk+1+Qk+2+Qk+3Qi+Qi+1,i≥1.It is easy to find that∣ci∣≤Qii=0,1,⋯. Thus, the series Q=Qz=∑i=0∞Qizi is the majorant series of (33). Next, we show that the series Q=Qz has positive radius of convergence:
(42)Qz=Q0+Q1z+Q2z2+∑i=1∞Qi+2zi+2=Q0+Q1z+Q2z2+M∑i=1∞∑k=0j1i−1∑+⋯+jn=nQj1⋯QjnQk+1zi+2+∑i=1∞∑k=0j1i−1∑+⋯+jn=nQj1⋯QjnQk+2zi+2+∑i=1∞∑k=0j1i−1∑+⋯+jn=nQj1⋯QjnQkzi+2+3∑i=1∞Qizi+1+∑i=1∞Qi+1zi+2=−Q−Q2Q02+4Q0+Q3+Q2Q2−3Q+Q1Mz2+Q−Q0Q2−Q02+1M+Q1z+Q0.Now, we construct the implicit function with respect to the independent variablez:
(43)λz,φ=φ−M−Q−Q2Q02+4Q0+Q3+Q2Q2−3Q+Q1z2−Q−Q0Q2−Q02+1M+Q1z−Q0,and we verify that λz,φ is analytic in the neighborhood of 0,Q0 where λ0,Q0=0 and λφ′0,Q0=1≠0. By the implicit function theorem [20, 21], we see that Q=Qz is analytic in a neighborhood of the point 0,Q0 and with a positive radius. This implies that the power series Qz=∑i=0∞Qizi converges in a neighborhood of the point 0,Q0.Hence, an explicit power series solution of equation (1) is given by
(44)fμ,v,t=∑i=0∞ci−v+μtit−1/nepμ/nt,where the coefficients cii≥2 depend on (37) and (38) completely.


## 2.1. Determination of Lie Symmetry

Consider a local one-parameter Lie group of point transformation:(3)μ∗=μ+εξμ,v,t,f+Oε2,v∗=v+εζμ,v,t,f+Oε2,t∗=t+ετμ,v,t,f+Oε2,f∗=f+εημ,v,t,f+Oε2,where ε∈ℝ is the group parameter.In Lie’s framework, the following infinitesimal operator characterises the one-parameter Lie group (3) completely [2]:
(4)X=ξμ,v,t,f∂∂μ+ζμ,v,t,f∂∂v+τμ,v,t,f∂∂t+ημ,v,t,f∂∂f.Thus, if Lie group (3) leaves equation (1) invariant, then on the solution space of equation (1), operator (4) must satisfy the infinitesimal invariance criterion below:
(5)Pr2X∂f∂t+v∂f∂μ−epvfn∂2f∂v2=0,where the second prolongation Pr2X is given by
(6)Pr2X=X+ηt1∂∂ft+ημ1∂∂fμ+ηvv2∂∂fvv,in which ημ1 and ηvv2 are determined by classical formulae [1–3].For the sake of determining Lie group (3) admitted by equation (1), inserting (6) into condition (5) and making the coefficients of different order derivatives of f equal to zero, we obtain a linear overdetermined system of PDEs about ξ=ξμ,v,t,f, ζ=ζμ,v,t,f, τ=τμ,v,t,f, and η=ημ,v,t,f:
(7)∂ζ∂μ=∂ζ∂v=∂ζ∂t=∂ζ∂f=0,∂η∂μ=∂η∂v=∂η∂t=0,∂ξ∂f=∂ξ∂v=0,∂τ∂μ=∂τ∂v=∂τ∂f=0,η−f∂η∂f=0,ζ−∂ξ∂t=0,pfζ+nη+f∂ξ∂μ=0,pfζ+nη+f∂τ∂t=0.By solving equation (7), we have
(8)ξ=c2−pμ+t−c1nμ+c4,ζ=c2,τ=−c1nt−c2pt+c3,η=c1f,where c1, c2, c3, and c4 are arbitrary constants.Consequently, the infinitesimal generators admitted by equation (1) are given by
(9)X1=−pμ+t∂∂μ+∂∂v−pt∂∂t,X2=−nμ∂∂μ−nt∂∂t+f∂∂f,X3=∂∂t,X4=∂∂μ.In the wake of these infinitesimal generatorsXii=1,2,3,4, we obtain four Lie groups of point transformation admitted by equation (1):
(10)G1:μ,v,t,f⟶e−pεμ+tε,v+ε,e−pεt,f,G2:μ,v,t,f⟶e−nεμ,v,e−nεt,eεf,G3:μ,v,t,f⟶μ,v,t+ε,f,G4:μ,v,t,f⟶μ+ε,v,t,f.where ε∈ℝ is the group parameter.That is to say, iff=θμ,v,t satisfies equation (1), then fii=1,2,3,4 are also solutions of equation (1):
(11)f1=θepεμ−tε,v−ε,epεt,f2=eεθe−nεμ,v,e−nεt,f3=θμ,v,t−ε,f4=θμ−ε,v,t.


## 2.2. Optimal System of One-Dimensional Subalgebras

In this subsection, we will find a one-dimensional optimal system of Lie subalgebras admitted by equation (1) up to adjoint representation. First of all, the commutator table of Xii=1,⋯,4 is shown in Table 1 where the i,j entry means Xi,Xj=XiXj−XjXi.Table 1
Commutator table.
X1
X2
X3
X4
X1
0
0
pX3−X4
pX4
X2
0
0
nX3
nX4
X3
−pX3+X4
−nX3
0
0
X4
−pX4
−nX4
0
0EachXi generates an adjoint representation AdeεXiXj defined by [3]
(12)AdeεXiXj=Xj−εXi,Xj+ε22!Xi,Xi,Xj−⋯.Using (12) in conjunction with Table 1, we get Table 2 where the i,j-th entry represents AdeεXiXj.Proposition 1.
An optimal system of one-dimensional subalgebras spanned byX1, X2, X3, and X4 admitted by equation (1) is
(13)X1+aX2,X2,X3+X4,X3,X4,where a is an arbitrary constant.Proof.
Consider an arbitrary element spanned byX1~X4:
(14)X=a1X1+a2X2+a3X3+a4X4,where, hereinafter, aii=1,⋯,4 are arbitrary constants. Our target is to simplify as many of the coefficient ai as possible through the adjoint maps to X. We start with the coefficient of X1 and consider two cases about a1.
Case 1.a1≠0.
Without loss of generality, we assume thata1=1. According to Table 2, we act X by Adea3/a2n+pX3:
(15)X′=Ade−a3/a2n+pX3X=X1+a2X2+a4′X3,where a4′=a4+a3/a2n+p. With the adopted adjoint action, the coefficient of X4 disappears.
In order to cancel coefficienta4′, we further act on X′ by Ade−a4′/a2′n+pX4, then we get X′′=Ade−a4′/a2′n+pX4X′=X1+a2X2,
To summarize,X with a1≠0 is equivalent to X1+aX2, where a is an arbitrary constant.
Case 2.a1=0.
Considera2≠0. Following the above procedure, we act on X by Adea3/nX3 and Adea4′/nX4 successively to make the coefficient a3 and a4 zero. Thus, every one-dimensional subalgebra generated by X with a1=0 and a2≠0 is equivalent to the subalgebra spanned by X2.
Fora1=0, a2=0, and a3≠0, acting AdeεX1 on X, we obtain
(16)X′=AdeεX1X=e−nεX3+a4X4,which depends on the sign of a4. In fact, we can simplify the coefficient of X4 to either +1, −1, or 0. Moreover, we adopt the discrete symmetry μ,v,t,f⟶−μ,v,t,f, which maps X3−X4 to X3+X4. Thus, the one-dimensional subalgebra spanned by X3+a4X4 is equivalent to one spanned by either X3+X4 or X3.
Therefore, an optimal system of one-dimensional subalgebras admitted by equation (1) is determined by
(17)X1+aX2,X2,X3+X4,X3,X4,where a is an arbitrary constant. It completes the proof.Table 2
The adjoint representation ofX1, X2, X3, and X4.
Ad
X1
X2
X3
X4
X1
X1
X2
e−pεX3+εX4
e−pεX4
X2
X1
X2
e−nεX3
e−nεX4
X3
X1−εX4−pX3
X2+nX3
X3
X4
X4
X1+pεX4
X2+nX4
X3
X4


## 2.3. Similarity Reductions

In this subsection, we perform similarity reductions and construct invariant solutions for equation (1) based on the optimal system calculated in the preceding subsection.Case 3.
Reduction byX1+aX2.
Fora≠0, the characteristic equation for the generator aX1+X2, is
(18)dμ−anμ−pμ+t=dv1=dt−an−pt=dfaf,which gives the similarity variables x=ean+pvt, y=v−μ/t, and Fx,y=eavfμ,v,t. Substituting them into equation (1), we obtain
(19)an+p2x3Fn∂2F∂x2+an+px2Fnan+2a+p∂F∂x+2∂2F∂x∂y+xFna2F+2a∂F∂y+∂2F∂y2−x∂F∂x+y∂F∂y=0.
Fora=0, repeating the above procedure, the reduced equation corresponds to equation (19) with a=0.Case 4.
Reduction byX2.
Similarly, the similarity variables forX2 are y=μ/t and Fv,y=μ1/nfμ,v,t. Then, the corresponding reduced equation is
(20)vnF+y2−vy∂F∂y+epvFn∂2F∂v2=0.
Next, we use its symmetry to perform further symmetry reductions for equation (20). The infinitesimal operator of symmetry admitted by equation (20) is
(21)Y=∂∂v+∂∂y+Fny1−py∂∂F,whose similarity variables are x=y−v and Fv,y=gxy1/nepy/n. In fact, we can transform X1 to Y via y=μ/t and Fv,y=μ1/nfμ,v,t, that is to say, Y inherited symmetries of equation (1). Substituting the similarity variables into equation (20) leads to g=gx, satisfying the following reduced ordinary differential equation (ODE):
(22)2npgndgdx+n2gnd2gdx2+p2gn+1+n2xdgdx+ng=0.
Obviously, it is difficult to directly solve equation (22) by integration; thus, in the next subsection, we will construct power series solutions.Case 5.
Reduction byX3+X4.
Via the symmetryX3+X4, we get the group-invariant solution of the form fμ,v,t=Fv,y in which y=μ−t and F=Fv,y satisfy
(23)v−1∂F∂y=epvFn∂2F∂v2.
Again, using the symmetryY=−ny∂/∂y+F∂/∂F for equation (23), which actually is the inherited symmetry of equation (1) (X2⟶Y), we obtain the group-invariant solution Fv,y=y−1/ngv, where g=gv satisfies
(24)epvgn−1d2gdv2=1n1−v.
In particular, forn=1, we obtain a solution of equation (1):
(25)fμ,v,t=p−pv−2p3e−pv+c1v+c2μ−t−1,where c1 and c2 are arbitrary constants.
The evolutionary procedure of solution (25) is shown in Figure 1 by choosing appropriate parameters from different perspectives.Case 6.
Reduction byX3.
For the generatorX3, we have fμ,v,t=Fμ,v, where F=Fμ,v satisfies a 1+1-dimensional PDE:
(26)v∂F∂μ=epvFn∂2F∂v2.
Similarly, by means of the corresponding infinitesimal operatorY=−nμ∂/∂μ+F∂/∂F of equation (26), we reduce equation (26) into an ODE:
(27)v=epvgvn−1d2gvdv2,where Fμ,v=μ−1/ngv.
In particular, forn=1, we get a solution of equation (1):
(28)fμ,v,t=μ−1/n−pv+2p3e−pv+c1v+c2,where c1 and c2 are arbitrary constants.Case 7.
Reduction byX4.
ForX4, the group-invariant solution is fμ,v,t=Fμ,v, where F=Fμ,v satisfies
(29)∂F∂t=epvFn∂2F∂v2.
Equation (29) admits two Lie symmetries with infinitesimal operators Y1=−nt∂/∂t+F∂/∂F and Y2=∂/∂v−pt∂/∂t. The symmetry Y1 produces the group-invariant solution with the form Fv,t=t−1/ngv, where g=gv satisfies
(30)−1ng=epvgnd2gdv2.
In particular, forn=1, equation (1) has a solution:
(31)fμ,v,t=t1/n−e−pvp2+c1v+c2.
On the other hand, the similarity variables ofY2 are x=texppv and g=F. Then, the group-invariant solution is Fv,t=gx, where g=gx satisfies
(32)dgdx=p2xgnd2gdx.Figure 1
Solution (25) of Equation (1) by choosing suitable parameters: p=1, n=1, and μ=0.8. (a) Perspective view. (b) Overhead view. (c) Propagation pattern of the wave along the t axis.
(a)
(b)
(c)


## 2.4. Power Series Solutions via the Reduced Equation byX2

In this subsection, we will seek a power series solution of equation (22). Suppose that equation (22) has a power series solution of the form
(33)Fx=∑i=0∞Cixi,where ci are undetermined constants.Observe that(34)F′x=∑i=0∞i+1ci+1xi,F″x=∑i=0∞i+2i+1ci+2xi.Substituting expressions (33) and (34) into equation (22), we have
(35)∑i=0∞cixin2np∑i=0∞i+1ci+1xi+n2∑i=0∞i+2i+1ci+2xi+p2∑i=0∞cixi+n2x∑i=0∞i+1ci+1xi+n∑i=0∞cixi=0.Equating the coefficients of different powers ofx gives rise to the explicit expressions of ci. For i=0, one has
(36)c0n2npc1+2n2c2+p2c0+nc0=0,which leads to
(37)c2=−pnc1−p22n2c0−12nc0n−1.Generally fori≥1, we have
(38)ci+2=−1n2i+2i+1c0n∑k=0i−1∑j1+j2+⋯+jn=i−kcj1cj2⋯cjn×2npk+1ck+1+n2k+2k+1ck+2+p2ck−1n2i+2i+1c0nn2ici+nci+c0n2npi+1ci+1+p2cic0n.Therefore, for the chosen constantsc0 and c1, the sequence cii=0∞ can be determined by equations (37) and (38) successively.Now, we show that the power series solution (33) with ci given by (37) and (38) is convergent. As a matter of fact,
(39)∣c2∣≤M2∣c0∣+∣c1∣,and for i≥1,
(40)ci+2≤M∑k=0i−1∑j1+⋯jn=i−kcj1cj2⋯cjnck+1+ck+2+ck+3ci+ci+1,where M=max2p/nc0n,1/nc0n,p2/n2c0n,2p/n,p2/n2.Consider another power seriesQz=∑i=0∞Qizi, where
(41)Q0=c0,Q1=c1,Q2=M2c0+c1=MQ0+2Q1,Qi+2=M∑k=0i−1∑j1+⋯+jn=i−kQj1Qj2⋯QjnQk+1+Qk+2+Qk+3Qi+Qi+1,i≥1.It is easy to find that∣ci∣≤Qii=0,1,⋯. Thus, the series Q=Qz=∑i=0∞Qizi is the majorant series of (33). Next, we show that the series Q=Qz has positive radius of convergence:
(42)Qz=Q0+Q1z+Q2z2+∑i=1∞Qi+2zi+2=Q0+Q1z+Q2z2+M∑i=1∞∑k=0j1i−1∑+⋯+jn=nQj1⋯QjnQk+1zi+2+∑i=1∞∑k=0j1i−1∑+⋯+jn=nQj1⋯QjnQk+2zi+2+∑i=1∞∑k=0j1i−1∑+⋯+jn=nQj1⋯QjnQkzi+2+3∑i=1∞Qizi+1+∑i=1∞Qi+1zi+2=−Q−Q2Q02+4Q0+Q3+Q2Q2−3Q+Q1Mz2+Q−Q0Q2−Q02+1M+Q1z+Q0.Now, we construct the implicit function with respect to the independent variablez:
(43)λz,φ=φ−M−Q−Q2Q02+4Q0+Q3+Q2Q2−3Q+Q1z2−Q−Q0Q2−Q02+1M+Q1z−Q0,and we verify that λz,φ is analytic in the neighborhood of 0,Q0 where λ0,Q0=0 and λφ′0,Q0=1≠0. By the implicit function theorem [20, 21], we see that Q=Qz is analytic in a neighborhood of the point 0,Q0 and with a positive radius. This implies that the power series Qz=∑i=0∞Qizi converges in a neighborhood of the point 0,Q0.Hence, an explicit power series solution of equation (1) is given by
(44)fμ,v,t=∑i=0∞ci−v+μtit−1/nepμ/nt,where the coefficients cii≥2 depend on (37) and (38) completely.


## 3. Conclusion

In this paper, Lie symmetry analysis is employed to study a new nonlinear equation describing the growing cell populations. An optimal system of one-dimensional subalgebra is constructed and used to construct reduced equations and invariant solutions. Moreover, we obtain a new power series solution of equation (1). Such results provide positive potential roles for analyzing cell population growth with equation (1).



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