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.
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: where and are nonzero constants. This model describes the changes of cell population density with the maturation of the cell populations , the maturation velocity , and time . Equation (1) incorporates an exponential function in and power function in in the diffusion coefficient and extends the model proposed by Rotenberg [16]: where 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 Section 2, 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: where is the group parameter.
In Lie’s framework, the following infinitesimal operator characterises the one-parameter Lie group (3) completely [2]:
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: where the second prolongation is given by in which and 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 equal to zero, we obtain a linear overdetermined system of PDEs about , , , and :
By solving equation (7), we have where , , , and are arbitrary constants.
Consequently, the infinitesimal generators admitted by equation (1) are given by
In the wake of these infinitesimal generators , we obtain four Lie groups of point transformation admitted by equation (1): where is the group parameter.
That is to say, if satisfies equation (1), then are also solutions of equation (1):
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 is shown in Table 1 where the entry means .
Each generates an adjoint representation defined by [3]
Using (12) in conjunction with Table 1, we get Table 2 where the -th entry represents .
Proposition 1. An optimal system of one-dimensional subalgebras spanned by , , , and admitted by equation (1) is where is an arbitrary constant.
Proof. Consider an arbitrary element spanned by :
where, hereinafter, are arbitrary constants. Our target is to simplify as many of the coefficient as possible through the adjoint maps to . We start with the coefficient of and consider two cases about .
Case 1. .
Without loss of generality, we assume that . According to Table 2, we act by :
where . With the adopted adjoint action, the coefficient of disappears.
In order to cancel coefficient , we further act on by , then we get ,
To summarize, with is equivalent to , where is an arbitrary constant.
Case 2. .
Consider . Following the above procedure, we act on by and successively to make the coefficient and zero. Thus, every one-dimensional subalgebra generated by with and is equivalent to the subalgebra spanned by .
For , , and , acting on , we obtain
which depends on the sign of . In fact, we can simplify the coefficient of to either , , or . Moreover, we adopt the discrete symmetry , which maps to . Thus, the one-dimensional subalgebra spanned by is equivalent to one spanned by either or .
Therefore, an optimal system of one-dimensional subalgebras admitted by equation (1) is determined by
where is an arbitrary constant. It completes the proof.
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 by .
For , the characteristic equation for the generator , is
which gives the similarity variables , , and . Substituting them into equation (1), we obtain
For , repeating the above procedure, the reduced equation corresponds to equation (19) with .
Case 4. Reduction by .
Similarly, the similarity variables for are and . Then, the corresponding reduced equation is
Next, we use its symmetry to perform further symmetry reductions for equation (20). The infinitesimal operator of symmetry admitted by equation (20) is
whose similarity variables are and . In fact, we can transform to via and , that is to say, inherited symmetries of equation (1). Substituting the similarity variables into equation (20) leads to , satisfying the following reduced ordinary differential equation (ODE):
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 by .
Via the symmetry , we get the group-invariant solution of the form in which and satisfy
Again, using the symmetry for equation (23), which actually is the inherited symmetry of equation (1) (), we obtain the group-invariant solution , where satisfies
In particular, for , we obtain a solution of equation (1):
where and are arbitrary constants.
The evolutionary procedure of solution (25) is shown in Figure 1 by choosing appropriate parameters from different perspectives.
Case 6. Reduction by .
For the generator , we have , where satisfies a -dimensional PDE:
Similarly, by means of the corresponding infinitesimal operator of equation (26), we reduce equation (26) into an ODE:
where .
In particular, for , we get a solution of equation (1):
where and are arbitrary constants.
Case 7. Reduction by .
For , the group-invariant solution is , where satisfies
Equation (29) admits two Lie symmetries with infinitesimal operators and . The symmetry produces the group-invariant solution with the form , where satisfies
In particular, for , equation (1) has a solution:
On the other hand, the similarity variables of are and . Then, the group-invariant solution is , where satisfies
(a)
(b)
(c)
2.4. Power Series Solutions via the Reduced Equation by
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 where are undetermined constants.
Observe that
Substituting expressions (33) and (34) into equation (22), we have
Equating the coefficients of different powers of gives rise to the explicit expressions of . For , one has which leads to
Generally for , we have
Therefore, for the chosen constants and , the sequence can be determined by equations (37) and (38) successively.
Now, we show that the power series solution (33) with given by (37) and (38) is convergent. As a matter of fact, and for , where .
Consider another power series , where
It is easy to find that . Thus, the series is the majorant series of (33). Next, we show that the series has positive radius of convergence:
Now, we construct the implicit function with respect to the independent variable : and we verify that is analytic in the neighborhood of where and . By the implicit function theorem [20, 21], we see that is analytic in a neighborhood of the point and with a positive radius. This implies that the power series converges in a neighborhood of the point .
Hence, an explicit power series solution of equation (1) is given by where the coefficients 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).
Data Availability
Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.
Conflicts of Interest
The author declares that he has no conflicts of interest.
Acknowledgments
This paper is supported by the National Natural Science Foundation of China (No. 11601541) and the National Social Science Fund Youth Project of China (No.18CRK012).