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

Jia Zheng

doi:10.1155/2020/4975943

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.

Highlights

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 symmetrypreserving difference schemes [1,2,3,4,5]
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
An optimal system of one-dimensional subalgebras spanned by X1, X2, X3, and X4 admitted by equation (1) is fX1 + aX2, X2, X3 + X4, X3, X4g, ð13Þ

Summary

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 symmetrypreserving difference schemes [1,2,3,4,5]. 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 = ξðμ, v, t, f Þ, ζ = ζðμ, v, t, f Þ, τ = τðμ, v, t, f Þ, and η = ηðμ, v, t, f Þ: : In the wake of these infinitesimal generators Xi ði = 1, 2, 3, 4Þ, we obtain four Lie groups of point transformation admitted by equation (1): G1 : ðμ, v, t, f Þ ⟶ ðe−pεðμ + tεÞ, v + ε, e−pεt, f Þ, G2 : ðμ, v, t, f Þ ⟶ ðe−nεμ, v, e−nεt, eε f Þ, ð10Þ G3 : ðμ, v, t, f Þ ⟶ ðμ, v, t + ε, f Þ, G4 : ðμ, v, t, f Þ ⟶ ðμ + ε, v, t, f Þ: where ε ∈ R is the group parameter.

Results

Conclusion