Exact solutions of a reaction–diffusion system for Proteus mirabilis bacterial colonies

An Application of Equivalence Transformations to Reaction Diffusion Equations

“…With respect to the application in biomathematical models, equivalence and weak equivalence transformations were applied not only to study tumor models [26,27], but also the population dynamics in [1,3,4].…”

“…It is worthwhile to note that a symmetry transforms invariant solutions into invariant solutions that are not essentially different (see Ovsiannikov [11]), but, having a different form, they could satisfy different suitable initial/boundary conditions. The aim of this paper is an improvement of the results that we have shown in [12], bearing in mind some generalization of the special form assumed from the constitutive functions f , g and h already used in some previous papers about [4,[12][13][14]. In this paper, we use the infinitesimal generator of equivalence transformations derived in [12] for the class (1) in order to obtain some extensions of the principal Lie algebra for the following subclass:…”

On a generalized variable‐coefficient Gardner equation with linear damping and dissipative terms

“…For instance, it can be used to determine exact solutions, to reduce PDEs or to construct conservation laws. ()…”

Local conservation laws, symmetries, and exact solutions for a Kudryashov‐Sinelshchikov equation

“…[32][33][34] The essential basis of this method is that, when a differential equation is invariant under a Lie group of transformations, a reduction transformation exists. Several papers 30,[35][36][37] have been devoted to obtain symmetries of a PDE.Kudryashov 38 showed a method to obtain exact solutions of nonlinear differential equations. The idea of this method is to take into account an equation which has lower order than the original equation, for example, the Riccati equation or the equation for the Weierstrass elliptic function, and then, to apply the simplest method to this equation.…”

“…[32][33][34] The essential basis of this method is that, when a differential equation is invariant under a Lie group of transformations, a reduction transformation exists. Several papers 30,[35][36][37] have been devoted to obtain symmetries of a PDE.…”