Enhanced group classification of nonlinear diffusion–reaction equations with gradient-dependent diffusivity

Abstract

We carry out the enhanced group classification of a class of (1+1)-dimensional nonlinear diffusion-reaction equations with gradient-dependent diffusivity using the two-step version of the method of furcate splitting. For simultaneously finding the equivalence groups of an unnormalized class of differential equations and a collection of its subclasses, we suggest an optimized version of the direct method. The optimization includes the preliminary study of admissible transformations within the entire class and the successive splitting of the corresponding determining equations with respect to arbitrary elements and their derivatives depending on auxiliary constraints associated with each of required subclasses. In the course of applying the suggested technique to subclasses of the class under consideration, we construct, for the first time, a nontrivial example of finite-dimensional effective generalized equivalence group. Using the method of Lie reduction and the generalized separation of variables, exact solutions of some equations under consideration are found.

Group Classification and Conservation Laws of a Class of Hyperbolic Equations

“…The proposed method could be used to review some of the relevant cases of group classification already appearing in the literature and often cited in this paper, such as those given in papers [13] to [17] for instance. This will entail in particular the consideration of families of equations depending on more than one arbitrary function, each with its own argument, as well as cases where the classifying equations consist of a system of several equations.…”

“…This includes the well-known algebraic method which can be traced back to Lie's work on symmetry algebras of ordinary differential equations (ODEs), and which has been upgraded or applied in several papers [2,4,[8][9][10][11]. Another classification method that has emerged in recent years, often called furcate splitting, originated in [12], and its refinements and generalizations have been applied in other papers (see [13][14][15] and the references therein). Other methods suggested in the literature tend to be variants of the direct method, or adapted to a specific type of equations such as the one proposed for linear pdes in [6].…”

Group classification of the two-dimensional Green-Naghdi equations with a time dependent bottom topography

“…An alternative method for analyzing the classifying equations consists of an algebraic approach, taking the algebraic properties of an admitted Lie algebra into account, thus allowing for a significant simplification of the group classification. This algebraic approach for group classification has been applied in [24,25,26,27,28] 4 .…”

“…The equations are considered in Eulerian coordinates. For the group classification we use the algebraic approach applied earlier to different types of systems (see, e.g., [24,25,26,27,28] and references therein). The algebraic approach takes the algebraic properties of an admitted Lie algebra into account and allows one a significant simplification of the group classification.…”

First-Order Sign-Invariants and Exact Solutions of the Radially Symmetric Nonlinear Diffusion Equations with Gradient-Dependent Diffusivities

“…It is well known that this equation occur, for instance, in the theory of non-Newtonian liquids and in some turbulence problems [11][12][13]. The descriptions of Lie symmetries of a class of nonlinear diffusion equations in one and two dimensions are presented in [14][15][16][17][18][19].…”