An Application of Equivalence Transformations to Reaction Diffusion Equations

In this paper, we consider a quite general class of advection reaction diffusion systems. By using an equivalence generator, derived in a previous paper, the authors apply a projection theorem to determine some special forms of the constitutive functions that allow the extension by one of the two-dimensional principal Lie algebra. As an example, a special case is discussed at the end of the paper

Reductions and Conservation Laws of a Generalized Third-Order PDE via Multi-Reduction Method

In this work, we consider a family of nonlinear third-order evolution equations, where two arbitrary functions depending on the dependent variable appear. Evolution equations of this type model several real-world phenomena, such as diffusion, convection, or dispersion processes, only to cite a few. By using the multiplier method, we compute conservation laws. Looking for traveling waves solutions, all the the conservation laws that are invariant under translation symmetries are directly obtained. Moreover, each of them will be inherited by the corresponding traveling wave ODEs, and a set of first integrals are obtained, allowing to reduce the nonlinear third-order evolution equations under consideration into a first-order autonomous equation

Applications of Solvable Lie Algebras to a Class of Third Order Equations

A family of third-order partial differential equations (PDEs) is analyzed. This family broadens out well-known PDEs such as the Korteweg-de Vries equation, the Gardner equation, and the Burgers equation, which model many real-world phenomena. Furthermore, several macroscopic models for semiconductors considering quantum effects—for example, models for the transmission of electrical lines and quantum hydrodynamic models—are governed by third-order PDEs of this family. For this family, all point symmetries have been derived. These symmetries are used to determine group-invariant solutions from three-dimensional solvable subgroups of the complete symmetry group, which allow us to reduce the given PDE to a first-order nonlinear ordinary differential equation (ODE). Finally, exact solutions are obtained by solving the first-order nonlinear ODEs or by taking into account the Type-II hidden symmetries that appear in the reduced second-order ODEs

Symmetries and Solutions for Some Classes of Advective Reaction–Diffusion Systems

In this paper, we consider some reaction–advection–diffusion systems in order to obtain exact solutions via a symmetry approach. We write the determining system of a general class. Then, for particular subclasses, we obtain special forms of the arbitrary constitutive parameters that allow us to extend the principal Lie algebra. In some cases, we write the corresponding reduced system and we find special exact solutions

Symmetries and Solutions for a Class of Advective Reaction-Diffusion Systems with a Special Reaction Term

This paper is devoted to apply the Lie methods to a class of reaction diffusion advection systems of two interacting species u and v with two arbitrary constitutive functions f and g. The reaction term appearing in the equation for the species v is a logistic function of Lotka-Volterra type. Once obtained the Lie algebra for any form of f and g a Lie classification is carried out. Interesting reduced systems are derived admitting wide classes of exact solutions

Symmetries and Solutions for Some Classes of Advective Reaction–Diffusion Systems

In this paper, we consider some reaction–advection–diffusion systems in order to obtain exact solutions via a symmetry approach. We write the determining system of a general class. Then, for particular subclasses, we obtain special forms of the arbitrary constitutive parameters that allow us to extend the principal Lie algebra. In some cases, we write the corresponding reduced system and we find special exact solutions

An Application of Equivalence Transformations to Reaction Diffusion Equations

In this paper, we consider a quite general class of advection reaction diffusion systems. By using an equivalence generator, derived in a previous paper, the authors apply a projection theorem to determine some special forms of the constitutive functions that allow the extension by one of the two-dimensional principal Lie algebra. As an example, a special case is discussed at the end of the paper

Reductions and Conservation Laws of a Generalized Third-Order PDE via Multi-Reduction Method

In this work, we consider a family of nonlinear third-order evolution equations, where two arbitrary functions depending on the dependent variable appear. Evolution equations of this type model several real-world phenomena, such as diffusion, convection, or dispersion processes, only to cite a few. By using the multiplier method, we compute conservation laws. Looking for traveling waves solutions, all the the conservation laws that are invariant under translation symmetries are directly obtained. Moreover, each of them will be inherited by the corresponding traveling wave ODEs, and a set of first integrals are obtained, allowing to reduce the nonlinear third-order evolution equations under consideration into a first-order autonomous equation