Group classification of quasilinear elliptic-type equations. II. Invariance under solvable Lie algebras

We study the problem of group classification of quasilinear elliptic equations in a two-dimensional space. The list of all equations of this type admitting solvable Lie algebras of symmetry operators is obtained. Together with the results obtained earlier by the authors, these results give a complete solution of the problem of group classification of quasilinear elliptic equations.     

Lie symmetries and form-preserving transformations of reaction-diffusion-convection equations

New Lie symmetry classification of the known class of reaction-diffusion-convection equations is presented. The classification method is based on combining the standard group classification method and the form-preserving transformation approach.     

Complete group classification of systems of two linear second-order ordinary differential equations

We give a complete group classification of the general case of linear systems of two second-order ordinary differential equations excluding the case of systems which are studied in the literature. This paper gives the initial step in the study of nonlinear systems of two second-order ordinary differential equations. It can also be extended to systems of equations with more than two equations. Furthermore the complete group classification of a system of two linear second-order ordinary differential equations is done. Four cases of linear systems of equations with inconstant coefficients are obtained.     

Complete group classification of systems of two linear second‐order ordinary differential equations: the algebraic approach

We give a complete group classification of the general case of linear systems of two second‐order ordinary differential equations. The algebraic approach is used to solve the group classification problem for this class of equations. This completes the results in the literature on the group classification of two linear second‐order ordinary differential equations including recent results which give a complete group classification treatment of such systems. We show that using the algebraic approach leads to the study of a variety of cases in addition to those already obtained in the literature. We illustrate that this approach can be used as a useful tool in the group classification of this class of equations. A discussion of the subsequent cases and results is given. Copyright © 2014 John Wiley & Sons, Ltd.     

Invariant solutions of one-dimensional equations of two-temperature relaxation gas dynamics

The group analysis method is applied to the plane one-dimensional equations of two-temperature gas dynamics. The complete classification of the equations with respect to admitted Lie group is studied. All invariant solutions are analyzed and their comparisons with the known invariant solutions of the ideal gas dynamics system are presented.     

Group properties of generalized quasi-linear wave equations

In this paper, complete group classification of a class of (1+1)-dimensional generalized quasi-linear wave equations is performed by using the Lie-Ovsiannikov method, additional equivalent transformation and furcate split method. Lie reductions of some truly ‘variable coefficient’ wave equations which are singled out from the classification results are investigated. Some classes of exact solutions of these ‘variable coefficient’ wave equations are constructed by means of both the reductions and the additional equivalent transformations. The nonclassical symmetries to the generalized quasi-linear wave equation are also studied. This enabled to obtain some exact solutions of the wave equations which are invariant under certain conditional symmetries.     

A property of the defining equations for the Lie algebra in the group classification problem for wave equations

We solve the group classification problem for nonlinear hyperbolic systems of differential equations. The admissible continuous group of transformations has the Lie algebra of dimension less than 5. This main statement follows from the principal property of the defining equations of the admissible Lie algebra: the commutator of two solutions is a solution. Using equivalence transformations we classify nonlinear systems in accordance with the well-known Lie algebra structures of dimension 3 and 4.     

一类微分-差分方程组的内禀对称和等价群变换

研究一类微分-差分方程组的对称和等价群变换.采取内禀的无穷小算子方法,给出了方程组的内禀对称和等价群变换.为结合抽象Lie代数结构,给方程完全分类提供了理论基础.     

Group classification of projective type second-order ordinary differential equations

Group classification with respect to admitted point transformation groups is carried out for second-order ordinary differential equations with cubic nonlinearity of the first-order derivative. The result is obtained with use of the invariants of the equivalence transformation group of the family of equations under consideration. The corresponding Riemannian metric is found for the equations that are the projection of the system of geodesics to a two-dimensional surface.     

Application of group analysis to classification of systems of three second‐order ordinary differential equations

Here, we give a complete group classification of the general case of linear systems of three second‐order ordinary differential equations excluding the case of systems which are studied in the literature. This is given as the initial step in the study of nonlinear systems of three second‐order ordinary differential equations. In addition, the complete group classification of a system of three linear second‐order ordinary differential equations is carried out. Four cases of linear systems of equations are obtained. Copyright © 2015 John Wiley & Sons, Ltd.     

Group classification of quasilinear elliptic-type equations. I. Invariance with respect to Lie algebras with nontrivial Levi decomposition

The problem of group classification of quasilinear elliptic-type equations in the two-dimensional space is considered. We obtain the lists of all equations of this class that admit the semisimple Lie algebras of symmetry operators and the Lie algebras of symmetry operators with nontrivial Levi decomposition. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 11, pp. 1532–1545, November, 2007.     

Bifurcations of completely integrable first-order ordinary differential equations

We consider an implicit first-order ordinary differential equation with complete integral. In [3], the authors give a generic classifications of first-order ordinary differential equations with complete integral with respect to the equivalence relation which is given by the group of point transformations. The classification problem is reduced to the classification of a certain class of divergent diagrams of mapping germs. In this paper, we give a generic classifications of bifurcations of such differential equations as an application of the Legendrian singularity theory. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 33, Suzdal Conference-2004, Part 1, 2005.     

Modified classification of non-self-adjoint second-order difference equations

This paper is concerned with the classification of non-self-adjoint second-order difference equations. The relationship between the number of summable solutions of non-self-adjoint difference equations and that of the discrete linear Hamiltonian system is discussed. A classification for non-self-adjoint second-order difference equations is established, which is independent of the choice of the rotated half plane and the fixed point.     

Zeroth-order conservation laws of two-dimensional shallow water equations with variable bottom topography

We classify zeroth-order conservation laws of systems from the class of two-dimensional shallow water equations with variable bottom topography using an optimized version of the method of furcate splitting. The classification is carried out up to equivalence generated by the equivalence group of this class. We find additional point equivalences between some of the listed cases of extensions of the space of zeroth-order conservation laws, which are inequivalent up to transformations from the equivalence group. Hamiltonian structures of systems of shallow water equations are used for relating the classification of zeroth-order conservation laws of these systems to the classification of their Lie symmetries. We also construct generating sets of such conservation laws under action of Lie symmetries.     

Classification of equations of state for viscous fluids

This paper presents a classification of equations of state for viscous fluids (or gases) whose motion is governed by the Navier–Stokes equations. The classification is based on an analysis of admissible symmetries.     

Bifurcations of completely integrable 2-variable first-order partial differential equations

In this paper, we consider an implicit 2-variable first-order partial differential equation with complete integral. As an application of the Legendrian singularity theory, we give a generic classification of bifurcations of such differential equations with respect to the equivalence relation which is given by the group of point transformations following S. Lie?s view. Since two one-parameter unfoldings of such differential equations are equivalent if and only if induced one-parameter unfoldings of integral diagrams are equivalent for generic equations, our normal forms are represented by one-parameter integral diagrams.     

A bibliography of graph equations

Graph equations are equations in which the unknowns are graphs. Many problems and results in graph theory can be formulated in terms of graph equations. Here we offer a classification and a large bibliography of graph equations.     

Symmetries and invariants of the systems of two linear second-order ordinary differential equations

For arbitrary systems of two linear second-order ordinary differential equations, the symmetry Lie algebra is described in terms of invariant theory, resulting in eleven non-equivalent symmetry types. The result is compared with the group classification approach recently obtained by different authors.     

On the group classification of systems of two linear second-order ordinary differential equations with constant coefficients

The completeness of the group classification of systems of two linear second-order ordinary differential equations with constant coefficients is delineated in the paper. The new cases extend what has been done in the literature. These cases correspond to the type of equations where the commutative property of the coefficient matrices with respect to the dependent variables and the first-order derivatives in the considered system does not hold. A discussion of the results as well as a note on the extension to linear systems of second-order ordinary differential equations with more than two equations are given.     

Group classification of a family of second-order differential equations

We find the group of equivalence transformations for equations of the form y^″=A(x)y^′+F(y), where A and F are arbitrary functions. We then give a complete group classification of this family of equations using a direct method of analysis, together with the equivalence transformations.