## About

65

Publications

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Introduction

My research is mainly related with application of algebraic methods for solving different problems of mathematical physics

Additional affiliations

January 2009 - January 2011

September 2004 - January 2014

## Publications

Publications (65)

In this series we list some important dates and events in history of mathematics and mathematical culture. Pure mathematicians just love to try unsolved problems-they love a challenge.

In this series we list some important dates and events in history of mathematics and mathematical culture. Pure mathematics is, in its way, the poetry of logical ideas.

Let $\mathfrak{h}_3$ be the Heisenberg algebra and let $\mathfrak g$ be the 3-dimensional Lie algebra having $[e_1,e_2]=e_1\,(=-[e_2,e_1])$ as its only non-zero commutation relations. We describe the closure of the orbit of a vector of structure constants corresponding to $\mathfrak{h}_3$ and $\mathfrak g$ respectively as an algebraic set giving in...

We investigate degenerations of $n$-dimensional algebras over an arbitrary infinite field paying particular attention to algebras satisfying the identity $[x,x]=0$ (where $[x_1,x_2]$ denotes the product of the elements $x_1,$ $x_2$ of the algebra). We show, for $n\ge3$, that there are precisely two non-isomorphic $n$-dimensional algebras which sati...

A full symmetry classification is given for models of energy transport in
radiant plasma when the mass density is spatially variable and the diffusivity
is nonlinear. A systematic search for conservation laws also leads to some
potential symmetries, and to an integrable nonlinear model. Classical point
symmetries, potential symmetries and nonclassi...

In this paper, we explain in more details the modern treatment of the problem
of group classification of (systems of) partial differential equations (PDEs)
from the algorithmic point of view. More precisely, we revise the classical
Lie--Ovsiannikov algorithm of construction of symmetries of differential
equations, describe the group classification...

We provide a group classification of a class of nonlinearisable evolution partial differential equations which arise in Financial Mathematics. Sixteen different cases are identified for the general problem and another seven for a restricted version. In the cases for which the algebra is suitable we determine the solution to the problem u(0,x)=U, wh...

Nonclassical symmetries of a class of generalized Huxley equations of form $u_t=u_{xx}+k(x)u^2(1-u)$ are found. More precisely, for the class under consideration we completely classify reduction operators with $\tau=1$ and give a wide number of examples of equations admitting reduction operators with $\tau=0$. Comment: contribution to the Proceedin...

The modern group analysis of differential equations is used to study a class of two-dimensional variable coefficient Burgers
equations. The group classification of this class is performed. Equivalence transformations are also found that allow us to
simplify the results of classification and to construct the basis of differential invariants and oper...

In the present paper we perform the complete group classification of a class of three-dimensional variable-coefficient Burgers equation. We construct the optimal system of three-dimensional subalgebras of Lie symmetry algebra for an equation from this class. We give an example of construction of exact solutions via reduction to ordinary differentia...

A generalization of the usual procedure for constructing potential systems for systems of partial differential equations with
multidimensional spaces of conservation laws is considered. More precisely, for the construction of potential systems with
a multi-dimensional space of local conservation laws, instead of using only basis conservation laws,...

We consider the (1+3)-dimensional Burgers equation ut=uxx+uyy+uzz+uux which has considerable interest in mathematical physics. We complete the list of similarity reductions that are obtained from the Lie symmetries admitted by this equation. To achieve this goal we employ two- and three-dimensional subalgebras of the Lie symmetry algebra, in additi...

We show that the so-called hidden potential symmetries considered in a recent paper [M.L. Gandarias, New potential symmetries for some evolution equations, Physica A 387 (2008) 2234–2242] are ordinary potential symmetries that can be obtained using the method introduced by Bluman and collaborators [G.W. Bluman, S. Kumei, Symmetries and Differential...

In this paper we consider generalization of procedure of construction of potential systems for systems of partial differential equations with multidimensional spaces of conservation laws. More precisely, for construction of potential systems in cases when dimension of the space of local conservation laws is greater than one, instead of using only b...

We perform the complete Lie group classification of a (2+1)- and a (3+1)-dimensional classes of non-linear diffusion–convection equations. This classification generalizes and completes existing results in the literature. The derived Lie symmetries are used for construction of similarity reductions and exact solutions of certain equations from both...

We classify local first-order conservation laws for a class of systems of nonlinear diffusion equations. The derived conservation laws are used to construct the set of inequivalent potential systems for the class under consideration. Four potential systems are investigated from the Lie point of view and new potential symmetries are obtained. An exa...

This paper completes investigation of symmetry properties of nonlinear variable coefficient diffusion-convection equations of the form $f(x)u_t=(g(x)A(u)u_x)_x+h(x)B(u)u_x$. Potential symmetries of equations from the considered class are found and the connection of them with Lie symmetries of diffusion-type equations is shown. Exact solutions of th...

In the presented paper known (up to the beginning of 2008) Lie- and non-Lie exact solutions of different $(1+1)$-dimensional diffusion-convection equations of form $f(x)u_t=(g(x)A(u)u_x)_x+h(x)B(u)u_x$ are collected.

The notions of generating sets of conservation laws of systems of differential equations with respect to symmetry groups and equivalence groups are introduced and applied. This allows us to generalize essentially the procedure of finding potential symmetries for the systems with multidimensional spaces of conservation laws. A class of variable coef...

This is the second part of the series of papers on symmetry properties of a class of variable coefficient (1+1)-dimensional nonlinear diffusion-convection equations of general form $f(x)u_t=(g(x)A(u)u_x)_x+h(x)B(u)u_x$. At first, we review the results of Part 1 of the series on equivalence transformations and group classification of the class under...

We discuss the classical statement of group classification problem and some its extensions in the general case. After that, we carry out the complete extended group classification for a class of (1+1)-dimensional nonlinear diffusion--convection equations with coefficients depending on the space variable. At first, we construct the usual equivalence...

We study conservation laws and potential symmetries of (systems of) differential equations applying equivalence relations
generated by point transformations between the equations. A Fokker–Planck equation and the Burgers equation are considered
as examples. Using reducibility of them to the one-dimensional linear heat equation, we construct complet...

Classifications of symmetries and conservation laws are presented for a variety of physically and analytically interesting wave equations with power nonlinearities in n spatial dimensions: a radial hyperbolic equation, a radial Schrödinger equation and its derivative variant, and two proposed radial generalizations of modified Korteweg–de Vries equ...

We carry out an extensive investigation of conservation laws and potential symmetries for the class of linear (1+1)-dimensional second-order parabolic equations. The group classification of this class is revised by employing admissible transformations, the notion of normalized classes of differential equations and the adjoint variational principle....

The fast diffusion equation ut=x(u−1ux) is investigated from the symmetry point of view in development of the paper by Gandarias [M.L. Gandarias, Phys. Lett. A 286 (2001) 153]. After studying equivalence of nonclassical symmetries with respect to a transformation group, we completely classify the nonclassical symmetries of the corresponding potenti...

A complete group classification of a class of variable coefficient (1+1)-dimensional telegraph equations $f(x)u_{tt}=(H(u)u_x)_x+K(u)u_x$, is given, by using a compatibility method and additional equivalence transformations. A number of new interesting nonlinear invariant models which have non-trivial invariance algebras are obtained. Furthermore,...

We consider the variable coefficient diffusion–convection equation of the form f(x)ut=[g(x)D(u)ux]x+h(x)K(u)ux which has considerable interest in mathematical physics, biology and chemistry. We present a complete group classification for this class of equations. Also we derive equivalence transformations between equations that admit Lie symmetries....

Any partial differential equation PDE system can be effectively analyzed through consideration of its tree of nonlocally related systems. If a given PDE system has n local conservation laws, then each conservation law yields potential equations and a corresponding nonlocally related potential system. Moreover, from these n con-servation laws, one c...

Any PDE system can be effectively analyzed through consideration of its tree of non-locally related systems. If a given PDE system has n local conservation laws, then each conservation law yields potential equations and a corresponding nonlocally related potential system. Moreover, from these n conservation laws, one can directly construct 2 n − 1...

All possible linearly independent local conservation laws for n-dimensional diffusion–convection equations u
t=(A(u))ii
+(B
i(u))i
were constructed using the direct method and the composite variational principle. Application of the method of classification of conservation laws with respect to the group of point transformations [R.O.~Popovych, N.M....

We study local conservation laws of variable coefficient diffusion-convection equations of the form $f(x)u_t=(g(x)A(u)u_x)_x+h(x)B(u)u_x$. The main tool of our investigation is the notion of equivalence of conservation laws with respect to the equivalence groups. That is why, for the class under consideration we first construct the usual equivalenc...

We introduce notions of equivalence of conservation laws with respect to Lie symmetry groups for fixed systems of differential equations and with respect to equivalence groups or sets of admissible transformations for classes of such systems. We also revise the notion of linear dependence of conservation laws and define the notion of local dependen...

We investigate conservation laws of diffusion-convection equations to construct first-order potential systems corresponding to these equations. We do two iterations of the construction procedure, looking, in the second step, for the first-order conservation laws of the potential systems obtained in the first step.

We perform the complete group classification in the class of nonlinear Schrödinger equations of the form iψ t +ψ xx +|ψ| γ ψ+V (t, x)ψ = 0 where V is an arbitrary complex-valued potential depending on t and x, γ is a real non-zero constant. We construct all the possible inequivalent potentials for which these equations have non-trivial Lie symmetri...

Potential equivalence transformations (PETs) are effectively applied to a class of nonlinear diffusion-convection equations. For this class all possible potential symmetries are classified and a theorem on connection of them with point ones via PETs is also proved. It is shown that the known non-local transformations between equations under conside...

We perform the complete group classification in the class of nonlinear Schr\"odinger equations of the form $i\psi_t+\psi_{xx}+|\psi|^\gamma\psi+V(t,x)\psi=0$ where $V$ is an arbitrary complex-valued potential depending on $t$ and $x,$ $\gamma$ is a real non-zero constant. We construct all the possible inequivalent potentials for which these equatio...

We perform the complete group classification in the class of cubic Schr\"odinger equations of the form $i\psi_t+\psi_{xx}+\psi^2\psi^*+V(t,x)\psi=0$ where $V$ is an arbitrary complex-valued potential depending on $t$ and $x$. We construct all possible inequivalent potentials for which these equations have non-trivial Lie symmetries using algebraic...

Using a new method and additional (conditional and partial) equivalence transformations, we performed group classification in a class of variable coefficient $(1+1)$-dimensional nonlinear diffusion-convection equations of the general form $f(x)u_t=(D(u)u_x)_x+K(u)u_x.$ We obtain new interesting cases of such equations with the density $f$ localized...

In the presented paper known (up to the beginning of 2008) Lie- and non-Lie exact solutions of dieren t (1+1)-dimensional diusion{con vection equations of form f(x)ut = (g(x)A(u)ux)x + h(x)B(u)ux are collected.

A complete group classification of a class of reaction-diffusion equations of the form u t =u xx +k(x)u 2 (1-u) is given. Lie symmetries are used to reduce these reaction-diffusion equations to ordinary differential equations.

We employ the infinitesimal method for calculating invariants of families of differential equations using equivalence groups. We apply the method to the class of semilinear wave equations u tt −u xx = f (x, u, u t , u x). We show that this class of equations admits four functionally independent differential invariants of second order. We employ the...

## Projects

Projects (2)

In this series we list some important dates and events in history of mathematics and mathematical culture. A day-by-day version of this calendar can be found at https://www.facebook.com/This-day-in-history-of-mathematics-103991334328402/