Sergey V. Meleshko

Symmetries of Integro-Differential Equations

This book aims to coherently present applications of group analysis to integro-differential equations in an accessible way. The book will be useful to both physicists and mathematicians interested in general methods to investigate nonlinear problems using symmetries. Differential and integro-differential equations, especially nonlinear, present the most effective way for describing complex processes. Therefore, methods to obtain exact solutions of differential equations play an important role in physics, applied mathematics and mechanics. This book provides an easy to follow, but comprehensive, description of the application of group analysis to integro-differential equations. The book is primarily designed to present both fundamental theoretical and algorithmic aspects of these methods. It introduces new applications and extensions of the group analysis method. The authors have designed a flexible text for postgraduate courses spanning a variety of topics.

On linearization of third-order ordinary differential equations

A new algorithm for linearization of a third-order ordinary differential equation is presented. The algorithm consists of composition of two operations: reducing order of an ordinary differential equation and using the Lie linearization test for the obtained second-order ordinary differential equation. The application of the algorithm to several ordinary differential equations is given.

Linearization of fourth-order ordinary differential equations by point transformations

The solution of the problem on linearization of fourth-order equations by means of point transformations is presented here. We show that all fourth-order equations that are linearizable by point transformations are contained in the class of equations which is linear in the third-order derivative. We provide the linearization test and describe the procedure for obtaining the linearizing transformations as well as the linearized equation. For ordinary differential equations of order greater than 4 we obtain necessary conditions, which separate all linearizable equations into two classes.

Group classification of the equations of two-dimensional motions of a gas☆

Abstract A generalization of the definition of an equivalence group is proposed and a group classification of a system of equations describing the two-dimensional flows of an ideal gas [1] is given. Plane flows, which were earlier investigated from the group point of view in [2–4], are a special case of such motions. The algebraic approach employed rests on an analysis which has recently been developed [5, 6].

Reduction Procedure and Generalized Simple Waves for Systems Written in Riemann Variables

In this article, the method of differential constraintsis applied for systems written in Riemann variables. Westudied generalized simple waves. This class of solutions can beobtained by integrating a system of ordinary differentialequations. Two models from continuum mechanics are studied:traffic flow and rate-type models.

Linearization of Second-Order Ordinary Differential Equations by Generalized Sundman Transformations

The linearization problem of a second-order ordinary differential equation by the generalized Sundman transformation was considered earlier by Duarte, Moreira and Santos using the Laguerre form. The results obtained in the present paper demonstrate that their solution of the linearization problem for a second-order ordinary differential equation via the generalized Sundman transformation is not complete. We also give examples which show that the Laguerre form is not sufficient for the linearization problem via the generalized Sundman transformation.

Homogeneous autonomous systems with three independent variables

Abstract Double-wave solutions of equations with three independent variables are studied. The case when a homogeneous autonomous system consisting of four independent quasilinear first-order differential equations can be formed to study compatibility is considered. All such systems having solutions with an arbitrary function that cannot be reduced to invariant ones are given and their solutions are found.

On definition of an admitted Lie group for functional differential equations

Abstract The manuscript is devoted to applications of group analysis to functional differential equations. It is given a definition of an admitted Lie group for such type of equations and some examples of applications of this definition are studied. The way for constructing an admitted Lie group is similar to the way developed for differential equations: first, one has to construct determining equations, then to split these equations with respect to arbitrary elements, and then to find the general solution of these equations. Particularly, for delay differential equations the process of splitting determining equations and solving them is similar to the case of differential equations. The proposed approach can also be applied for finding an equivalence group, contact and Lie–Backlund transformations.

APPLICATION OF THE GENERALISED SUNDMAN TRANSFORMATION TO THE LINEARISATION OF TWO SECOND-ORDER ORDINARY DIFFERENTIAL EQUATIONS

In the literature, the generalized Sundman transformation has been used for obtaining necessary and sufficient conditions for a single second- and third-order ordinary differential equation to be equivalent to a linear equation in the Laguerre form. As far as we are aware, the generalized Sundman transformation has not been applied to a system of equations. The motivation of this work is then to expand the application of the generalized Sundman transformation to a system of ordinary differential equations, in particular, to a system of two second-order ordinary differential equations.

Classification of invariant solutions of the Boltzmann equation

An isomorphism of the Lie algebras L11 admissible by the full Boltzmann kinetic equation with an arbitrary differential cross section and by the Euler gas dynamics system of equations with a general state equation is set up. The similarity is also proved between extended algebras L12 admissible by the same equations for specified power-like intermolecular potentials and for polytropic gas. This allows the solution of the problem of classification of the full Boltzmann equation invariant H-solutions using an optimal system of subalgebras known for the Euler system. Representations of essentially different H-solutions of the spatially inhomogeneous Boltzmann equation with one and two independent invariant variables in the explicit form are obtained on this basis.