Yurii N. Grigoriev

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.

Application of group analysis to the spatially homogeneous and isotropic Boltzmann equation with source using its Fourier image

Group analysis of the spatially homogeneous and molecular energy dependent Boltzmann equations with source term is carried out. The Fourier transform of the Boltzmann equation with respect to the molecular velocity variable is considered. The correspondent determining equation of the admitted Lie group is reduced to a partial differential equation for the admitted source. The latter equation is analyzed by an algebraic method. A complete group classification of the Fourier transform of the Boltzmann equation with respect to a source function is given. The representation of invariant solutions and corresponding reduced equations for all obtained source functions are also presented.

Plasma Kinetic Theory: Vlasov–Maxwell and Related Equations

This chapter is devoted to a group analysis of the Vlasov–Maxwell and related type equations. The equations form the basis of the collisionless plasma kinetic theory, and are also applied in gravitational astrophysics, in shallow-water theory, etc. Nonlocal operators in these equations appear in the form of the functionals defined by integrals of the distribution functions over momenta of particles.

Symmetries of Stochastic Differential Equations

This chapter deals with applications of the group analysis method to stochastic differential equations. These equations are often obtained by including random fluctuations in differential equations, which have been deduced from phenomenological or physical view. In contrast to deterministic differential equations, only few attempts to apply group analysis to stochastic differential equations can be found in the literature. It is worth to note that this theory is still developing.

Introduction to Group Analysis of Differential Equations

The first chapter is a brief, but a sufficiently comprehensive introduction to the methods of Lie group analysis of ordinary and partial differential equations. The chapter presents basic concepts from the theory: continuous transformation groups, their generators, Lie equations, groups admitted by differential equations, integration of ordinary differential equations using their symmetries, group classification and invariant solutions of partial differential equations. New trends in modern group analysis such as the theory of Lie–Backlund transformations groups and approximate groups are also reflected. The intention of the chapter is to give the basic ideas of classical and modern group analysis to beginner readers and provide useful materials for advanced specialists.

The Boltzmann Kinetic Equation and Various Models

The chapter deals with applications of the group analysis method to the full Boltzmann kinetic equation and some similar equations. These equations form the foundation of the kinetic theory of rarefied gas and coagulation. They typically include special integral operators with quadratic nonlinearity and multiple kernels which are called collision integrals.

Delay Differential Equations

In this chapter, applications of group analysis to delay differential equations are considered. Many mathematical models in biology, physics and engineering, where there is a time lag or aftereffect, are described by delay differential equations. These equations are similar to ordinary differential equations, but their evolution involves past values of the state variable.

Introduction to Group Analysis and Invariant Solutions of Integro-Differential Equations

In this chapter an introduction into applications of group analysis to equations with nonlocal operators, in particular, to integro-differential equations is given. The most known integro-differential equations are kinetic equations which form a mathematical basis in the kinetic theories of rarefied gases, plasma, radiation transfer, coagulation. Since these equations are directly associated with fundamental physical laws, there is special interest in studies of their solutions.