| 1. |
- Grigoriev, Yurii, et al.
(author)
-
Delay differential equations
- 2010
-
In: Lecture Notes in Physics. - Dordrecht : Springer. - 0075-8450. ; 806, s. 251-292
-
Journal article (peer-reviewed)abstract
- 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. For the sake of completeness the chapter is started with a short introduction into the theory of delay differential equations. The mathematical background of these equations is followed by the section which deals with the definition of an admitted Lie group for them and some examples. The purpose of the next section is to give a complete group classification with respect to admitted Lie groups of a second-order delay ordinary differential equation. The reasonable generalization of the definition of an equivalence Lie group for delay differential equations is considered in the next section. The last section of the chapter is devoted to application of the developed theory to the reaction-diffusion equation with a delay.
|
|
| 2. |
- Grigoriev, Yurii, et al.
(author)
-
Introduction to group analysis and invariant solutions of integro-differential equations
- 2010
-
In: Lecture Notes in Physics. - Dordrecht : Springer. - 0075-8450. ; 806, s. 57-111
-
Journal article (peer-reviewed)abstract
- 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. The first section of this chapter contains a retrospective survey of different methods for constructing symmetries and finding invariant solutions of such equations. The presentation of the methods is carried out using simple model equations of small dimensionality, allowing the reader to follow the calculations in detail. In the next section, the classical scheme of the construction of determining equations of an admitted Lie group is generalized for equations with nonlocal operators. In the concluding sections of this chapter, the developed regular method of obtaining admitted Lie groups is illustrated by applications to some known integro-differential equations.
|
|
| 3. |
- Grigoriev, Yurii, et al.
(author)
-
Introduction to group analysis of differential equations
- 2010
-
In: Lecture Notes in Physics. - Dordrecht : Springer. - 0075-8450. ; 806, s. 1-55
-
Journal article (peer-reviewed)abstract
- 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-Bäcklund 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
|
|
| 4. |
- Grigoriev, Yurii, et al.
(author)
-
Plasma kinetic theory : Vlasov-maxwell and related equations
- 2010
-
In: Lecture Notes in Physics. - Dordrecht : Springer. - 0075-8450. ; 806, s. 145-208
-
Journal article (peer-reviewed)abstract
- 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. In the beginning sections the plasma kinetic theory equations are introduced and the way of looking at the symmetries of nonlocal equations is described. Much of the importance of the approach used in this chapter for calculating symmetries stems from the procedure of solving determining equations using variational differentiation. The set of symmetries obtained in the sections that follow comprises symmetries for the Vlasov-Maxwell equations of the non-relativistic and relativistic electron and electron-ion plasmas in both one- and three-dimensional cases, and symmetries for Benney equations. In the concluding sections of this chapter the procedure for symmetry calculation and the renormalization group algorithm go hand in hand to present illustrations from plasma kinetic theory, plasma dynamics, and nonlinear optics, which demonstrate the potentialities of the method in construction of analytic solutions to nonlocal problems of nonlinear physics.
|
|
| 5. |
- Grigoriev, Yurii, et al.
(author)
-
Symmetries of stochastic differential equations
- 2010
-
In: Lecture Notes in Physics. - Dordrecht : Springer. - 0075-8450. ; 806, s. 209-250
-
Journal article (peer-reviewed)abstract
- 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. Before defining an admitted symmetry for stochastic differential equations an introduction into the theory of this type of equations is given. The introduction includes the discussion of a stochastic integration, a stochastic differential and a change of the variables (Itô formula) in stochastic differential equations. Applications of the Itô formula are considered in the next section which deals with the linearization problem. The Itô formula and the change of time in stochastic differential equations are the main tools of defining admitted transformations for them. After introducing an admitted Lie group and supporting material of the introduced definition, some examples of applications of the given definition are studied.
|
|
| 6. |
- Grigoriev, Yurii, et al.
(author)
-
The Boltzmann kinetic equation and various models
- 2010
-
In: Lecture Notes in Physics. - Dordrecht : Springer. - 0075-8450. ; 806, s. 113-144
-
Journal article (peer-reviewed)abstract
- 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. Calculations of the 11-parameter Lie group G 11 admitted by the full Boltzmann equation with arbitrary intermolecular potential and its extensions for power potentials are presented. The found isomorphism of these Lie groups with the Lie groups admitted by the ideal gas dynamics equations allowed one to obtain an optimal system of admitted subalgebras and to classify all invariant solutions of the full Boltzmann equation. For equations similar to the full Boltzmann equation complete admitted Lie groups are derived by solving determining equations. The corresponding optimal systems of admitted subalgebras are constructed and representations of all invariant solutions are obtained.
|
|
