Regular, partially invariant solutions of rank 1 and defect 1 of equations of plane motion of a viscous heat-conducting gas

A system of the Navier-Stokes equations of two-dimensional motion of a viscous heat-conducting perfect gas with a polytropic equation of state is considered. Regular, partially invariant solutions of rank 1 and defect 1 are studied. A sufficient condition of their reducibility to invariant solutions of rank 1 is proved. All solutions of this class with a linear dependence of the velocity-vector components on spatial coordinates are examined. New examples of solutions that are not reducible to invariant solutions are obtained. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 6, pp. 23–33, November–December, 2006.     

One class of partially invariant solutions of the Navier-Stokes equations

A family of partially invariant solutions of the Navier-Stokes equations of rank 2 and defect 2 is considered. These solutions describe the three-dimensional unsteady motions of a viscous incompressible fluid in which the vertical velocity component and the pressure are independent of the horizontal coordinates. In particular, they can be interpreted as flows in a horizontal layer, one boundary of which is the free surface. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 2, pp. 24–33, March–April, 1999.     

Regular submodels of types (1,2) and (1,1) of the equations of gas dynamics

Partially invariant solutions of types (1, 2) and (1, 1) for gas-dynamic equations are regularly divided into two classes: for the first class, the invariant independent variable is the time, i.e., this class contains barochronic solutions, and for the second class, the invariant variable necessarily depends on spatial coordinates. The barochronic submodel of gas-dynamic equations, as well as a passive subsystem for solutions of the second class, is integrated in finite form. In the latter case, the invariant subsystem is reduced to an ordinary differential equation and quadratures. Integration of the submodels is illustrated by a number of examples. The following common properties of barochronic gas flows are described: rectilinear trajectories of gas particles, the possibility of collapse of density on a manifold, and stratification of the space of events. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 2, pp. 40–49, March–April, 1999.     

Exact solutions of the equations of motion for an incompressible viscoelastic Maxwell medium

The unsteady plane-parallel motion of a incompressible viscoelastic Maxwell medium with constant relaxation time is considered. The equations of motion of the medium and the rheological relation admit an extended Galilean group. The class of solutions of this system which are partially invariant with respect to the subgroup of the indicated group generated by translation and Galilean translation along one of the coordinate axes is studied. The system does not have invariant solutions, and the set of partially invariant solutions is very narrow. A method for extending the set of exact solutions is proposed which allows finding solutions with a nontrivial dependence of the stress tensor elements on spatial coordinates. Among the solutions obtained by this method, the solutions describing the deformation of a viscoelastic strip with free boundaries is of special interest from a point of view of physics. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 2, pp. 16–23, March–April, 2009.     

Galilean-invariant axisymmetric self-similar submodel of gas dynamics without swirling

This paper deals with one insufficiently studied submodel of invariant solutions of rank 1 of the equations of gas dynamics. It is shown that, in cylindrical coordinates, the submodel without swirling reduces to a system of two ordinary differential equations. For the equation of state with additional invariance, a self-similar system is obtained. A pattern of phase trajectories is constructed, and particle motion is studied using asymptotic methods. The obtained solutions describe unsteady flows over axisymmetric bodies with possible strong discontinuities. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 2, pp. 46–52, March–April, 2009.     

Gas flows with helical surfaces of the level

Some properties of the invariant gas-dynamic submodel of rank 2 with spiral surfaces of the level are reported. Invariant and isobaric solutions of the submodel are considered. Institute of Mechanics, Ufa Scientific Center, Russian Academy of Sciences, Ufa 450000. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 2, pp. 34–39, March–April, 1999.     

Relationship between invariant sets of systems of differential equations and the corresponding difference equations

We study the relationship between invariant sets of systems of differential equations and the corresponding difference equations in terms of sign-constant Lyapunov functions. For systems of differential equations, we obtain a converse result concerning the existence of a positive-definite Lyapunov function whose zeros coincide with a given invariant manifold. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 2, pp. 280–285, April–June, 2006.     

Invariant and Partially Invariant Solutions of the Green-Naghdi Equations

All invariant and partially invariant solutions of the Green-Naghdi equations are obtained that describe the second approximation of shallow water theory. It is proved that all nontrivial invariant solutions belong to one of the following types: Galilean-invariant, stationary, and self-similar solutions. The Galilean-invariant solutions are described by the solutions of the second Painleve equation, the stationary solutions by elliptic functions, and the self-similar solutions by the solutions of the system of ordinary differential equations of the fourth order. It is shown that all partially invariant solutions reduce to invariant solutions. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 6, pp. 26–35, November–December, 2005.     

Invariant recording of elasticity theory equations

An invariant (with respect to rotations) formalization of equations of linear and nonlinear elasticity theory is proposed. An equation of state (in the form of a convex generating potential) for various crystallographic systems is written. An algebraic approach is used, which does not require any geometric constructions related to the analysis of symmetry in crystals. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 5, pp. 127–142, September–October, 2008.     

Solving initial–boundary-value creep problems

The Galerkin–Bubnov method with global approximations is used to find approximate solutions to initial–boundary-value creep problems. It is shown that this approach allows obtaining solutions available in the literature. The features of how the solutions of initial–boundary-value problems for oneand three-dimensional models are found are analyzed. The approximate solutions found by the Galerkin–Bubnov method with global approximations is shown to be invariant to the form of the equations of the initial–boundary-value problem. It is established that solutions of initial–boundary-value creep problems can be classified according to the form of operators in the mathematical problem formulation     

On invariant tori of weakly connected systems of differential equations

We consider a family of systems of differential equations depending on a sufficiently small parameter, whose zero value corresponds to a couple of independent systems. We use the method of Green-Samoilenko function for the construction of an invariant manifold of the perturbed system and present some examples of application. Published in Neliniini Kolyvannya, Vol. 8, No. 4, pp. 468–489, October–December, 2005.     

Nonlinear differential equations with solutions bounded on ℝ

We present assertions on bounded solutions of nonlinear differential equations. __________ Translated from Neliniini Kolyvannya, Vol. 11, No. 1, pp. 96–111, January–March, 2007.     

Existence of positive solutions of systems of Volterra nonlinear difference equations

In this paper, a classification scheme for eventually positive solutions of a class of two-dimensional Volterra nonlinear difference equations is given in terms of asymptotic magnitudes. Some necessary as well as sufficient conditions for the existence of such solutions are provided without any monotonicity conditions on the nonlinear term. Published in Neliniini Kolyvannya, Vol. 9, No. 1, pp. 37–47, January–March, 2006.     

Asymptotic properties of solutions of systems of nonlinear functional differential equations of neutral type

We establish sufficient conditions for systems of nonlinear functional differential equations of neutral type to have solutions that are continuously differentiable and bounded for t ∈ ℝ (together with their first derivatives) and investigate the asymptotic properties of these solutions. Translated from Neliniini Kolyvannya, Vol. 12, No. 1, pp. 20–26, January–March, 2009.     

Asymptotics of solutions of nonautonomous second-order ordinary differential equations close to linear equations

We find asymptotic representations for certain classes of solutions of nonautonomous second-order differential equations that are close, in a certain sense, to linear equations. __________ Translated from Neliniini Kolyvannya, Vol. 11, No. 2, pp. 230–241, April–June, 2008.     

Existence of an invariant torus for one class of systems of differential equations

We consider the problem of the existence of an asymptotically stable toroidal set for a system of linear differential equations defined on an m-dimensional torus. We establish conditions under which a nonlinear system of differential equations has an invariant toroidal manifold. Translated from Neliniini Kolyvannya, Vol. 11, No. 4, pp. 520–529, October–December, 2008.     

Two-dimensional gas vortices and twisted gas jets

This paper studies an invariant solution of rank one of the equations of motion of a polytropic gas that describes two-dimensional gas vortices and twisted gas jets. Flow types are classified according to the governing parameter: vortices in the form of sources and sinks, unlimited expansion, and collapse. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 3, pp. 71–81, May–June, 2009.     

Generalized solitonary and periodic solutions for nonlinear partial differential equations by the Exp-function method

In this paper, the Exp-function method with the aid of the symbolic computational system Maple is used to obtain the generalized solitonary solutions and periodic solutions for nonlinear evolution equations arising in mathematical physics, namely, (2+1)-dimensional Konopelchenko–Dubrovsky equations, the (3+1)-dimensional Jimbo–Miwa equation, the Kadomtsev–Petviashvili (KP) equation, and the (2+1)-dimensional sine-Gordon equation. It is shown that the Exp-function method, with the help of symbolic computation, provides a powerful mathematical tool for solving other nonlinear evolution equations arising in mathematical physics.     

Bounded solutions of linear differential equations in a banach space

For a linear inhomogeneous differential equation in a Banach space, we find a criterion for the existence of solutions that are bounded on the entire real axis under the assumption that the homogeneous equation admits an exponential dichotomy on the semiaxes. This result is a generalization of the Palmer lemma to the case of infinite-dimensional spaces. We consider examples of countable systems of ordinary differential equations that have bounded solutions. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 1, pp. 3–14, January–March, 2006.     

Invariant submodels of rank two of the equations of gas dynamics

Invariant submodels of rank two of systems of gas-dynamic equations with a general equation of state are described. All submodels (26 representatives) are divided into two, classes—evolutionary and stationary. New relations and independent variables and the coefficients and right sides of the corresponding systems of equations are given. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 2, pp. 50–55, March–April, 1999.