Group properties and invariant solutions of equations of an electric field in the case of nonlinear ohm's law

The group properties of one-dimensional nonstationary equations of an electric field in homogeneous isotropic media with nonlinear conductivity are considered. The nonlinear Ohm's laws for which these equations have the broadest symmetry properties are determined. Ordinary differential equations determining invariants solutions are obtained; the order of the equations is lowered or they are integrated to the end.Translated from Zhurnal Prikladnoi Mekhanika i Tekhnicheskaya Fizika, No. 3, pp. 28–36, May–June, 1972.     

Numerical simulation of flow of multicomponent mixtures in porous media

A study is made of the isothermal flow of multicomponent mixtures in a porous medium, accompanied by phase transitions, interphase mass exchange, and change in the physicochemical properties of the phases [1–3], It is assumed that at each point of the flow region, phase equilibrium is established instantaneously and the flow velocities of the separate phases conform to Darcy's law. Approximate solutions of problems of displacing oil by high-pressure gas were obtained in [1]. By generalizing the theory developed in [4], a study is made in [5] of the structure of the exact solutions of the problems of the flow of three-component systems which describe the displacement of oil by different reactants (gases, solvents, micellar solutions). The numerical solutions of the problems of multicomponent system flow are considered in [2, 3, 6, 7]. This paper presents a numerical method which is distinguished from the well-known ones [2, 3, 6, 7] by the following characteristics. The flow equations are approximated by a completely conservative finite-difference scheme of the implicit pressure-explicit saturation type, the calculation being carried out using Newton's method of iteraction with spect to both the pressure and the composition of the mixture. The minimum derivative principle [8] is used in the approximation of the divergence terms of the equations. The phase equilibrium is calculated using the equation of state.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 101–110, July–August, 1985.     

Oscillation of Nonlinear Hyperbolic Differential Equations with Impulses

We study oscillatory properties of solutions of nonlinear impulsive hyperbolic differential equations and find new necessary and sufficient conditions for the existence of oscillations.__________Published in Neliniini Kolyvannya, Vol. 7, No. 4, pp. 439–445, October–December, 2004.     

The motion of snow avalanches in the conditions of limiting friction

We investigate the equations of motion of large snow avalanches, and in contrast with [1–3] we take into account the fact that the dry friction can reach a critical value above which the snow in the avalanche or the underlaying material cannot sustain the friction. We find asymptotic solutions for long times after the beginning of motion. These solutions describe the avalanche motion in which a part of the snow moves in the conditions of limiting friction over a tilted plane with a uniform layer of snow. The equations which are used to find these asymptotic solutions have the property that for certain depths the flow velocity of small perturbations decreases with increasing depth. This is related to a number of unusual features (from the hydraulic point of view) of the solutions. In particular, on relatively gentle slopes two zones are formed in the avalanche: the forward part, with a large velocity and thickness of the moving layer, and the rear part, which is significantly slower and thinner. The two parts are separated by a narrow region characterized by a sharp decline in velocity and thickness of the moving layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 30–37, September–October, 1977.     

On a class of exact particular solutions of the equations of transonic gas flows

The paper presents exact particular solutions of the equations of transonic gas flows, analogous to the solutions derived in [1–3] for the case of short waves. These solutions are used to construct the flow around a body in a supersonic stream with an attached shock.     

Screw flow of a liquid in a cylindrical tube

A study is made of the flow of a viscous incompressible liquid with helical streamlines in an infinite cylindrical tube within which a screw rotates (auger). Generalized linearized Oseen equations are derived, and one class of the exact solutions of these is identical with the corresponding class of exact solutions of the complete Navier-Stokes equations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Shidkosti i Gaza, No. 6, pp. 3–7, November–December, 1979.     

Some exact solutions of the ferrohydrodynamics equations

The theoretical study of nonisothermal flows of magnetizable liquids presents serious matheical difficulties, which are intensified as compared to to the study of normal liquids by the necessity of simultaneous solution of both the hydrodynamics and Maxwell's equations, with corresponding boundary conditions for the magnetic field. Thus, in most cases problems of this type are solved by neglecting the effect of the liquid's nonisothermal state on the field distribution within the liquid, and also, as a rule, with use of an approximate solution for Maxwell's equations and fulfillment of the boundary conditions for the field [1–3]. The present study will present easily realizable practical formulations of the problem which permit exact one-dimensional solutions of the complete system of the equations of thermomechanic s of electrically nonconductive incompressible Newtonian magnetizable liquids with constant transfer coefficients. A common feature of the formulations is the presence of a longitudinal temperature gradient along the boundaries along which liquid motion is accomplished. Plane-parallel convective flows of this type in a conventional liquid and their stability were considered in [4–6].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 126–133, May–June, 1979.     

Remarks on the Perturbation Methods in Solving the Second-Order Delay Differential Equations

The paper presents a study on the validity of perturbation methods, suchas the method of multiple scales, the Lindstedt–Poincaré method and soon, in seeking for the periodic motions of the delayed dynamic systemsthrough an example of a Duffing oscillator with delayed velocityfeedback. An important observation in the paper is that the method ofmultiple scales, which has been widely used in nonlinear dynamics, worksonly for the approximate solutions of the first two orders, and givesrise to a paradox for the third-order approximate solutions of delaydifferential equations. The same problem appears when theLindstedt–Poincaré method is implemented to find the third-orderapproximation of periodic solutions for delay differential equations,though it is effective in seeking for any order approximation ofperiodic solutions for nonlinear ordinary differential equations. Apossible explanation to the paradox is given by the results obtained byusing the method of harmonic balance. The paper also indicates thatthese perturbation methods, despite of some shortcomings, are stilleffective in analyzing the dynamics of a delayed dynamic system sincethe approximate solutions of the first two orders already enable one togain an insight into the primary dynamics of the system.     

A realization of the moment method

The difficulties involved in solving a system of moment equations using two-sided distributions are analyzed. The properties of these distributions do not permit the realization of the moment method (in the case of a collision integral in Boltzmann form) in specific transport boundary-value problems. A method of obtaining analytic solutions of the system of moment equations for linearized transport problems is proposed. The accuracy of the method is analyzed with reference to a classical transport problem.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 185–188, May–June, 1991.     

Group-invariant solutions of kinetic equations

In the development of analytic methods of solution of kinetic equations, it is expedient to use group raetliods. The establishment of a symmetry group makes it possible to justify the choice of a definite model of kinetic equation corresponding to the physical formulation of the problem, to solve the Cauchy problem in a number of cases, and to obtain classes of new exact solutions that can be used as standards in the construction of numerical algorithms for solving kinetic equations. Bobylev [1–4] and Krook and Wu [5, 6] used group methods to analyze the spatially homogeneous Boltzmann equation in the case of isotropy with respect to the velocities and Maxwellian molecules. They obtained exact solutions and investigated the asymptotic behavior of the main equation. In the present paper, group methods are used to find and analyze exact solutions of the Bhatnagar-Gross-Krook kinetic equation, which successfully simulates the basic properties of the Boltzmann equation. Conclusions are drawn about the symmetries of the Boltzmann equation. To simplify the calculations, the exposition is presented for the case of the one-dimensional Bhatnagar-Gross-Krook equation with constant effective collision frequency.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 135–140, July–August, 1982.     

An invariant solution of the turbulent boundary-layer equations

The problem of the group stratification of the system of equations describing motion in the laminar sublayer and the turbulent core is considered. The fundamental group admissible by the initial system is constructed; invariant solutions constructed on one of the subgroups lead to a system of ordinary differential equations. Joining of the solutions and interchange of the equations occur at the boundary of the laminar sublayer. A class of power-law flows of a turbulent boundary layer is investigated. In the region of decelerated motion a double-valued solution is found corresponding to attached or separated flow. The commonly used integral characteristics are calculated and presented in the form of an interpolation polynomial.Translated from Zhurnal Prikladnoi Mekhaniki i Teknicheskoi Fiziki, No. 4, pp. 126–132, July–August, 1975.     

Quasi-one-dimensional theory of peristaltic flows

General properties of the quasi-one-dinensional equations describing peristaltic transport of a fluid along a tube are considered. Possibilities of obtaining analytic solutions are indicated.Translated from Izvestiya Akadenii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 89–97, September–October, 1984.     

Lie Symmetry Analysis of Differential Equations in Finance

Lie group theory is applied to differential equations occurring as mathematical models in financial problems. We begin with the complete symmetry analysis of the one-dimensional Black–Scholes model and show that this equation is included in Sophus Lie's classification of linear second-order partial differential equations with two independent variables. Consequently, the Black–Scholes transformation of this model into the heat transfer equation follows directly from Lie's equivalence transformation formulas. Then we carry out the classification of the two-dimensional Jacobs–Jones model equations according to their symmetry groups. The classification provides a theoretical background for constructing exact (invariant) solutions, examples of which are presented.     

Almost Periodic Solutions and Global Attractors of Non-autonomous Navier–Stokes Equations

The article is devoted to the study of non-autonomous Navier–Stokes equations. First, the authors have proved that such systems admit compact global attractors. This problem is formulated and solved in the terms of general non-autonomous dynamical systems. Second, they have obtained conditions of convergence of non-autonomous Navier–Stokes equations. Third, a criterion for the existence of almost periodic (quasi periodic, almost automorphic, recurrent, pseudo recurrent) solutions of non-autonomous Navier–Stokes equations is given. Finally, the authors have derived a global averaging principle for non-autonomous Navier–Stokes equations.     

Dynamic surface properties of solutions of colloidal surfactants

The dilatational dynamic properties of surfactant solutions are determined. In part 1 the boundary conditions at the free surface, corresponding to small perturbations, are formulated for the equations of motion of a viscous incompressible fluid. The exchange of surfactant between the surface layer and the volume phase and the kinetics of micelle formation in the volume phase are taken into account. In part 2 a solution of the boundary-value problem is proposed. The spectral density of the fluctuations of the interphase boundary displacement is found on the basis of the known static correlation function. Finally, part 3 gives expressions for the complex dynamic surface elasticity corresponding to the first and second stages of the process of micelle formation. The discussion is mainly confined to the particular case of a liquid-gas interphase boundary. Limiting relations for high and low frequencies are presented.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 105–114, March–April, 1989.     

Nonlinear interactions and chaotic dynamics of suspended cables with three-to-one internal resonances

Nonlinear planar oscillations of suspended cables subjected to external excitations with three-to-one internal resonances are investigated. At first, the Galerkin method is used to discretize the governing nonlinear integral–partial-differential equation. Then, the method of multiple scales is applied to obtain the modulation equations in the case of primary resonance. The equilibrium solutions, the periodic solutions and chaotic solutions of the modulation equations are also investigated. The Newton–Raphson method and the pseudo-arclength path-following algorithm are used to obtain the frequency/force–response curves. The supercritical Hopf bifurcations are found in these curves. Choosing these bifurcations as the initial points and applying the shooting method and the pseudo-arclength path-following algorithm, the periodic solution branches are obtained. At the same time, the Floquet theory is used to determine the stability of the periodic solutions. Numerical simulations are used to illustrate the cascades of period-doubling bifurcations leading to chaos. At last, the nonlinear responses of the two-degree-of-freedom model are investigated.     

Self-similar solutions of three-dimensional laminar magnetohydrodynamic boundary-layer equations

Self-similar solutions of three-dimensional boundary-layer equations of an incompressible fluid in ordinary hydrodynamics were considered in [1–3] et al. The present work looks for self-similar solutions of three-dimensional magnetohydrodynamic boundary-layer equations.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 9, No. 4, pp. 10–17, July–August, 1968.     

The Method of R-Functions in the Solution of Elastic Problems on the Basis of Reissner's Mixed Variational Principle

A method is presented for solving boundary-value elastic problems on the basis of the variational–structural method of R-functions and Reissner's mixed variational principle. A mathematical formulation is given to problems on the deformation of elastic bodies under mixed boundary conditions and bodies interacting with smooth rigid dies. Solutions satisfying all the boundary conditions are proposed. For undetermined components of these solutions, the resolving equations are derived and their properties are studied. A posteriori estimation of numerical solutions is made. As examples, solutions are found to a problem on the stress–strain state of a short cylinder and to a contact problem on a cylinder interacting with a smooth die. A numerical method of solving such problems is analyzed for convergence, and the accuracy of the solutions is estimated.     

Construction of Dynamically Equivalent Time-Invariant Forms for Time-Periodic Systems

In this study dynamically equivalent time-invariant forms are obtained for linear and non-linear systems with periodically varying coefficients via Lyapunov–Floquet (L–F) transformation. These forms are equivalent in the sense that the local stability and bifurcation characteristics are identical for both systems in the entire parameter space. It is well known that the L–F transformation converts a linear periodic first order system into a time-invariant one. In the first part of this study a set of linear second order periodic equations is converted into an equivalent set of time-independent second order equations through a sequence of linear transformations. Then the transformations are applied to a time-periodic quadratic Hamiltonian to obtain its equivalent time-invariant form. In the second part, time-invariant forms of nonlinear equations are studied. The application of L–F transformation to a quasi-linear periodic equation converts the linear part to a time-invariant form and leaves the non-linear part with time-periodic coefficients. Dynamically equivalent time-invariant forms are obtained via time-periodic center manifold reduction and time-dependent normal form theory. Such forms are constructed for general hyperbolic systems and for some simple critical cases, including that of one zero eigenvalue and a purely imaginary pair. As a physical example of these techniques, a single and a double inverted pendulum subjected to periodic parametric excitation are considered. The results thus obtained are verified by numerical simulation.