Uniform stabilization of a nonlinear structural acoustic model with a Timoshenko beam interface

This paper is concerned with a nonlinear model which describes the interaction of sound and elastic waves in a two‐dimensional acoustic chamber in which one flat ‘wall’, the interface, is flexible. The composite dynamics of the structural acoustic model is described by the linearized equations for a gas defined on the interior of the chamber and the nonlinear Timoshenko beam equations on the interface. Uniform stability of the energy associated with the interactive system of partial differential equations is achieved by incorporating a nonlinear feedback boundary damping scheme in the equations for the gas and the beam. Copyright © 2006 John Wiley & Sons, Ltd.     

Stability in mean square of a linear stochastic differential equation

The stability of the zero solution of a first-order linear differential equation with a random right-hand side is investigated using moment equations. Transformations of moment equations are considered. Conditions for reducing the order of the moment equations are derived.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 61, pp. 119–126, 1987.     

Motion of a container as a nonholonomous system in a curved section of a horizontal pipeline

The problem of the motion of a container in a curved section of a horizontal pipeline is solved using second-order Lagrange equations in the presence of nonholonous couplings. The special case of the motion of a container in a circular curve is examined.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 25, pp. 90–95, 1987.     

Stability and convergence of a system of nonlinear difference equations

We investigate stability and convergence of solutions of a system of nonlinear difference equations approximating a system of nonlinear parabolic equations. A linear system of similar structure is also considered. An energy norm is constructed for the linear system, and stability and convergence in this norm are proved under certain necessary conditions. Stability and convergence of solutions of the nonlinear system of difference equations are proved in a similar norm.Translated from Metody Matematicheskogo Modelirovaniya, Avtomatizatsiya Obrabotki Nablyudenii i Ikh Primeneniya, pp. 120–127, 1986.     

Axisymmetric indentation of a circular stamp with adhesion into an elastic half-space with a spherical cavity

On the basis of the expansion formulas of the vector solutions of the Lame equations in cylindrical and spherical coordinates, the problem of a circular stamp is formulated in the form of an integro-algebraic system of equations. By the method of orthogonal polynomials, it is reduced to a collection of infinite systems of linear algebraic equations, for which the method of reduction is justified. Formulas for the normal and tangential stresses under the stamp are given.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 18, pp. 14–20, 1987.     

Nonlinear oscillations of a variable composition liquid in a cylindrical vessel

The problem of wave motions of an ideal liquid of variable composition in a cylindrical vessel is reduced to an infinite system of nonlinear differential equations. It is shown that in the case of axisymmetric oscillations the equations obtained refine the linear analog of the problem, even if the vessel filling depth is significantly greater than the amplitude of the liquid free surface oscillations.Donetsk. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 20, pp. 104–107, 1989.     

On a new method of solving a class of problems of applied mechanics usingR-functions,weighted discrepancies,and least squares

We propose a new approach to solving a class of problems of applied mechanics using R-functions and boundary integral equations. We give the results of numerical analysis of the solution of second-order differential equations with constant coefficients using the example of the torsion of orthotropic rods.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 36, 1992, pp. 25–30.     

Vibrations of a fractal elastic string

We construct the solution of the fractional space-time equations that describe the vibrations of a quasi-one-dimensional fractal elastic string. We give the solution of the Cauchy problem for fractional differential equations with initial conditions. We carry out a numerical analysis and construct the graphic variation of the displacement function of a fractal elastic string. Three figures. Bibliography: 7 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 26, 1996, pp. 142–147     

Strain of a flexible orthotropic cylindrical shell as a function of orthotropy parameters

A procedure is proposed for calculating the stress-strain state of flexible orthotropic cylindrical shells of constant thickness with unsymtnetric load and nonhomogeneous boundary conditions. The system of nonlinear partial differential equations is solved by the method of lines. The system of nonlinear ordinary differential equations is reduced by linearization to a sequence of linear systems. The sequence of linear boundary-value problems is solved by the discrete orthogonalization method.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 59, pp. 57–61, 1986.     

Existence of solutions for models of shallow water in a basin with a degenerate varying bottom

We prove the existence of solutions for the great lake equations. These equations are obtained from the three-dimensional Euler equations in a basin with a free upper surface and a spatially varying bottom topography by taking a low aspect ratio, i.e., low wave speed and small wave amplitude expansion. These equations are rewritten in an abstract form by considering generalized Euler equations as in Levermore et?al. (Indiana Univ Math J 45:479?C510, 1996). This paper is an extension of Levermore et?al. (Indiana Univ Math J 45:479?C510, 1996), where the varying bottom was assumed to be nondegenerate. Here, we discuss the degenerate case and obtain similar results as in Levermore et?al. (Indiana Univ Math J 45:479?C510, 1996).     

Equation for the Complex Energy of a Bound Particle in a Laser Radiation Field in the Presence of Strong Constant Electromagnetic Fields

We construct exact solutions of the Schrodinger and Pauli equations for charged particles in the external field of the Redmond generalized configuration. We calculate the Green's functions of scalar and spinning particles in this field. Using them, we calculate the equations for the complex quasienergy of a bound particle (bound by a short-range potential) two different ways. In the example of an external constant electric field, we discuss the applicability domain of the obtained equations and the differences between their solutions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 145, No. 2, pp. 198–211, November, 2005.     

Tubuloglomerular feedback in a dynamic nephron

A dynamic model for a short-looped mammalian nephron is developed to study tubuloglomerular feedback (TGF). Evolution equations for salt and urea concentrations and for fluid flux in the nephron are derived and coupled to a resistance network that serves as a schematic model of the glomerulus and associated structures. The evolution equations, which are semi-linear hyperbolic partial differential equations, are solved by the method of flux-corrected transport. The implementation and testing of this method is described and numerical results are presented. This investigation suggests that: (i) the concentrating nephron exhibits high gain, i.e., a small increase in single nephron glomerular filtration rate produces a large increase in the salt concentration of tubular fluid in the cortical thick ascending limb at the macula densa; (ii) the nephron, as a concentrating system, acts as a low-pass filter, i.e., high frequency pressure oscillations (1 Hz) of a prescribed amplitude at the proximal tubule produce relatively low amplitude oscillations in tubular concentrations, while low frequency oscillations (1/30 Hz) produce relatively high amplitude oscillations in tubular concentrations; and (iii) as a consequence of long time delay in TGF, some perturbations in afferent arteriolar blood pressure induce sustained periodic oscillations similar to those observed in recent experiments.     

Boundary and variational problems for a combined mathematical model of the theory of elasticity

We construct a combined mathematical model of the theory of elasticity that describes the stress-strain state of an elastic body using the equations of the theory of elasticity in one part of the body and the equations of the theory of shells of Timoshenko type in the other part. We write the resolvent equations and conditions for elastic coupling. We study the variational formulation of the boundary-value problems of the combined model.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 32, 1990, pp. 92–95.     

New example of a nonlinear hyperbolic equation possessing integrals

We discover an important new case in the classical problem of the classification of nonlinear hyperbolic equations possessing integrals. In the general (least degenerate) case, in addition, we obtain a formula describing the splitting of the right-hand side of such equations with respect to the first derivatives. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 120, No. 1, pp. 20–26, July, 1999.     

The macroscopic system of Einstein-Maxwell equations for a system of interacting particles

We derive the macroscopic Einstein—Maxwell equations up to the second-order terms, in the interaction for systems with dominating electromagnetic interactions between particles (e.g., radiation-dominated cosmological plasma in the expanding Universe before the recombination moment). The ensemble averaging of the microscopic Einstein and Maxwell equations and of the Liouville equations for the random functions of each type of particle leads to a closed system of equations consisting of the macroscopic Einstein and Maxwell equations and the kinetic equations for one-particle distribution functions for each type of particle. The macroscopic Einstein equations for a system of electromagnetically and gravitationally interacting particles differ from the classical Einstein equations in having additional terms in the lefthand side due to the interaction. These terms are given by a symmetric rank-two traceless tensor with zero divergence. Explicitly, these terms are represented as momentum-space integrals of the expressions containing one-particle distribution functions for each type of particle and have much in common with similar terms in the left-hand side of the macroscopic Einstein equations previously obtained for a system of self-gravitating particles. The macroscopic Maxwell equations for a system of electromagnetically and gravitationally interacting particles also differ from the classical Maxwell equations in having additional terms in the left-hand side due to simultaneous effects described by general relativity and the interaction effects. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 125, No. 1, pp. 107–131, October, 2000.     

Integral equations for correlation functions of a quantum one-dimensional Bose gas

The large-time, long-distance behavior of the temperature correlation functions of a quantum one-dimensional Bose gas is considered. We obtain integral equations, which are closely related to the thermodynamic Bethe ansatz equations and whose solutions describe asymptotic expressions. In the low-temperature limit, the solutions of these equations are expressed through observables of the model. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 121, No. 1, pp. 117–138, October, 1999.     

Discretizing a system of parabolic nonlinear equations with a discontinuous solution

A new initial— boundary-value problem is posed for a system of two nonlinear parabolic equations with convective transport terms that allow a discontinuity in the solution. Discretization on classes of discontinuous functions is proposed.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 73, pp. 33–42, 1992.     

Asymptotics for the solution of averaged equations for a system of coupled oscillators

The equations obtained by averaging a system of three weakly nonlinear oscillators are considered. Distinctive properties of asymptotics at infinity for the solutions of averaged equations for both constant and variable proper frequencies are studied. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 6, pp. 99–113, 2006.     

The equations of mechanothermoelectric processes in semiconductors alloyed with impurities of a single type

In the context of the theory of mutually interpenetrating continua we formulate a system of equations to describe jointly mechanical, thermal, and electromagnetic processes in semiconductors with monopolar conductivity. The system of equations obtained is used to study the stress-deformed state of a semiconductor in an electrostatic field.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya. Issue 28, 1988, pp. 14–18.     

Fractal oscillations for a class of second order linear differential equations of Euler type

The s-dimensional fractal oscillations for continuous and smooth functions defined on an open bounded interval are introduced and studied. The main purpose of the paper is to establish this kind of oscillations for solutions of a class of second order linear differential equations of Euler type. Next, it will be shown that the dimensional number s only depends on a positive real parameter α appearing in a singular term of the main equation. It continues some recent results on the rectifiable and unrectifiable oscillations given in Paši? [M. Paši?, Rectifiable and unrectifiable oscillations for a class of second-order linear differential equations of Euler type, J. Math. Anal. Appl. 335 (2007) 724-738] and Wong [J.S.W. Wong, On rectifiable oscillation of Euler type second order linear differential equations, Electron. J. Qual. Theory Differ. Equ. 20 (2007) 1-12].