Asymptotic stability of solutions of systems of stochastic differential equations in the critical case

We consider the problem of mean square stability and instability of trivial solutions of systems of stochastic differential equations with random operators in the critical case.Translated from Ukrainskii Matematicheskii Zhurnal, Vol.47, No. 11, pp. 1558–1565, November, 1995.     

An investigation of stresses in composite shells on the basis of methods of asymptotic synthesis due to various local actions

It is assumed that the shells are infinite or semi-infinitely long. The fundamental differential equations of twodimensional shell theory are employed in order to obtain a solution of the problem. Also, effective methods of an asymptotic synthesis (MAS) are suggested to avoid difficulties in calculations. For these methods various approximate equations are used, which are: Vlasov-Donnell's, of the semi-membrane theory, of the edge effect theory, the bending or planning states. Solutions of the local thermoelastic problem were obtained in the form of Fourier's series along the contour and in the form of Fourier's integral to the axis direction. Convenient expressions for stresses are obtained by using the MAS, which permits us to reduce the calculation to a calculation of the elementary functions and a very rapidly convergent series.Presented at the Ninth International Conference on the Mechanics of Composite Materials, Riga, October 1995.Published in Mekhanika Kompozitnykh Materialov, Vol. 31, No. 6, pp. 783–791, November–December, 1995.     

Numerical computation of weak solutions of nonlinear motion equations of stiffened cylindrical shells

An algorithm is presented for the numerical solution of nonlinear equations of motion of stiffened cylindrical shells described by a Timoshenko-type theory. This algorithm is constructed using a weak solution of nonlinear motion equations of stiffened cylindrical shells. A difference scheme is constructed using the approximation of integral identities. A dependence was found between the steps of the time and space variables for the linearized difference scheme.Presented at the Ninth International Conference on the Mechanics of Composite Materials, Riga, Latvia, October, 1995.Translated from Mekhanika Kompozitnykh Materialov, Vol. 31, No. 6. pp. 808–815, November–December, 1995.     

Theory of self-adjoint extensions of symmetric operators. entire operators and boundary-value problems

This is a brief survey of M. G. Krein's contribution to the theory of self-adjoint extensions of Hermitian operators and to the theory of boundary-value problems for differential equations. The further development of these results is also considered.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, Nos. 1–2, pp. 55–62, January–February, 1994.     

A property of stable systems of linear stochastic equations

The relationship between exponential mean square stability of systems of linear ordinary differential equations with Gaussian coefficients and the same stability of the corresponding linear stochastic Ito differential equations is obtained.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 2, pp. 147–152, February, 1990.     

Partial averaging of quasidifferential equations in metric spaces

We justify partial averaging schemes for quasidifferential equations that generalize the corresponding results for differential equations and inclusions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 10, pp. 1442–1447, October, 1995.     

A method for solution of nonlinear ordinary differential equations

We present a method for the solution of nonlinear second-order differential equations by using a system of Fredholm equations of the second kind.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 9, pp. 1254–1260, September, 1995.     

Majorant estimates for solutions of quadratic difference systems with lag

Systems of differential equations with lag with quadratic right-hand sides are considered. A compact matrix form for representing these systems in the general form is suggested. Upper and lower bounds for solutions of systems of quadratic differential equations with arbitrary lag are obtained by using quadratic Lyapunov functions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 4, pp. 542–545, April, 1995.     

Estimation of the conditional stability of a two-contour boundary-value problem

We obtain estimates for the conditional stability of a two-contour boundary-value problem for fourth-order elliptic partial differential equations.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 10, pp. 1418–1420, October, 1995.     

Runge–Kutta–collocation methods for systems of functional–differential and functional equations

Systems of functional–differential and functional equations occur in many biological, control and physics problems. They also include functional–differential equations of neutral type as special cases. Based on the continuous extension of the Runge–Kutta method for delay differential equations and the collocation method for functional equations, numerical methods for solving the initial value problems of systems of functional–differential and functional equations are formulated. Comprehensive analysis of the order of approximation and the numerical stability are presented.     

Fourier–Bessel-type series: The fourth-order case

The structured higher-order Bessel-type linear ordinary differential equations were first discovered in 1994. There is a denumerable infinity of these higher-order equations, all of then of even-order.These differential equations possess many of the properties of the classical second-order Bessel differential equation, but these higher-order cases bring remarkable new analytic structures. In many ways it is sufficient to study the properties of the fourth-order Bessel-type differential equation to be able to assess the corresponding properties of the sixth-and higher-order cases.This paper follows a number of earlier papers devoted to the study of the fourth-order case. These publications show the connections between the special function properties of solutions of the differential equation, and the properties of linear differential operators generated by the associated linear differential expression in certain weighted Lebesgue, and Lebesgue–Stieltjes function spaces.To follow the earlier papers on the study of the fourth-order Bessel-type differential equation, this present paper determines the form of the Fourier–Bessel-type series which best extends the classical theory of the second-order Fourier–Bessel series.In fact the Fourier–Bessel-type series are based on a new orthogonal system in terms of the regular eigensolutions of the fourth-order Bessel-type equation. The corresponding eigenvalues are obtained by restricting the spectral parameter to the zeros of an analytic function arising already in the Dini boundary conditions.     

Averaging method applicable to the dynamics of not fully elastic bodies

Using well-known methods, various problems of dynamics are reduced to a standard system of integral differential equations, accounting for heredity effects in the form of multiple integrals according to the theory of Volterra. An averaging scheme is applied to this standard system, converting it to a system of differential equations which are much simpler that the initial equations. A theorem is proved which establishes the similarity of the solutions of these systems.V. I. Lenin Tashkent State University. Translated from Mekhanika Polimerov, No. 5, pp. 940–942, September–October, 1972.     

Nonlinear evolution equations with (1,1)-supersymmetric time

A generalization of the Cauchy-Kowalewsky theorem is obtained for nonlinear evolution equations with (1,1)-supersymmetric time. This theorem ensures the existence and uniqueness of a solution for a large class of superanalytic functions. A generalization of Cartan's technique to the supersymmetric case is also obtained, and by means of it the problem of integrating a system of partial differential equations is transformed into the problem of finding a sequence of integral supermanifolds of lower dimension by means of a succession of integrations based on the Cauchy-Kowalewsky theorem. Evolution equations with (1, 1) time are important for applications to supersymmetric quantum mechanics and field theory, namely, square roots of the Schrödinger and heat-conduction equations. We consider nonlinear generalizations of such equations.Moscow Institute of Electronic Technology; University of Genoa. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 97, No. 2, pp. 238–246, November, 1993.     

On one problem of stochastic control

We prove a comparison theorem for solutions of nonlinear stochastic differential equations with Poisson perturbations and delay. We consider one problem of stochastic control.Translated from Ukrainskii Matematicheskii Zhurnal, Vol.47, No. 11, pp. 1566–1573, November, 1995.     

Necessary and sufficient conditions for oscillation of solutions of nonlinear pulse systems with multiplicatively separated right-hand side

Necessary and sufficient conditions for oscillation of solutions of nonlinear differential equations with fixed times of pulse influence and multiplicatively separated right-hand sides are established.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 3, pp. 381–389, March, 1995.This paper was supported by the Ukrainian State Committee on Science and Technology.     

Analytical properties of solutions to a system of nonlinear partial differential equations

Certain analytical properties of self-similar solutions to a system of N partial differential equations, which is the generalized Liouville equation, are investigated.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 105, No. 2, pp. 208–213, November, 1995.     

Stability of Solutions of a Quasilinear Index-2 Tractable Differential Algebraic Equation by the Lyapunov Second Method

The Lyapunov second method is an important tool in the qualitative theory of ordinary differential equations. In this paper, we consider the behavior of solutions of quasilinear index-2 tractable differential algebraic equations. Using the Lyapunov second method, we prove sufficient conditions for the stability of zero solution of such equations.__________Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 10, pp. 1321 – 1334, October, 2004.     

Elliptic curves and soliton matrix differential equations

A systematic exposition is given of recent results of the general algebraic theory of soliton equations with an elliptic sheaf which generalize a number of integrable two-dimensional matrix differential equations, starting with the equations of principal chiral fields. Local conservation laws, Bäcklund-Darboux transformations, algebra-geometric solutions, quantum integrals, and Hamiltonian structure are studied.Translated from Itogi Nauki i Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 22, pp. 205–265, 1984.     

Confluence of fuchsian second-order differential equations

Ordinary linear homogeneous second-order differential equations with polynomial coefficients including one in front of the second derivative are studied. Fundamental definitions for these equations: of s-rank of the singularity (different from Poincaré rank), of s-multisymbol of the equation, and of s-homotopic transformations are proposed. The generalization of Fuchs' theorem for confluent Fuchsian equations is proved. The tree structure of types of equations is shown, and the generalized confluence theorem is proved.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 104, No. 2, pp. 233–247, August, 1995.     

Variant of the geometrically nonlinear theory of anisotropic shells with allowance for transverse shear

A very simple variant of the geometrically nonlinear theory of anisotropic shells with allowance for the high compliance of the material in transverse shear is proposed. From this theory there follow, as a special case, the equations for an isotropic shell; these differ from the relations of [2] with respect to terms of the order of the ratio of the thickness of the shell to the radii of curvature small as compared with unity. The equations obtained are used to solve the problem of the stability of orthotropic shells of revolution relative to the starting axisymmetric state of stress.Translated from Mekhanika Polimerov, No. 5, pp. 863–871, September–October, 1969.