Local buckling of compressed elements of layered viscoelastic material

The problem of local surface buckling of compressed elements of layered material is solved using the equations of the theory of multilayer plates [1, 2, 5]. The local buckling deformations are described by means of the solutions of differential-difference equations damping into the interior of the medium. The effect of compressive forces, buckling wavelength, and the elastic and viscoelastic constants of the material on the behavior of the deformations and on the thickness of the layer to which significant buckling extends is investigated. Numerical estimates are presented for the buckling parameters of compressed laminated-plastic elements.Moscow Power Engineering Institute. Translated from Mekhanika Polimerov, Vol. 4, No. 5, pp. 816–821, September–October, 1968.     

Semilinear Kolmogorov Equations and Applications to Stochastic Optimal Control

Semilinear parabolic differential equations are solved in a mild sense in an infinite-dimensional Hilbert space. Applications to stochastic optimal control problems are studied by solving the associated Hamilton–Jacobi–Bellman equation. These results are applied to some controlled stochastic partial differential equations.     

Numerical solutions for some coupled nonlinear evolution equations by using spectral collocation method

In this paper, we introduce a spectral collocation method based on Lagrange polynomials for spatial derivatives to obtain numerical solutions for some coupled nonlinear evolution equations. The problem is reduced to a system of ordinary differential equations that are solved by the fourth order Runge–Kutta method. Numerical results of coupled Korteweg–de Vries (KdV) equations, coupled modified KdV equations, coupled KdV system and Boussinesq system are obtained. The present results are in good agreement with the exact solutions. Moreover, the method can be applied to a wide class of coupled nonlinear evolution equations.     

Partitioned Runge–Kutta Methods in Lie-Group Setting

We introduce partitioned Runge–Kutta (PRK) methods as geometric integrators in the Runge–Kutta–Munthe-Kaas (RKMK) method hierarchy. This is done by first noticing that tangent and cotangent bundles are the natural domains for the differential equations to be solved. Next, we equip the (co)tangent bundle of a Lie group with a group structure and treat it as a Lie group. The structure of the differential equations on the (co)tangent-bundle Lie group is such that partitioned versions of the RKMK methods are naturally introduced. Numerical examples are included to illustrate the new methods.     

Dynamic Optimization for Reachability Problems

This paper uses dynamic programming techniques to describe reach sets andrelated problems of forward and backward reachability. The original problemsdo not involve optimization criteria and are reformulated in terms ofoptimization problems solved through the Hamilton–Jacobi–Bellmanequations. The reach sets are the level sets of the value function solutionsto these equations. Explicit solutions for linear systems with hard boundsare obtained. Approximate solutions are introduced and illustrated forlinear systems and for a nonlinear system similar to that of theLotka–Volterra type.     

Exact travelling wave solutions for the generalized nonlinear Schrödinger (GNLS) equation with a source by Extended tanh–coth, sine–cosine and Exp-Function methods

The capability of Extended tanh–coth, sine–cosine and Exp-Function methods as alternative approaches to obtain the analytic solution of different types of applied differential equations in engineering mathematics has been revealed. In this study, the generalized nonlinear Schrödinger (GNLS) equation is solved by three different methods. To obtain the single-soliton solutions for the equation, the Extended tanh–coth and sine–cosine methods are used. Furthermore, for this nonlinear evolution equation the Exp-Function method is applied to derive various travelling wave solution. Results show that while the first two procedures easily provide a concise solution, the Exp-Function method provides a powerful mathematical means for solving nonlinear evolution equations in mathematical physics.     

Numerical solutions of the first boundary-value problem for the telegraph equation on open contours

The first plane initial—boundary-value problem for the telegraph equation is reduced by a Chebyshev—Laguerre temporal integral transform to a sequence of stationary boundary-value problems for elliptic equations. Their solutions are sought in integral form. This leads to a recursive sequence of integral equations of the first kind that are solved by the collocation method with isolation of singularities. The sought function is determined by the inverse transform.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 72, pp. 57–62, 1990.     

Stability of transversely isotropic annular plates with low shear resistance

The stability differential equations of a cylindrically orthotropic circular plate are obtained on the assumption of an axisymmetric buckling mode with allowance for transverse shears. These equations are solved for the case of a transversely isotropic material when the inner and outer edges of the plate are identically loaded by uniformly distributed radial forces. The transcendental equations for the critical load parameter are constructed for various edge conditions. The dependence of this parameter on the boundary conditions and the relative thickness of the plate, Poisson's ratio, and the ratio of the Young's and shear moduli of the material are investigated. Certain conclusions are reached concerning the design of reinforced-plastic plates.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 5, pp. 872–880, September–October, 1969.     

Testing glass-reinforced plastic rings by means of rigid sectors

The problem of the extension (compression) of a cylindrically orthotropic ring by means of rigid sectors is solved on the basis of the corrected equations of the theory of bending of bars. Test results are given. The calculated and experimental data are compared.Moscow Power Engineering Institute. Translated from Mekhanika Polimerov, No. 6, pp. 1132–1134, November–December, 1973.     

Classical solutions to Navier–Stokes equations for nonhomogeneous incompressible fluids with non-negative densities

We study the Navier–Stokes equations for nonhomogeneous incompressible fluids in a bounded domain Ω of R³. We first prove the existence and uniqueness of local classical solutions to the initial boundary value problem of linear Stokes equations and then we obtain the existence and uniqueness of local classical solutions to the Navier–Stokes equations with vacuum under the assumption that the data satisfies a natural compatibility condition.     

Local Constraints on Einstein–Weyl Geometries: The 3-Dimensional Case

Following an earlier study [3], we consider the Einstein–Weyl equations on a fixed (complex) background metric as an equation for a 1-form and its first few derivatives. If the background is flat then we conclude that the only solutions are conformal rescalings of constant curvature metrics. If the background is a homogeneous 3-geometry in Bianchi class A (i.e., with unimodular isometry group), we find necessary and sufficient conditions on the 3-geometry for solutions of the Einstein–Weyl equations to exist. The solutions we find are complexifications of known ones. In particular, we find that the general left-invariant metric on S³ and the metric 'Sol' admit no local solutions of the Einstein–Weyl equations.     

The theory of transverse bending of plates with asymmetric elasticity

The problems of bending plates made of various types of composites can be solved efficiently using Cosserat's theory. An approximate theory, reducing three-dimensional equations of the asymmetric theory of elasticity of solids to two-dimensional equations of plate theory has been formulated.Institute of Mechanics, Academy of Sciences, Erevan, Armenia. Translated from Mekhanika Kompozitnykh Materialov, No. 1, pp. 42–52, January–February, 1996.     

Stability in the mean of solutions of difference equations

We consider the membership of solutions of stochastic difference equations in special sets. The notion of stability in the mean is introduced for nonlinear equations. Some assertions about practical stability are proved under certain restrictions. The problem of optimization of regions of practical stability is solved in a particular case.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 72, pp. 78–82, 1990.     

On the determining equations for the nonclassical reductions of the heat and Burgers' equation

The determining equations for the nonclassical reductions of the heat and Burgers' equations are considered. It is shown that both systems belong to a Burgers' equation hierarchy. Each system is written in terms of the same matrix Burgers' equation that is linearized via a matrix Hopf–Cole transformation. In essence, it is shown that both systems can be solved simultaneously. Their respective solutions are then presented in a very compact form.     

Load transmission in layered materials

Load transmission in reinforced plastic heterogeneous media is investigated on the basis of the equations of the theory proposed in [1–3]. The stress distribution problem is solved for a layered half-space with loads applied to one of the "hard" layers in the plane of that layer. Simplified stress formulas are presented. The corresponding error is estimated by working a numerical example. The results are compared with the corresponding problem of the theory of elasticity for a homogeneous orthotropic body.Mekhanika Polimerov, Vol. 4, No. 2, pp. 322–327, 1968     

On the theory of orientation of linear amorphous polymers

The previously derived orientation equations are solved for uniaxial extension at constant true stress, unloading, stress relaxation, and biaxial orientation at constant strain rate. The dependence of the birefringence of biaxially oriented PMMA on the magnitude and conditions of preliminary orientation has been experimentally investigated. There is good qualitative agreement between the theoretical and experimental results.For communication 1 see [1].Lenin Moscow State Pedagogic Institute. Translated from Mekhanika Polimerov, No. 1, pp. 14–21, January–February, 1973.     

Postcritical stage of plates buckling beyond the elastic limit

The theory of small elastoplastic deformations is used to construct differential equations for investigating the behavior of plates in the postcritical stage. The problem of the cylindrical bending of a rectangular plate compressed in one direction is solved by way of example.Tartu State University. Translated from Mekhanika Polimerov, Vol. 4, No. 5, pp. 881–886, September–October, 1968.     

Anisotropic ring stretched by rigid half-disks

A theoretical basis is provided for the experimental method of determining the elastic and strength properties of materials by stretching ring specimens with a pair of rigid half-disks. The equilibrium equations of the plane problem for an anisotropic ring with mixed boundary conditions are solved by means of an expansion in Fourier series with respect to the circumferential coordinate. Numerical results obtained on a computer are analyzed. The dependence of the circumferential stress concentration coefficient on various parameters is investigated.Moscow Power Engineering Institute. Translated from Mekhanika Polimerov, No. 6, pp. 1113–1116, November–December, 1970.     

Stability of elastoplastic flat arches

The stability of flat arches with pinned and fixed ends under a uniformly distributed load is investigated on the assumption that the material obeys a linear hardening law and the geometric relations are nonlinear. The equations obtained are solved for a sinusoidal arch with a rectangular cross section using the Bubnov-Galerikn method and the method of elastic solutions. Equations are obtained for the load and deflection at which transition from the purely elastic to the elastoplastic state occurs.Tartu State University. Translated from Mekhanika Polimerov, Vol. 4, No. 5, pp. 887–896, September–October, 1968.     

Creep stability of glass-reinforced plastic cylindrical shells under long-term external pressure

The behavior of a glass-reinforced plastic cylindrical shell under long-term hydrostatic pressure is investigated using the geometrically nonlinear equations of Timoshenko-type shell theory, which permit transverse shear strains to be taken into account. A system of nonlinear differential equations for describing the variation of the state of the shell with time under load is obtained and solved on a BÉSM-3M computer using a program written in Algol-60 and a "Signal" translator. Values of the critical time are obtained for various load levels.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 1, pp. 81–85, January–February, 1970.