An asymptotic solution of an initial value problem for a non-linear viscoelastic rod

Stress wave propagation in a one-dimensional materai, described by the non-linear constitutive law

$\text{?σ}\text{?t}\text{? E}\text{?ε}\text{?t}\text{+ k[σ ? E'g3 + βε}{}^2\text{] = 0}$

, is analyzed by a two-time variable perturbation method. Secular terms are “cast out” by employing special orthogonality conditions. The resulting expansion reveals a “mode coupling”, due to the non-linearity, and displays the significance of the three parameters present in the constitutive law. Numerical values for the parameters are assumed and the resulting displacements computed.     

Exact solution of an automodel (self-sustaining) problem regarding the motion of a nonviscous heat-conducting gas bounded by a vacuum

The one-dimensional transient motion of a nonviscous, compressible, heat-conducting, gas bordered on one side by free space and on the other by a solid wall is considered. The boundary conditions at the wall are taken in a form which ensures an automodel (self-sustaining) state. An exact solution is obtained for the problem, and this is then compared with a numerical calculation.     

On the initial stage of a point explosion in a radiating gas

A study is made of the initial stage of a point explosion in a radiating gray gas whose absorption coefficient is approximated by the dependenceK=x()e ^–n ,where is the density and e is the internal energy of the gas. It is shown that for n > —1/3 the initial stage of the process differs significantly from the solution of the problem in not only the classical adiabatic case [1, 2] but also in the case of a medium with nonlinear thermal conductivity [2–4]. The supply of energy to the medium at a point leads to instantaneous heating of the complete medium. The form of this heating is found analytically. The method of matched asymptotic expansions is used to investigate the behavior of the solution in the neighborhood of the center. It is found that for definite conditions at the center of the perturbed region there are formed a shock wave and a region of reverse flow of the gas.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 75–82, May–June, 1980.I should like to thank V. P. Korobeinikov for interest in the work and a helpful discussion of it.     

Construction of an asymptotic solution of one optimal control problem

We construct an asymptotic solution of a singularly perturbed optimal control problem in the case of a regular pencil of boundary matrices.     

Calculation of an ejection explosion in a radial approximation

Equations are derived describing the motion of a medium with an ejection explosion, under the assumption that the medium is incompressible and moves in a radial direction away from the center of the explosion. Here account is taken of the tangential stresses between the moving layers of the medium. A comparison of the calculations of the velocities of the motion of the dome and the dimensions of the craters formed showed good agreement, both with model experiments on the ejection of sand, and with large-scale ejection explosions.     

The asymptotic solution of a kind of Stefan problem

In this paper, a kind of Stefan problem subject to general initial condition is studied. The slab considered is divided into three regions. There is a different time scale in each one. By means of PLK method or like multi-scales method, the asymptotic solution of each one is obtained. Finally we discuss the asymptotic solution and draw some conclusions.     

Construction of an asymptotic solution of the Cauchy problem for a degenerate linear system in the singular case

We propose an algorithm for the construction of an asymptotic solution of the Cauchy problem for a singularly perturbed linear system of differential equations with degenerate matrix of coefficients of derivatives in the case where the limit pencil of matrices is singular. __________ Translated from Neliniini Kolyvannya, Vol. 11, No. 2, pp. 271–288, April–June, 2008.     

The asymptotic solution of a dynamic buckling problem in elastic columns

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On the uniformly valid asymptotic solution of non-linear Dirichlet problem

In this paper, Dirichlet problem for second order quasilinear elliptic equation with a small parameter at highest derivatives is studied. In case degenerate equation has no singular point and parameter is sufficiently small, the existence and uniqueness of solution are proved, and the uniformly valid asymptotic solution is derived on the entire domain.     

Development of dynamic perturbations in initial stage of a point explosion in a thermally conducting gas

The one-dimensional perturbations originating in a cold homogeneous gas (T₁ = 0, ₁ = const) by the instantaneous release of finite energy at the origin of the coordinates are considered. The starting equations are compiled for a gas in which the heat-transfer mechanism is simulated by a nonlinear thermal conductivity with coefficient Tⁿ. Transformation of the equations to the dimensionless form by the introduction of natural variable allows the simplest path for investigating the process as a whole to be shown by means of the method of perturbations. The initial approximation corresponds to the well-known solution for a thermal wave [1], while subsequent approximations describe the joint development of both, thermal and dynamic perturbations. An investigation of the properties of the solutions and an example of the calculation of the first two approximations (without taking account of the starting approximation) for the case of a point spherical explosion with n = 5 gives a representation of the formation of the shock wave.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 55–62, January–February, 1978.     

Numerical and asymptotic solution of the problem concerning the complete stabilization of a boundary layer

The numerical method given in [1] is used here for calculating the temperatures of complete stabilization for a supersonic boundary layer at a flat plate with the boundary condition (0) = 0, where denotes the amplitude of temperature perturbations. According to the results, the conclusion in [2] that there exist two regions of complete stabilization is wrong. The asymptotic method used in [2] is analyzed here. It is shown that two regions of complete stabilization appear to exist, because the equations used in [2] had been set up for the viscous case and, therefore, are not applicable at low surface temperatures. The results of this analysis are confirmed by direct numerical integration.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 60–64, May–June, 1972.The authors thank V. Ya. Levchenko for his continued interest in this study, and A. S. Dryzhova for her valuable comments.