Vol 14, No 2 (2010)

Some subjects of the well-formed initial value problems for ordinary differential equa- tions with Riemann-Liouville derivatives are discussed. As an example the simplest linear homogeneous differential equation with two fractional derivatives is cited. It's shown, that the requirement of the highest derivative summability influence the value of the lowest derivative order or the initial values in Cauchy type conditions. The specific class of functions, allowing the non-summability of the highest derivative, is intro- duced. The correctness of the modified Cauchy type problem and initial value problems with local and nonlocal conditions is substantiated.

The homogeneous thermoconvection problem of the non-compressible viscoelastic Kelvin-Voight fluid of the non-zero order is considered. The conducted research is based on the results of the semilinear Sobolev type equations theory, because the first initial value problem for the corresponding system of the differential equations in private derivatives is reduced to the abstract Cauchy problem for the specified equation. The concepts of the p-sectorial operator and the resolving semigroup of operators of the Cauchy problem for the corresponding linear homogeneous Sobolev type equation are used. The existence and uniqueness theorem of the solution which is a quasi-stationary semi-trajectory is proved. The complete description of the phase space is obtained.

The residual stress definition problem was solved for stripe after rolling in one mill. Lengthwise residual stresses are basic cause of the defect initiation of stripe geometry of the buckles or corrugation type. Lengthwise residual stresses exist in the stripe with transversal residual stresses. These stresses induce the exfoliation of multi-layer stripes. The qualitative model for the two-ply stripe exfoliation under pressure of extension transversal residual stresses field was obtained subject to equality of the stripe materials stress-strain properties. The cause of the stepwise (unexpected) exfoliation was defined. The layer may be depressed essentially wanting the destruction (exfoliation) binding stripe.

Procedure of calculation of critical buckling time of shells under creep with use of Rabotnov's variation equation is presented. Comparison of critical times for the reinforced shells is calculated with use of three-layer model with experimental data and results obtained by other authors is also presented.

Using decomposition of hoop and radial components of displacement vector to the trigonometrical and generalized power series, the new precise analytical solution to problem on equilibrium state of thick-walled heavy transversally-isotropic central-symmetric body, which is fixed on the interior surface and is subject to the action of uniform external lateral pressure, is obtained. This can set a pattern for precise solutions in particular cases of the relations for displacements, stresses and strains at the points inside thick-walled heavy isotropic sphere, the interior surface of which is fixed, while the exterior one being under the uniform pressure. The estimation of an initial strength of solid-cast reinforced concrete sphere is carried out on the basis of a multicriteria approach taking into account real damage mechanisms (i. e. damage from tension or compression in radial, hoop and axial directions, and from transversal and antiplane shear) of anisotropic central-symmetric bodies.

In this work the self-similar solution of a problem of propagation of hydraulic fracture crack is presented. It is proposed that a crack is focused vertically and research of of its development process is analyzed within the coupled problem of the hydrodynamics theory (flow of viscous fluids) and the linear theory of elasticity (fracture crack growth). Asymptotics of the self-similar solution at the long and small distances from tip of a crack were obtained. Simple approximations of the solution in all length range are offered.

Numerical technique for determination of the differential operator paramaters for the systems with turbulent friction is considered. Mean-square estimation of coefficients of stochastic difference equations describing observation results of the impulse characteristic of the system form the basis of this method. The results of numerical analytical treatments of recommended symbolic models and algorithms for calculation on their basis are presented. These results confirm high efficiency of the developed method for the parametric identification on the base of finite-difference equations.

The results of cylindrical layered silicate glass based material shell manufacturing process mathematical modeling are presented in the article. Problems of cooling of a com-posit shell and axisymmetric deformation in the course of its formation are considered. The received results testify the possibility of manufacturing of three-layer cylindrical glass-metalic envelope shells which can be used for creation of strong cases for deep-water technical devices. Limiting values of residual pressure on interface surfaces of metal coverings with a glass layer which it is essential less strength of connected materials of a composit cover are received.

On the basis of use of a method of division of variables and an orthogonal method of Bubnov-Galyorkin the exact analytical decision of the hyperbolic equation of heat conductivity for an infinite plate under boundary conditions of the first sort is obtained. It is shown that having warmed up (or cooled) a body it is defined by movement of front of a shock thermal wave on which there is a breakage temperature curve (temperature jump). The front of a thermal wave divides investigated area on two subareas - revolted where the temperature changes from wall temperature (a boundary condition of the first sort) to the temperature at the front waves and not revolted where the temperature is equal to reference temperature.

The structural algorithm of a method of final integral transformations constructs closed solution of the task on axes symmetrical oscillation of the spirally-multilayered non-uniform cylinder with the account of forces of viscous resistance. The cylinder is formed by winding of preliminary tense metal tape on elastic pliable tube. The spectrum of circular frequencies free axes symmetrical oscillations is analyzed.

Properties of a task graph of an optimum investment are considered the additional limited resource for reduction of a critical way of the network project at non-uniform marking change of arches graph of the project. The algorithm construction of the count of the project on the set matrix of precedence of works is offered. The algorithm reduction lists of technological precedence of works of the project to lists of direct precedence by correct ordering is developed. Examples of addition fictitious works are resulted and reception minimization of necessary number of fictitious works for simplification of the count the project is specified. It is shown that, on an extreme measure, with addition of fictitious works, columns of the project it will be constructed.

The influence of multicycle fatigue actions and thermal exposition, previously hardened in the rate of the air shot-blasting, on such characteristics of cylindrical samples made from alloys V95 and D16T as roughness and microhardness was researched. In fact, surface plastic hardening significantly affects the growth of microhardness in comparison with samples unhardened. Microhardness decreases to values corresponding to sample unhardened after thermal exposition. Surface hardening increases the roughness layer and further thermal exposition and fatigue tests have no effect on these parameters. Results of experimental measurements are presented in tables and graphs.

Generalization of integral-differential Riemann-Liouville operator on the matrix order-is reviewed and its properties are studied. Theorem of the composition of operators of the matrix of integration and differentiation can be proved. The necessary and sufficient conditions for the unique solvability of the matrix Abel equation in a special class of functions are obtained.

It is well known that differential equations with an operator are used for study of the processes connected with appearances of vibration and other mechanics problems, and also play an essential role in the theory of approximation and mapping. In the present work a unique solution for the mixed problem of the full hyperbolic equation of the third order with constant factors, in a three-dimensional Euclidean space, was obtain with the Riemann method, which then becomes considerably simpler at the expense of integral representation of one of boundary conditions. Owing to this it can be used for statement and a solution of new boundary value problems.

In work necessary and sufficient conditions of uniqueness of the decision of a regional problem for the loaded equation mixed parabolic hyperbolic type in rectangular area are established. The problem decision is constructed in the form of the number sum on own functions of a corresponding one-dimensional problem on own values.

It is researched the residual stresses into the surface hardening details with V-type notches. It is established that depth of the notches exerts significant effect on the distribution of behavior of residual stresses distribution within notches region.

The paper considers the problem of vibrations in flexible link with bending hardness when one of the basis is a roller or a pulley. When vibrating the touch point link moves along the basis. Movement of touch point depends on the vibrating process that makes the problem non-linear. Using decomposition function describing vibrations in the number of touch points, boundary conditions are received when link and basis are interesting. Rating of dependence of frequency vibrations on amplitude is received. Stated, that at low amplitude vibrations boundary conditions can be written in the form of equality of angle to zero and displace of flexible link. Error when using such boundary conditions increases with increasing amplitude.

The field of obtaining approximated analytical solutions of mathematical physics boundary problems is developed. It is based on additional boundary conditions, allowing to define any item number of model (approximated) analytical solutions and to obtain solutions with the required accuracy degree. Physical meaning of additional boundary conditions lies in implementation of the basic differential equation and its different order derivatives at boundary points of the field. It leads to implementation of differential equation within the given field with number of approximations dependant on the accuracy.

Deformed copper samples were burned at different temperatures. This procedure formed different dislocation density in the samples. Then such prepared samples were used as the anodes in acidic C11SO4 solution. The voltage for electrolyse was 10 mV. The experiment has shown that the increase of the dislocation density in the anode leads to the increase of the density of anode current. It was discovered that the density of anode current could become 2,5 times bigger. This can be explained by the fact that dislocations decrease the potential barrier during the transfer of copper ions from the grid into the electrolyte.

The iterative projection algorithm for solving ill-posed systems of linear algebraic equations is examined. This algorithm is based on transforming the regularized normal equations to the equivalent augmented regularized normal system of equations. The proposed algorithm can effectively solve ill-posed problems of large dimension.

The developed software package for calculating residual stresses relaxation in the surface hardened layer of structural elements under creep conditions is announced. Block of software package for calculating the fields of residual stresses and strains in the hardened layer of constructions (cylindrical samples) after strain hardening is described. The calculation is produced for one or two components of the experimentally measured residual stresses.