Vol 16, No 3 (2012)

In this paper, we deal with free boundary problem with nonlocal boundary condition for quasilinear parabolic equation. For the solutions of the problem apriory estimates of Shauder’s type are established. On the base of apriory estimations the existence and uniqueness theorems are proved.

Two special functions, concerning Mittag–Leffler type functions, are considered. The first is the modification of generalized Mittag–Leffler type function, introduced by A. A. Kilbas and M. Saigo; the second is the special case of the first one. The solutions of the integral equation with the Kober operator and the generalized power series as the free term are presented. The existence and uniqueness of these solutions are proved. The explicit solutions of the integral equations above are found out in terms of introduced special functions. The correctness of initial value problems for linear homogeneous differential equations with Riemann–Liouville and Kober fractional derivatives is investigated. The solutions of the Cauchy type problems are found out in the special classes of functions with summable fractional derivative via the reduction to the considered above integral equation and also are written in the explicit form in terms of the introduced special functions. The replacement of the Cauchy type initial values to the modified (weight) Cauchy conditions is substantiated. The particular cases of parameters in the differential equations when the Cauchy type problems are not well-posed in sense of the uniqueness of solutions are considered. In these cases the unique solutions of the Cauchy weight problems are existed. It is noted in this paper that the weight Cauchy problems allow to expand the acceptable region of the parameters values in the differential equations to the case when the fractional derivative has the nonsummable singularity in zero.

We consider the damping problem for the hyperbolic system with mixed derivative as the special case of boundary control problem. For different cases the given system is transformed to the triangular or diagonal form, allowing separation of equations. The corresponding transformation is applied to the initial and final data. Two components of boundary control vectors are constructed by solving the Cauchy problem for homogeneous or inhomogeneous equation. The inverse transformation gives the desired control functions.

Medium which strains are described by diagonal components of the strain tensor is considered (in spherical coordinate system). It is assumed that the first invariant of the strain tensor is not positive. Under these restrictions Hencky defining relations with regard to softening of material are written. These defining relations are represented as map of strain space in the stress space. Jacobi matrix of this map is singular in some points in strain space. It is shown that using this map it is possible to find the objective number of deformed states corresponding to a given strain tensor. Also the equations of incremental plasticity law are written. These equations allow us to find the inelastic strain by the total strain.

The work is devoted to investigation of the dynamic behavior of elastic coaxial cylindrical shells interacting with an ideal compressible fluid. The shell behavior is examined in the framework of the classical shell theory, using the variational principle of virtual displacements as a mathematical formulation. The fluid is described in terms of potential theory. With this approach, the stated problem reduces to simultaneous solving of four sets of equations using the finite element method. For shells with different boundary conditions the numerical investigations have been carried out to explore the effects of the annular gap on the boundary of hydroelastic stability.

Queuing systems with distinct channels are considered. Channels may have different capacities (from each other) and distinct queues. The term “dispatch control” is introduced to optimize the system, considering the average time of service and failure probability minimization. These systems are treated as deterministic or nondeterministic finite state machines. State equations of these systems in the form of Zhegalkin polynomial are derived.

The analytical solution of the non-linear heat transfer problem for laminar flow of liquid in plane-parallel channel (Coutte flow) corrected for energy dissipation by the third kind boundary conditions on the moving wall is obtained by the Kantorovich and the orthogonal Bubnov–Galerkin methods. The solution allows to estimate the temperature state of liquid for the small values of longitudinal coordinate, where the displacement of temperature profile to the immovable wall takes place as the investigations have shown.

The conducted research compares the method of Taylor expansions and the finite difference method. The obtained method of Taylor expansions employs three, four and five series terms. For each modification a finite difference scheme is presented and its stability, convergence, and approximation are analyzed. For both methods the number of arithmetic operations is counted and compared as well as an error. The advantages of the method of Taylor expansions are revealed.

The peculiarities of the structure of polymers synthesized in a low-pressure glow discharge in vapors of adamantane on the electrodes and substrates at a “floating” potential are described. The structure of the product formed is conditioned by the ratio of deposits of the mechanisms of the surface and volume polymerization in the formation of the polymer on the surface. The determining factors are the type of discharge and its parameters — direct or alternating current discharge, pressure and current density, presence of the flow of gas or its absence in connection with the feed in the zone of formation of coatings and set out from her particles of the dispersed phase; the location of the substrate in the area of the discharge and outside of it. The kinetics of growth of films of adamantane at different discharge conditions on the electrodes of different materials — aluminum and copper was studied. The growth rate of polymers varied between 0.5–1.4 nm/s at the anode, and 4.2–7.5 nm/s at the cathode for different discharge modes and two electrode materials.

The solution of Cauchy problem for the system of Euler–Poisson–Darboux equations with nilpotent matrix coefficient of power m is obtained by the Riemann method. The Hadamard well-posedness theorem for the Cauchy problem solution is formulated.

In this paper the method of calculation of the stress strain state of a homogeneous isotropic body of arbitrary shape with a piecewise smooth surface is offered. The behavior of the body is described by an uncoupled quasistatic thermoelastic problem, boundary conditions of the first kind are considered. The offered method allows to find the analytical solution of a considered problem of thermoelasticity and to define components of a displacement vector and temperature as functions of body point’s coordinates and time. In order to obtain the solution the considered problem decomposed to an initial boundary value problem of heat conductivity and a boundary value problem of the linear theory of elasticity. The solution of a heat conductivity problem is built by support functions method. The non-uniform problem of the linear theory of elasticity is reduced to the homogeneous problem by means of Kelvin–Somigliana’s tensor; its solution is obtained by means of the theory of potential and Fourier’s transformation.

The analysis of errors of a method of integral characteristics measurement on instant values of harmonic signals divided both in space and in time, using characteristic points, is carried out. The received results allow choosing optimum parameters of measuring process for providing the least error.

Influence of density of dislocations in Ni and Fe on kinetics of anode process in chloride electrolyte was studied. To create the increased density of dislocation this material was deformed by 15 %. It was discovered that the density of anode current increases several times with the increase of the density of dislocation. It was observed that the unevenness of dissolution of Ni anode increases with the increase of the density of dislocations in it.

The problem of discrete optimal control which has m consistently applied objective functions is formulated. In this problem the optimal process, also called m-optimal, is sought as a pair of functions defined on a finite set of steps at the links by which one function is uniquely defines the other, with the constraints of these functions with inclusion “∈” of their values in the final multiple values of the functions of the known pair. A uniform representation of sets, forming the k-optimal processes for k not greater than m, is given with construction of nondecreasing sequence, upper limited by this pair by the “⊂” inclusions, on the basis of characterization of solvability of the problem.