## Vol 16, No 3 (2012)

**Year:**2012**Articles:**27**URL:**https://journals.eco-vector.com/1991-8615/issue/view/1251

Articles

The nonlocal stefan problem for quasilinear parabolic equation

#### Abstract

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.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(3):8-16

Inverse problem for nonlinear partial differential equation with high order pseudoparabolic operator

#### Abstract

We consider the questions of generalized solvability of inverse problem for nonlinear partial differential equations with high order pseudoparabolic operator by method of separation of variables. The mixed problem is reduced to the Volterra integral equation of the second kind, and the inverse problem — to the system of Volterra integral equations. The unique solvability of the inverse problem and the stability of its solution are proved.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(3):17-29

Two special functions, generalizing the Mittag–Leffler type function, in solutions of integral and differential equations with Riemann–Liouville and kober operators

#### Abstract

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.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(3):30-40

A problem with the operator M. Saigo in the boundary condition for a loaded heat conduction equation

#### Abstract

The existence of a unique solution of the non-classical boundary value problem for the heat equation, the loaded value of the desired function $u(x, y)$ on the boundary $x = 0$ of the rectangular area $\Omega= \{ (x, t) : 0$ < $x$ < $l$, $0$ < $t$ < $T \}$ was proved. One of the boundary conditions of the problem has a generalized operator of fractional integrodifferentiation in the sense of Saigo. Using the properties of the Green function of the mixed boundary value problem and the speciﬁed boundary condition, the problem reduces to an integral equation of Volterra type with respect to the trace of the desired function $u(0, t)$. It is shown that the equation is Volterra integral equation of the second kind with weak singularity in the kernel, which is unambiguously and unconditionally solvable. The main result is given in the form of the theorem. The special case is considered, where the generalized operator of fractional integro-differentiation of M. Saigo in the boundary condition reduces to the operator of Kober–Erdeyi. In this case, the existence of an unique solution of the boundary value problem is justiﬁed.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(3):41-46

Damping problem for the special class of the second order hyperbolic systems

#### Abstract

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.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(3):47-52

Evaluation of the reliability of structures under creep for stochastic generalized models

#### Abstract

The nonlinear stochastic model of uniaxial creep and creep rupture strength with three stages of deformation is suggested. The method for the identiﬁcation of stochastic parameters of model by series of experimental creep curves is developed. The stochastic linearization of model for analytical evaluation of the probability of no-failure for stretchable rod by deformation criterion is obtained. The checking of accordance of the method with the experimental data for the creep of samples made of 12Kh18N10T steel under temperature 850850~$^\circ$C is implemented. The generalization of the approach developed to describe the deformation of structural elements of constructions in terms “generalized load, generalized displacement, time” is obtained. The feature is considered as a unit (speciﬁc sample with complex structure). A complete analogy between the curves of uniaxial creep model and generalized creep curves in coordinates “generalized displacement – time” is established for ﬁxed values of the generalized displacement for a feature. Based on the analogy, the generalized stochastic model of rheological deformation of structural elements is proposed. The method for evaluating the reliability of structural elements under creep on parametric failure criteria, implemented in the model example of creep of thick-walled tubes under internal pressure, is developed. The results of the calculations and recommendations for operation life deﬁning are given.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(3):53-71

On defining relations for the Hencky environment of softening of the material under diagonal stress tensor

#### Abstract

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.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(3):72-80

Transition disorder–order–disorder in reaction-diffusion biophysical system

#### Abstract

The evolution of spatial pattern formation, which arises in the biophysical system of reaction-diffusion type in external noise, is researched analytically. The behavior of the probability density of the order parameter, its mean and the most probable values, susceptibility and second-order cumulant as a function of external noise intensity are studied in the statistically steady state. The boundary of transition “order-disorder” is deﬁned. It is shown that there is a sequence of noise-induced ordering and disordering transitions in this system.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(3):81-87

Investigation of boundary conditions on stability of coaxial cylindrical shells interacting with flowing fluid

#### Abstract

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.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(3):88-101

Parametrical identification of the mathematical model in the form of fraction-rational dependencies on the basis of difference equations

#### Abstract

The numerical method of parametrical identiﬁcation of the mathematical model in the form of fraction-rational functional dependencies is considered. The method is based on iteration procedure for mean-square estimation of coefficients of linear parametric discrete models in the form of stochastic difference equations. Such an approach to solving the problem of identiﬁcation of the fraction-rational functional dependencies can ensure a high adequacy of the models, and as a consequence, achieve high accuracy of estimating of the models parameters.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(3):102-113

A queuing system with distinct devices as the finite state machine

#### Abstract

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.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(3):114-124

Regression models construction for describing the thermal system state of two flat bodies in cyclic contact

#### Abstract

Sophisticated analytical solutions of cyclic contact heat transfer problems in the form of integral equations were reduced to criteria form and converted into polynomial models based on the application of experiment planning methodology with economical numerical analysis. Approximation of the desired functions was performed by discrete points using Bonnet formulas, calculations showed quite rapid convergence of approximations and in practical calculations the number of iterations was 7–11. Thirteen criteria equations of regression type were received; the equations contain the most important and diverse in composition and formation structure characteristics of quasi-steady stage of cyclic contact heat exchange. The evaluation of the adequacy of models was made using multiple correlation coefficient.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(3):125-135

Heat transfer in Couette flow corrected for energy dissipation by the third kind boundary conditions

#### Abstract

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.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(3):136-144

On a method of analytical solution of wave equation describing the oscillations sistem with moving boundaries

#### Abstract

The method of analytical solution of wave equation with the conditions, assigned on the moving boundaries, is described. With the aid of the change of variables in the system of functional equations the original boundary-value problem is brought to the system of difference equations with one ﬁxed bias, which can be solved using the Laplace integral transform. The expression for amplitude of oscillation corresponding to n-th dynamic mode is obtained for the ﬁrst kind boundary conditions. This method makes it possible to examine the broader class of boundary conditions in comparison with other exact methods of solving the boundary-value problems with the moving boundaries.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(3):145-151

Modification of finite differences method with use of Taylor expansions

#### Abstract

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.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(3):152-162

Reduction of the sum of the weighted equal powers to explicit combinatorial representation

#### Abstract

The paper contains the proof of the statement that the component of the sum of weighted powers with natural bases and equal parameters, dependent on weight coefficients, is equal to the sum of products of binomial and weight coefficients. It is proved also, that the component of this sum, independent of weight coefficients, is the algebraic sum of products of binomial coefficients and powers of natural numbers. Explicit combinatorial representation of the sum of the weighted equal powers contains the magnitudes taken from proved equalities.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(3):163-169

Structure and growth kinetics of films of adamantane in a glow discharge

#### Abstract

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.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(3):170-179

One characteristic problem for the general hyperbolic differential equation of the third order with nonmultiple characteristics

#### Abstract

In the paper we consider the well-posed characteristic problem for the general hyperbolic differential equation of the third order with nonmultiple characteristics. The solution of this problem is constructed in an explicit form. The illustrative example of the Hadamard ill-posedness of the Goursat problem for the hyperbolic differential equation of the third order with nonmultiple characteristics is given.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(3):180-183

On Cauchy problem for Euler–Poisson–Darboux system with nilpotent matrix coefficient

#### Abstract

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.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(3):184-187

On the character of nonlinearity discontinuities in eigenvalue problems for elliptic equations

#### Abstract

The eigenvalue problems for equations of elliptic type with discontinuous by the phase variable nonlinearities are considered. The character of nonlinearity discontinuities is investigated. In this paper the restrictions on discontinuity points of nonlinearity are weaker than in works of other authors.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(3):188-190

The solution of uncoupled thermoelastic problem with first kind boundary conditions

#### Abstract

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.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(3):191-195

Experimental proof of the rheological model of viscoelastic softening medium with exponential creep kernel

#### Abstract

Behavior of densiﬁed laminated wood DSP-G is described by relations of rheological model of viscoelastic medium providing consideration of eﬀects of material softening. Value of material’s long–term strength limit corresponding to boundary between domains of stable (asymptotically limited creep) and unstable (development of tertiary creep) deformation is found. Comparison of calculated and experimental creep data and long–term strength limit data is executed. Correlation between calculated and experimental data is observed.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(3):196-198

Error estimation of method of integral characteristics measurement using spatial and time division of harmonic signals instant values

#### Abstract

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.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(3):199-202

Interaction of shock waves with area of the non-equilibrium in vibrationally excited gas

#### Abstract

On the basis of numerical solution of the equations of gas dynamics of non-equilibrium medium the penetration of a shock wave in the area of non-equilibrium gas was investigated. The splitting of the shock wave front in a shock and heat waves was observed, it qualitatively coincides with the experimental results, obtained by A.I. Klimov.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(3):203-207

Influence of density of dislocations in Ni and Fe on kinetics of anode process in chloride electrolyte

#### Abstract

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.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(3):208-210

Research of the orbital evolution of asteroid 2012 DA14

#### Abstract

Research of the orbital evolution of asteroid 2012 DA14 on the time interval from 1800 to 2206 is made, an object close approaches with Earth and the Moon are detected, the probability of impact with Earth is calculated. The used mathematical model is consistent with the DE405, the integration was performed using a modiﬁed Everhart’s method of 27th order, the probability of collision is calculated using the Monte Carlo method.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(3):211-214

On the algorithms of dynamic programming for optimal processes

#### Abstract

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.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(3):215-218