## Vol 18, No 1 (2014)

**Year:**2014**Articles:**17**URL:**https://journals.eco-vector.com/1991-8615/issue/view/1243

Models of multiparameter bifurcation problems for the fourth order ordinary differential equations

###### Abstract

We consider the problem of computing the bifurcating solutions of nonlinear eigenvalue problem for an ordinary differential equation of the fourth order, describing the divergence of the elongated plate in a supersonic gas ﬂow, compressing (extending) by external boundary stresses on the example of the boundary conditions (the left edge is rigidly ﬁxed, the right one is free). Calculations are based on the representation of the bifurcation parameter using the roots of the characteristic equation of the corresponding linearized operator. This representation allows one to investigate the problem in a precise statement and to ﬁnd the critical bifurcation surfaces and curves in the neighborhood of which the asymptotics of branching solutions is being constructed in the form of convergent series in the small parameters. The greatest difficulties arise in the study of the linearized spectral problem. Its Fredholmness is proved by constructing the corresponding Green’s function and for this type of problems it is performed for the ﬁrst time.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(1):9-18

Generalized integral Laplace transform and its application to solving some integral equations

###### Abstract

We present integral transforms $\widetilde {\mathcal L}\left\{f(t);x\right\}$ and $\widetilde {\mathcal L}_{\gamma_1,\gamma_2,\gamma} \left\{f(t);x\right\}$, generalizing the classical Laplace transform. The $(\tau, \beta)$-generalized conﬂuent hypergeometric functions are the kernels of these integral transforms. At certain values of the parameters these transforms coincides with the famous classical Laplace transform. The inverse formula for the transforms is given. The convolution theorem for transform $\widetilde {\mathcal L}\left\{f(t);x\right\}$ is proven. Volterra integral equations of the ﬁrst kind with core containing the generalized conﬂuent hypergeometric function ${\mathstrut}_1\Phi{\mathstrut}_1^{\tau,\beta}(a;c;z)$ are considered. The above equation is solved by the method of integral transforms. The treatment of integral transforms is applied to get the desired solution of the integral equation. The solution is obtained in explicit form.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(1):19-24

Eigenvalue problem for differential Cauchy-Riemann operator with nonlocal boundary conditions

###### Abstract

We consider the reduced spectral problem for the Cauchy-Riemann operator with nonlocal boundary conditions to Fredholm linear integral equation of the second kind with a continuous kernel. The corresponding Fredholm determinant is deﬁned for all spectral parameters, excepting the points: two and one. Finding zeros of the Fredholm determinant recorded in this form is inefficient, because it is not an entire function of the spectral parameter and the main part of the determinant is not separated. Moreover, we study the structure of the kernel by the method shown above, and establish that the linear Fredholm integral equation could not be solved exactly. Therefore, for its approximate solution the results of I. Akbergenov have been applied, where the estimates of the magnitude of the difference between the exact and approximate solutions of the integral equation are given, main part of the kernel is separated. In this case, the spectral parameters are described under which the nonhomogeneous boundary value problem with shift for the Cauchy-Riemann equations is solvable everywhere in the class of continuous functions on the unit circle. Moreover, the design of the approximated solution of the nonhomogeneous boundary value problem is given.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(1):25-36

A boundary-value problem with shift for a hyperbolic equation degenerate in the interior of a region

###### Abstract

For a degenerate hyperbolic equation in characteristic region (lune) a boundary-value problem with operators of fractional integro-differentiation is studied. The solution of this equation on the characteristics is related point-to-point to the solution and its derivative on the degeneration line. The uniqueness theorem is proved by the modified Tricomi method with inequality-type constraints on the known functions. Question of the problem solution's existence is reduced to the solvability of a singular integral equation with Cauchy kernel of the normal type.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(1):37-47

Problems with conjunction on a characteristic plane for the third-order hyperbolic equation in the three-dimensional space

###### Abstract

In the present article the full equation of hyperbolic type of the third order with set of variable factors, in the area representing an infinite triangular prism, limited to the characteristic planes $z = 0$, $x = h$ of the given equation and two noncharacteristic planes $y = x$, $y = -x$ is considered. Two boundary-value problems with data on the edges of the prism, which are both characteristic and non-characteristic planes of the given equation, are solved. In connection with difficulties of a gluing together of considered type solutions of the hyperbolic equations and the representation of conditions of interface on performance integrals and fractional derivatives have been introduced into interface conditions. On the interior characteristic plane the matching conditions, containing fractional order derivatives of required function, are established in order to avoid troubles with intersection of solutions. For equation considered in this article we have obtained the solution of the Darboux problem by method of Riemann, taken for the basis solutions of both problems, which are reduced to uniquely solvable equations of Volterra and Fredholm respectively, that has allowed to obtain the solutions of problems in the explicit analytic form.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(1):48-55

Inverse problem for a Fredholm third order partial integro-diﬀerential equation

###### Abstract

The solvability of various problems for partial differential equations of the third order is researched in many papers. But, partial Fredholm integro-differential equations of the third order are studied comparatively less. Integro-differential equations have traits in their one-valued solvability. The questions of solvability of linear inverse problems for partial differential equations are studied by many authors. We consider a nonlinear inverse problem, where the restore function appears in the equation nonlinearly and with delay. This equation with respect to the restore function is Fredholm implicit functional integral equation. The one- valued solvability of the nonlinear inverse problem for a partial Fredholm integro-differential equation of the third order is studied. First, the method of degenerate kernel designed for Fredholm integral equations is modiﬁed to the case of partial Fredholm integro-differential equations of the third order. The nonlinear Volterra integral equation of the ﬁrst kind is obtained while solving the nonlinear inverse problem with respect to the restore function. This equation by the special non-classical integral transformation is reduced to a nonlinear Volterra integral equation of the second kind. Since the restore function, which entered into the integrodifferential equation, is nonlinear and has delay time, we need an additional initial value condition with respect to restore function. This initial value condition ensures the uniqueness of solution of a nonlinear Volterra integral equation of the ﬁrst kind and determines the value of the unknown restore function at the initial set. Further the method of successive approximations is used, combined with the method of contracting mapping.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(1):56-65

On nonlinear strain vectors and tensors in continuum theories of mechanics

###### Abstract

A non-linear mathematical model of hyperbolic thermoelastic continuum with ﬁne microstructure is proposed. The model is described in terms of 4-covariant ﬁeld theoretical formalism. Fine microstructure is represented by d-tensors, playing role of extra ﬁeld variables. A Lagrangian density for hyperbolic thermoelastic continuum with ﬁne microstructure is given and the corresponding least action principle is formulated. 4-covariant ﬁeld equations of hyperbolic thermoelasticity are obtained. Constitutive equations of microstructural hyperbolic thermoelasticity are discussed. Virtual microstructural inertia is added to the considered action density. It is also concerned to the thermal inertia. Variational symmetries of the thermoelastic action are used to formulate covariant conservation laws in a plane space-time. For micropolar type II thermoelastic Lagrangians following the usual procedure independent rotationally invariant functional arguments are obtained. Objective forms of the Lagrangians satisfying the frame indifference principle are given. Those are derived by using extra strain vectors and tensors.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(1):66-85

Reliability evaluation of stochastically heterogeneous thick-walled pipe by long-term strength criterion

###### Abstract

We have developed a method of probabilistic reliability evaluation of microheterogeneous thick-walled pipe, based on the already received solution of the stochastic creep boundary value problem. The rheological properties of the material were described using random function of one variable (radius $r$). Damage parameter $0< \omega(t) <1$ was introduced here to study the process of degradation of the material during creep stage. Also the power law of the rate of $\omega(t)$ change on the equivalent stress σe , determined by Sdobyrev criterion, is assigned. The reliability evaluation is made by the mean integral value of the equivalent stress. We have found a random time before destruction and its distribution function, which was approximated by lognormal law. The problem of the probability of failure-free operation was calculated for a thick-walled microheterogeneous pipe with the speciﬁed parameters. The obtained results allow to evaluate reliability of stochastically inhomogeneous axisymmetric structural elements if necessary statistical data are obtained from the experiment.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(1):86-92

Kinetics of the stress-strain state of surface hardened cylindrical specimen under complex stress state of creep

###### Abstract

The method for calculating the boundary-value problem of the evaluation of kinetics of the stress-strain state of surface-hardened solid cylindrical specimen under creep for three types of stress state (tension, pure torsion, the combined effect of tensile load and torque) is offered. The energy theory of creep and creep rupture strength is used as base of rheological model. The algorithm for numerical solution of the problem for calculating the relaxation of residual stresses in the surface-hardened layer of cylindrical specimen under creep for all three types of stress state is developed. The intensiﬁcation of relaxation of all residual stress tensor components is established. The signiﬁcant redistribution of stress state along the radius depending on time is observed. The results of variable-based calculations are presented.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(1):93-108

Stress field near the mixed mode crack tip under plane stress conditions

###### Abstract

The asymptotical solution to the problem of the mixed loading of the cracked specimen under plane stress conditions in materials with fractional-linear constitutive relations of steady -state creep is given. The stresses and creep strain rates in the vicinity of the mixed mode crack tip are obtained. The type of mixed loading is speciﬁed by the mixity parameter which is varying from 0 (this type of loading corresponds to pure shear) to 1 (the loading corresponds to tensile loading). The analytical presentation of the stress and the creep strain rate ﬁelds is found for all values of the mixity parameter. It is shown that the stress ﬁeld consists of different regions inside which the stress components are determined by different formulae. The boundaries of the regions are found numerically. The comparison of the analytical solution with the numeric solution obtained for the power-law material for large values of the exponent n is given.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(1):109-124

Analysis of the difference scheme of wave equation equivalent with fractional differentiation operator

###### Abstract

Analysis of the difference scheme of boundary-value problem for the wave equation analogue is made. Explicit and implicit difference schemes for numerical solution of the ﬁrst boundary-value problem for the wave equation analogue with Caputo fractional differentiation operator are investigated, and the stability criteria for these difference schemes are proved by the harmonic Fourier method. Estimates for eigenvalues of the operator of transition from one time layer to another are obtained. Computational experiment on the analysis of the given difference scheme has been performed for the example. The graphs of the numerical solution of the boundary-value problem for the wave equation with the operator of fractional differentiation having different values of parameters of fractional differentiation $\alpha$ and $\beta$ have been built. Change of the period of ﬂuctuations under transition to a fractional derivative is established. On an example it is shown that parameters $\alpha$ and $\beta$ become managing directors.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(1):125-133

Development of identification methods for fractional differential equations with riemann-liouville fractional derivative

###### Abstract

The methods for parametric identiﬁcation of fractional differential operators with $\alpha \on (1, 2)$ degree according to instantaneous values of experimental observations based on the Barrett differential equation example are suggested. The methods are based on construction of the linear parametrical discrete model for fractional differential equation. The coefficients of the model are associated with the required parameters of differentiation equation of fractional order. Different approaches to the determination of the relationships between the parameters of the differential equation and the discrete model coefficients are considered. Connection expressions for coefficients of linear parametrical discrete model and Cauchy type problem parameters to be identiﬁed are obtained. The algorithm of the method which let us reduce the problem to computation of mean-square estimates for coefficients of linear parametrical discrete model is described. Numerical investigations have been done; furthermore, their results let us conclude high efficiency of the methods.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(1):134-144

Discrete-continuous model for the problem of analysis critical level of exploitation of bioresources

###### Abstract

The article deals with the model developed in the framework of the research on the problem of replenishment of ﬁsh populations. A new approach takes into account the peculiarities of describing the changes in the number of generations by the system of ordinary differential equations. The method of models implementation in computing environment enables the simulation of various types of human impact, including the case of introduction of the new species to the environment. The author analyzes the behavior of scenarios of a population model with different levels of ﬁshing impact and concludes the most dangerous case for the systematic management of bioresources. Examples and compared data on the dynamics of some commercial ﬁsh populations of the Volga basin are discussed. It is noted that a dangerous scenario for managing the bioresources is realized with continued slight excess of quotas.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(1):145-155

On the accuracy of difference scheme for Navier-Stokes equations

###### Abstract

The article presents a study of difference schemes in time, which accuracy can be arbitrarily high. We present difference schemes in time for solving the Navier-Stokes equations, where series expansions are used to ﬁnd the singularities of solutions of the Euler equations. These methods are generalized in this article for the arbitrary order schemes and for solving the Burgers equation and the Navier-Stokes equations for an incompressible ﬂuid. The impact of the scheme on the calculation accuracy is examined. First, the method is applied to the test case associated with the Burgers equation, and then the problem of three-dimensional incompressible ﬂow ﬁnding by solving the Navier-Stokes equations is considered. It is shown that the ﬁnite-difference scheme used to calculate the time derivatives is the main source of deviations of the approximate solution from the exact solution.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(1):156-167

Asymptotic analysis of solutions of a nonlinear problem of unsteady heat conduction of layered anisotropic inhomogeneous shells under boundary conditions of the first kind on the front surfaces

###### Abstract

The heat conduction problem is formulated for the layered shells consisting of heatsensitive anisotropic inhomogeneous layers, with boundary conditions of general form. The heat sensitivity of the material layers is described by the linear dependence of their thermophysical characteristics on temperature. The equation of heat conduction, boundary conditions and conditions of thermal conjugations on the boundaries of the contact between the layers are written in the dimensionless form. Two small parameters in dimensionless ratios are deﬁned: thermophysical parameter characterizing the degree of thermal sensitivity of the material layers and geometrical parameter characterizing the relative shell thickness. Sequential recursion of dimensionless ratios is carry out, ﬁrst on thermophysical small parameter, and then on the geometrical parameter. The ﬁrst type of recursion allowed to linearize the problem of heat conduction. On the basis of the second type of recursion the exterior asymptotic expansion of the solution is built for the problem of nonstationary heat conduction of layered anisotropic heterogeneous shells with boundary conditions of the ﬁrst kind on the facial surfaces. The obtained two-dimensional governing equation is analyzed. The asymptotic properties of solutions of the problem of heat conductivity are investigated.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(1):168-185

On the dynamic programming algorithm under the assumption of monotonicity

###### Abstract

We formulate a discrete optimal control problem, which has not been considered earlier, which arises in the design of oil and gas networks. For this problem we set four theorems so that you can have a process, the optimal process and the optimum value. Necessary and sufficient conditions we give in Theorem 1. Under these conditions, by Theorem 1, we get not empty attainability intervals. For each interval, we choose the grid-a subset of its points, where by an arbitrary point of interval, we find the nearest point on the left. By means of such approximations, we define the Bellman functions on the grids. Using Bellman functions in Theorem 2 we give the process and we evaluate its deviation from the optimal process. In Theorem 2, we guarantee, that the given process is optimal when the attainability intervals and their grids coincide. In other cases, to get the optimal process, we use Theorem 3 and Theorem 4. In Theorem 3 we set that the process given in Theorem 2, is minimal in the lexicographical order which we introduce using Bellman functions. In Theorem 3 we give procedure that builds, if possible, in this order, the next process, skipping only the processes that are not optimal. We find the optimal process and the optimal value by Theorem 4, starting from the process given in Theorem 2, using one or more calls of the procedure given in Theorem 3.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(1):186-191

On construction of perfect ciphers

###### Abstract

K. Shannon in the 40s of the 20th century introduced the concept of a perfect cipher, which provides the best protection of plaintexts. Perfect secrecy means that cryptanalyst can obtain no information about the plaintext by observing the ciphertext. In the paper we study the problem of construction of perfect ciphers on a given set of plaintexts $X$, a set of keys $K$ and a probability distribution $P (K)$. We give necessary and sufficient conditions for a perfect ciphers on given $X$, $K$ and $P (K)$. It is shown that this problem is reduced to construction of the set of partitions of the set $K$ with certain conditions. As one of the drawbacks of the probability model of cipher are limitations on the power of sets of plaintexts, keys and ciphertexts we also study the problem of construction of substitution cipher with unbounded key on a given set of ciphervalues, a set of keys and a probability distribution on the set of keys.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(1):192-199