 # Vol 20, No 1 (2016)

### Goursat problem for loaded degenerate second order hyperbolic equation with Gellerstedt operator in principal part

Attaev A.H.

#### Abstract

In the paper we study a loaded degenerate hyperbolic equation of the second order with variable coefficients. The principal part of the equation is the Gellerstedt operator. The loaded term is given in the form of the trace of desired solution on the degenerate line. The latter is located in the inner part of the domain. We investigate a boundary value problem. The boundary conditions are given on a characteristics line of the equation under study. For the model equation (when all subordinated coefficients are zero) we construct an explicit representation for solution of the Goursat problem. In the general case, we reduce the problem to an integral Volterra equation of the second kind. We apply the method realized by Sven Gellerstedt solving the second Darboux problem. In both cases, model and general, we use widely properties of the Green-Hadamard function.
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2016;20(1):7-21 7-21  ### The nonlocal A. A. Desin's problem for an equation of mixed elliptichyperbolic type

Gushchina V.A.

#### Abstract

In this paper for the equation of mixed elliptic-hyperbolic type in rectangular area the task with the conditions of periodicity and the nonlocal problem of A. A. Desin was studied, the uniqueness criterion was set. The solution of the problem was constructed as a sum of orthogonal series in eigenfunctions of the corresponding one-dimensional spectral problem. The problem of small denominators arises in justifying the convergence of the series. Therefore the evaluation on the separation from zero of small denominators with the corresponding asymptotics was established. This assessment allowed under certain conditions relative to the set objectives and functions to prove convergence of the constructed series in the class of regular solutions and the stability of the solution.
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2016;20(1):22-32 22-32  ### On the determination of pure quantum states by the homodyne detection

Dnestryan A.I.

#### Abstract

The methods of reconstruction of the wave function of a pure state of a quantum system by quadrature distribution measured experimentally by the homodyne detection are considered. Such distribution is called optical tomogram of a state and containes one parameter θ. Wave function of a state is determined exactly by its optical tomogram if last one is known for all θ. But one can obtain optical tomogram from experiment of homodyne detection only for discrete number of θ. We introduce some approximate methods of reconstructing the state by such information about its optical tomogram.
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2016;20(1):33-42 33-42  ### An internal boundary value problem with the Riemann-Liouville operator for the mixed type equation of the third order

Repin O.A., Kumykova S.K.

#### Abstract

The unique solvability of the internal boundary value problem is investigated for the mixed type equation of the third order with Riemann-Liouville operators in boundary condition. The uniqueness theorem is proved for the diﬀerent orders of operators of fractional integro-diﬀerentiation when the inequality constraints on the known functions exist. The existence of solution is veriﬁed by the method of reduction to Fredholm equations of the second kind, which unconditional solvability follows from the uniqueness of the solution of the problem.
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2016;20(1):43-53 43-53  ### On optimal control problem for the heat equation with integral boundary condition

Tagiev R.K., Habibov V.M.

#### Abstract

In this paper we consider the optimal control problem for the heat equation with an integral boundary condition. Control functions are the free term and the coefficient of the equation of state and the free term of the integral boundary condition. The coefficients and the constant term of the equation of state are elements of a Lebesgue space and the free term of the integral condition is an element of Sobolev space. The functional goal is the final. The questions of correct setting of optimal control problem in the weak topology of controls space are studied. We prove that in this problem there exist at least one optimal control. The set of optimal controls is weakly compact in the space of controls and any minimizing sequence of controls of a functional of goal converges weakly to the set of optimal controls. There is proved Frechet differentiability of the functional of purpose on the set of admissible controls. The formulas for the differential of the gradient of the purpose functional are obtained. The necessary optimality condition is established in the form of variational inequality.
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2016;20(1):54-64 54-64  ### On problems with displacement in boundary conditions for hyperbolic equation

Utkina E.A.

#### Abstract

We consider three problems for hyperbolic equation on a plane in the characteristic domain. In these problems at least one of the conditions of the Goursat problem is replaced by nonlocal condition on the relevant characteristic. Non-local conditions are the linear combinations of the normal derivatives at points on opposite characteristics. In case of replacement of one condition we solve the problem by reduction to the Goursat problem for which it exists and is unique. To find the unknown Goursat condition author receives the integral equation, rewrite it in operational form and finds its unique solvability cases. To prove the unique solvability of the equation, the author shows the continuous linear operator and the fact, that some degree of the resulting operator is a contraction mapping. It is known that in this case the required Goursat condition can be written as Neumann series. We considered in detail only one of the tasks, but for both the unique solvability theorems are formulated. If the two conditions are changed, the uniqueness of the solution on the assumption that it exists, is proved by the method of a priori estimates. For this purpose, the inner product and the norm in $L_2$ are used. As a result, the conditions were obtained for the coefficients of a hyperbolic equation that ensure the uniqueness of the solution. An example is given, confirming that these conditions are essential. Namely, constructed an equation whose coefficients do not satisfy the conditions of the last theorem, given the conditions on the characteristics and nontrivial solution is built.
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2016;20(1):65-73 65-73  ### Cauchy problem for a parabolic equation with Bessel operator and Riemann-Liouville partial derivative

Khushtova F.G.

#### Abstract

In this paper Cauchy problem for a parabolic equation with Bessel operator and with Riemann-Liouville partial derivative is considered. The representation of the solution is obtained in terms of integral transform with Wright function in the kernel. It is shown that when this equation becomes the fractional diffusion equation, obtained solution becomes the solution of Cauchy problem for the corresponding equation. The uniqueness of the solution in the class of functions that satisfy the analogue of Tikhonov condition is proved.
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2016;20(1):74-84 74-84  ### A study of steady creep of layered metal-composite beams of laminated-fibrous structures with account of their weakened resistance to the transverse shift

Yankovskii A.P.

#### Abstract

Steady-state creep of a hybrid metal-composite laminated bending beams of irregular patterns is considered. Beams consist of walls and load-bearing layers, attached at the top and bottom (shelves). The shelves are reinforced with ﬁbers in the longitudinal direction, and the walls are reinforced either longitudinally or cross in the plane. Under the hypotheses of the Timoshenko theory the boundary value problem is formulated for the calculation of the considered beams, which allows taking into account the weakened resistance of the walls of the transverse shifts. The simple iteration method based on the idea of the se-cant modulus method is applied for linearization of the problem. The mechani-cal behavior of reinforced and unreinforced doubleseat beams and cantilevers in conditions of steady creep under the action of uniformly distributed trans-verse load is investigated. Cross sections of beams are I-shaped. It is shown that for homogeneous I-beams, the classical Bernoulli theory does not guaran-tee the calculated results in compliance within 20% accuracy, which is consid-ered to be acceptable if the width of the shelf is comparable with the height of cross sections of beams. In the cases of metal-composite beams, the classical theory becomes generally unacceptable, because it lowers by several orders of magnitude the compliance of such structures in conditions of steady creep. It is demonstrated that the rate of shear strain, actively developing in their walls, must be considered. The consideration of these strain rates within the frame-work of the Timoshenko theory led to the discovery of new mechanisms of de-formation of laminated beams which the classical theory does not ﬁnd. It is shown that the change in mechanism of deformation can occur by increasing the density of reinforcement of shelves or walls.
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2016;20(1):85-108 85-108  ### A method for solving problems of heat transfer during the flow of fluids in a plane channel

Eremin A.V., Kudinov I.V., Zhukov V.V.

#### Abstract

Using the integral method of heat-transfer with the additional boundary conditions we obtain the high precision approximate analytical solution of heat-transfer for a fluid, moving in plate-parallel channel with symmetric boundary conditions of the ﬁrst kind. Because of the inﬁnite speed of heat propagation described by a parabolic equation of heat-conduction, the temperature in the centre of channel would change immediately after the boundary conditions (of the ﬁrst kind) application. We receive the approximate analytical solution of boundary value problem using the representation of this temperature in the form of additional required function and introducing the additional boundary conditions to satisfy the original differential equation in boundary points by the desired function. Using of the integral of heat balance we reduce the solving of differential equation in partial derivatives to integration of ordinary differential equation with respect to additional required function, that changes depending on longitudinal variable. We note that fulﬁllment of the original equation at the boundaries of the area with increasing number of approximations leads to the fulﬁllment of that equation inside the area. No need to integrate the differential equation on the transverse spatial variable, so we are limited only by the implementation of the integral of heat-transfer (averaged original differential equation), that allows to apply this method to boundary value problems, unsolvable using classic analytical methods.
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2016;20(1):109-120 109-120  ### Comparison of the coordinates of the major planets, Moon, and Sun obtained based on a new principle of interaction and of the data bank DE405

Zausaev A.F.

#### Abstract

In this paper the comparison of orbit coordinates and elements of large planets, the Moon and the Sun obtained on the basis of a new principle of interaction and of data bank DE405 is made. The space environment is the physical vacuum, whose properties are currently still in the formative stage. Gravity is the result of the interaction of the physical vacuum with material bodies which are moving. Gravity explains by the properties of space compression in relation to moving material bodies. Differential equations of motion of the major planets, the Moon and the Sun have been obtained. It should be noted that the system of differential equations does not contain the mass of bodies and force interactions, in addition, the Earth is considered as a spheroid. By numerical integration of the equations of motion coordinates of the Moon, the Sun and major planets osculating elements of the orbits of the inner planets during 1602-2193 are computed. The results of calculations are compared with the coordinates and orbital elements determined according to the coordinates and velocities DE405. It is shown that in contrast to Newtonian mechanics and relativistic equations of motion, the coordinates of the major planets of the Moon and the Sun, based on the solution of a new system of differential equations, are in satisfactory agreement with the coordinates of these objects obtained using data bank DE405. The resulting equations do not contain terms that take into account the non-sphericity of the Earth and the Moon, being a non-relativistic equations. Based on the research the following conclusions are made: obtained differential equations of motion satisfactorily describe the motion of the major planets, of the Moon and Sun on the time interval of 600 years; these equations are much simpler and more accurate then the differential equations that take into account the relativistic effects.
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2016;20(1):121-148 121-148  ### Summation on the basis of combinatorial representation of equal powers

Nikonov A.I.

#### Abstract

In the paper the conclusion of combinatorial expressions for the sums of members of several sequences is considered. Conclusion is made on the basis of combinatorial representation of the sum of the weighted equal powers. The weighted members of a geometrical progression, the simple arithmeticgeometrical and combined progressions are subject to summation. One of principal places in the given conclusion occupies representation of members of each of the speciﬁed progressions in the form of matrix elements. The row of this matrix is formed with use of a gang of equal powers with the set weight factor. Besides, in work formulas of combinatorial identities with participation of free components of the sums of equal powers, and also separate power-member of sequence of equal powers or a geometrical progression are presented. All presented formulas have the general basis-components of the sums of equal powers.
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2016;20(1):149-157 149-157  ### On dynamic programming on the values in the semigroup

Ovchinnikov V.G.

#### Abstract

For not considered previously discrete optimal control problem with target function values in a linearly ordered Abelian semigroup given characteriza tion of the solvability and on its basis the algorithm seeks optimal process with the help of delivering Bellman values elements of limiting sets. We mark the modifications to this algorithm, when 1) P is nonempty subset of numbers with the natural ordering and the operation producing the maximum of two numbers; 2) P is set of nonnegative numbers with the natural ordering and the addition (or multiplication); 3) P is lexicographical product of m (not less than two) linearly ordered Abelian semigroups; 4) P is lexicographic product of m (not less than two) sets of real numbers with the natural ordering and the addition, and this algorithm gets m-optimal process easier than the previous author’s algorithm.
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2016;20(1):158-166 158-166  ### Mathematical modeling of hereditary elastically deformable body on the basis of structural models and fractional integrodifferentiation Riemann-Liouville apparatus

Ogorodnikov E.N., Radchenko V.P., Ungarova L.G.

#### Abstract

The standard one-dimensional generalized model of a viscoelastic body and some of its special cases-Voigt, Maxwell, Kelvin and Zener models are considered. Based on the V. Volterra hypothesis of hereditary elastically deformable solid body and the method of structural modeling the fractional analogues of classical rheological models listed above are introduced. It is shown that if an initial V. Volterra constitutive relation uses the Abel-type kernel, the fractional derivatives arising in constitutive relations will be the Rieman-Liouville derivatives on the interval. It is noted that in many works deal with mathematical models of hereditary elastic bodies, the authors use some fractional derivatives, convenient for the integral transforms, for example, the Riemann-Liouville derivatives on the whole real number line or Caputo derivatives. The explicit solutions of initial value problems for the model fractional differential equations are not given. The correctness of the Cauchy problem is shown for some linear combinations of functions of stress and strain for constitutive relations in differential form with Riemann- Liouville fractional derivatives. Explicit solutions of the problem of creep at constant stress in steps of loading and unloading are found. The continuous dependence of the solutions on the model fractional parameter is proved, in the sense that these solutions transform into a well-known solutions for classical rheological models when α → 1. We note the persistence of instantaneous elastic deformation in the loading and unloading process for fractional Maxwell, Kelvin and Zener models. The theorems on the existence and asymptotic properties of the solutions of creep problem are presented and proved. The computer system identifying the parameters of the fractional mathematical model of the viscoelastic body is developed, the accuracy of the approximations for experimental data and visualization solutions of creep problems is evaluated. Test data with constant tensile stresses of polyvinyl chloride tube were used for experimental veriﬁcation of the proposed models. The results of the calculated data based on the fractional analog of Voigt model are presented. There is a satisfactory agreement between the calculated and experimental data.
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2016;20(1):167-194 167-194  