Vol 20, No 1 (2016)

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.

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.

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.

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.

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.

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.

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.