No 2 (2016)

We have constructed an explicit local-cubic spline for the approximation of the smooth functions and have studied the behavior of the approximation. To solve numerically boundary value problems, a spline-scheme based on the properties of the local-cubic spline and the standard cubic spline collocation is proposed. The scheme is implemented by sequentially solving two tridiagonal systems, which allow to use the three-point sweep method and differ from each other only by matrix of the right-hand side of the equation. It indicates that this algorithm is efficient. The number of operations depends linearly on the number of grid nodes. It is proved that the constructed spline possesses the same approximation properties as the local-cubic spline. Thus, in this paper we actually considered the approximation of the solutions of the boundary value problems. The proposed scheme also allows to find the first and second derivatives of the solution of the boundary value problem on the uniform grid nodes of the fourth-order accuracy with respect to the step-size of the grid. The numerical experiments confirm the theoretical order of convergence. Due to good approximation properties and the simplicity of the algorithm implementation, the proposed method can be applied to solve numerically the boundary value problems for the second order ordinary differential equations, which often occur in mathematics, physics, and in the field of natural and engineering sciences.

Regression analysis has the task of finding a functional relationship between the observed values the studied process. The raw data is the realization of a random variable, it is therefore considered dependent on the expectation. This problem can be solved by “smoothing” the raw data. Smoothing is the process of removing the noise and insignificant fragments while preserving the most important properties of the data structure. It is similar to finding the expectation of data. Data smoothing usually attained by parametric and nonparametric regression. The nonparametric regression requires a prior knowledge of the regression equation form. However, most of the investigated data cannot be parameterized simply. From this point of view, nonparametric and semiparametric regression represents the best approach to smoothing data. The aim of the research is development and implementation of the fast smoothing algorithm of two-dimensional data. To achieve this aim previous works in this area have been analyzed and its own approach has been developed,improving the previous ones. As a result, this paper presents the algorithm that quickly and with minimal memory consumption cleanses the data from the “noise” and “insignificant” parts. To confirm the “efficiency” of the algorithm the comparisons with other generally accepted approaches were carried out on simulated and real data with other generally accepted approaches. The results of these comparisons are also shown in the paper.

Large amplitude (geometrically non-linear) vibrations of doubly curved shallow shells with rectangular boundary, simply supported at the four edges and subjected to harmonicexcitation normal to the surface in the spectral neighbourhood of the fundamental mode are subject of investigation in this paper. The first part of the study was presented by the authors in [M. Amabili et al. Nonlinear Vibrations of Doubly Curved Shallow Shells. Herald of Kazan Technological University, 2015, 18(6), 158-163, in Russian]. Two different non-linear strain-displacement relationships, from the Donnell’s and Novozhilov’s shell theories, are used to calculate the elastic strain energy. In-plane inertia and geometricimperfections are taken into account. The solution is obtained by Lagrangian approach. The non-linear equations of motion are studied by using (i) a code based on arclengthcontinuation method that allows bifurcation analysis and (ii) direct time integration. Numerical results are compared to those available in the literature and convergence of the solution is shown. Interaction of modes having integer ratio between their natural frequencies, giving rise to internal resonances, is discussed. Shell stability under dynamic load is also investigated by using continuation method, bifurcation diagram from direct time integration and calculation of the Lyapunov exponents and Lyapunov dimension. Interesting phenomena such as (i) snap-through instability, (ii) subharmonic response, (iii) period doubling bifurcations and (iv) chaotic behavior have been observed.

The physicochemical properties of titanium dioxide thin films prepared by the gel technology, doped with gold nanoparticles, were investigated. The differences betweentechnologies for the synthesis of titanium dioxide were compared. It is experimentally shown that the developed gel technology allows to get almost 100% phase of nanostructured anatase that was confirmed by high-resolution microscopy and X-ray results. The topography and morphology of the films samples were investigated. The photoactivity of the synthesized films was studied by EPR spectroscopy. It is shown that the photoactivity of the films is increased by the UV irradiation. Titanium dioxide was modified by nanoparticles of gold with various concentrations. Has been investigated the depending of the ratio of the solution components in the manufacture of gel films, as well as of the annealing temperature of their formation on transmission spectra. It is shown that the absorption spectra depend significantly on the parameters of the technology. A study of the absorption spectra of titanium dioxide films containing gold nanoparticles showed significant changes in the spectra, exactly, there is an additional absorption peaks of varying intensity and the observed shift in the passband region. These changes are caused, presumably, by changes of the film structure, and the aggregation of gold nanoparticles. Studies have shown the prospects of the gel method for the synthesis of titanium dioxide and its modification of nanoparticles.