Vol 59, No 3 (2019)

The problem of approximate calculation of the derivatives of functions with large gradients in the region of an exponential boundary layer is considered. It is known that the application of classical formulas of numerical differentiation to functions with large gradients in a boundary layer leads to significant errors. It is proposed to interpolate such functions by cubic splines on a Shishkin grid condensed in the boundary layer. The derivatives of a function defined on the grid nodes are found by differentiating the cubic spline. Using this approach, estimates of the relative error in the boundary layer and the absolute error outside of the boundary layer are obtained. These estimates are uniform in a small parameter. The results of computational experiments are discussed.

A relation for finding the harmonic barycentric coordinates for an arbitrary polygon is obtained. The solution is approximate analytical. In the proposed statement, the harmonic barycentric coordinates are determined in terms of the logarithmic potential of a double layer by solving the Dirichlet problem by the Fredholm method. The approximate nature of the solution is determined by the expansion of the kernel of the integral Fredholm equation of the second kind for the unknown density of potential on the boundary of the domain in the orthogonal Legendre polynomials and the expansion of Green’s function; these expansions are used for the calculation of the potential. An estimate of convergence rate and the error of the solution is obtained. The approximate solutions obtained by the proposed method are compared with the known exact solutions of some benchmark problems.

In the framework of the memory flow phenomenology, a model of nonstationary noise is constructed and applied for the stochastic simulation of the time series of the plasma density in a thermonuclear facility. A statistical test that makes it possible to check, at a certain level of significance, the validity of the proposed model based on its agreement with experimental data is presented.

For a thin straight wing satisfying a given constraint on the divergence speed (i.e., the speed above which the twist of the wing leads to its failure), the problem of determining an optimal skin thickness distribution that minimizes the skin mass is considered. The mathematical formulation of the problem is as follows: minimize a linear functional over a set of essentially bounded measurable functions for which the smallest eigenvalue of a Sturm–Liouville problem is no less than a preset value. It is proved that this problem has a unique solution. Since only piecewise smooth thickness distributions satisfy the requirements for applications, the regularity of the optimal solution is analyzed. It turns out that the optimal solution is a Lipschitz continuous function. Additionally, it is shown that the solution depends continuously on a parameter determining the lowest possible divergence speed, i.e., the considered problem is well-posed in the sense of Hadamard. Finally, an iteration method for constructing minimizing sequences converging to an optimal solution in Hölder spaces is proposed and numerical results are presented and discussed.

The basic algorithms for determining the thicknesses of layers of deposited multilayer optical coatings are discussed and compared. Using a series of model numerical experiments, the advantage of one of these algorithms—the modified T-algorithm—is demonstrated; this algorithm reduces the influence of the effect of error accumulation in the determined thicknesses of layers.

A mathematical model consisting of quasi-hydrodynamic equations and Dong’s outflow boundary conditions is proposed for solving fluid dynamics problems in a truncated computational domain. A solution algorithm based on finite-element and control-volume methods is developed. The Kovasznay flow and the flow over a backward-facing step in truncated computational domains are numerically simulated. A comparative analysis of the numerical results shows that the proposed mathematical model adequately describes the hydrodynamic flows in a truncated domain.

The free disposal hull (FDH) model of production technology assumes free disposability of all inputs and outputs, but does not make any convexity assumptions. In this paper, we develop an algorithm for the reconstruction of the frontier of the FDH technology that corresponds to its sections by different two-dimensional planes. From a practical perspective, the suggested algorithm is useful for the visualization and exploration of the efficient frontier of the FDH technology. Furthermore, based on the suggested algorithm, we develop a new procedure for the evaluation of returns to scale in the FDH technology. Compared to the existing methods, our approach does not require assessing the efficiency of productive units in the reference technologies, which is known to be a computationally intensive task. Our theoretical results are confirmed by computational experiments in applications with real-life data sets from different industries.