Vol 59, No 7 (2019)

Evaluation of a multidimensional integral up to a given accuracy is studied. The integrand is subject to certain constraints. The case when the integrand is the density of normal distribution, which is important for applications, is considered as a special case.

A new concept of \((\delta ,L)\)-model of a function that is a generalization of the Devolder–Glineur–Nesterov \((\delta ,L)\)-oracle is proposed. Within this concept, the gradient descent and fast gradient descent methods are constructed and it is shown that constructs of many known methods (composite methods, level methods, conditional gradient and proximal methods) are particular cases of the methods proposed in this paper.

We consider analytical systems of ODE with a real or complex time. Integration of such ODE is equivalent to an analytical continuation of a solution along some path, which usually belongs to the real axis. The problems that may appear along this path are often caused by singularities of the solution that lie outside the real axis. It is possible to circumvent problematic parts of the path (including singularities) by going on the Riemann surface of the solution (i.e., in the complex domain). A natural way to realize this program is to use the method of Taylor expansions, which does not require a formal complexification of the system (i.e., a change of variables). We use two classical problems, i.e., the Restricted Three-Body problem, and Van der Pol equation, for demonstration of how Taylor expansions can be used for integration of ODE with an arbitrary precision. We obtained some new results in these problems.

A periodic front-type solution of a singularly perturbed system of parabolic equations is considered. The system can be considered as a mathematical model describing a sharp change in the physical characteristics of spatially inhomogeneous media. Such models are used to describe processes in ecology, biophysics, chemical kinetics, combustion physics, and other fields. The existence of a front-type solution is proved, and the asymptotic stability of a periodic solution is established. An algorithm for constructing an asymptotic approximation of the solution is described.

The problem of a steady-state heat distribution in a domain consisting of two subdomains filled with different inhomogeneous materials is considered. Mathematically, the conditions on the boundary of the subdomains modeling heat transfer and a heat flux through their common boundary are transmission conditions. Additionally, the boundary conditions model a crack on the boundary of the subdomains, which leads to inhomogeneities in the boundary conditions. The problem is studied by constructing its solution using the Green’s function with the help of the WKB method and by comparing the singularities of the solution components with those of a specially chosen problem with constant coefficients and boundary functions of special form.

A stationary problem of radiative-conductive heat transfer in a three-dimensional domain is studied in the \({{P}_{1}}\)-approximation of the radiative transfer equation. A formulation is considered in which the boundary conditions for the radiation intensity are not specified but an additional boundary condition for the temperature field is imposed. Nonlocal solvability of the problem is established, and it is shown that the solution set is homeomorphic to a finite-dimensional compact. A condition for the uniqueness of the solution is presented.