Том 57, № 2 (2017)

The system of ordinary differential equations that in a specific case describes the cyclotron motion of a charged particle in an electromagnetic wave is considered. The capture of the particle into autoresonance when its energy undergoes a significant change is studied. The main result is a description of the capture domain, which is the set of initial points in the phase plane where the resonance trajectories start. This description is obtained in the asymptotic approximation with respect to the small parameter that in this problem corresponds to the amplitude of the electromagnetic wave.

We consider the classical Chazy equation, which is known to be integrable in hypergeometric functions. But this solution has remained purely existential and was never used numerically. We give explicit formulas for hypergeometric solutions in terms of initial data. A special solution was found in the upper half plane H with the same tessellation of H as that of the modular group. This allowed us to derive some new identities for the Eisenstein series. We constructed a special solution in the unit disk and gave an explicit description of singularities on its natural boundary. A global solution to Chazy equation in elliptic and theta functions was found that allows parametrization of an arbitrary solution to Chazy equation. The results have applications to analytic number theory.

A singularly perturbed parabolic equation \({\varepsilon ^2}\left( {{a^2}\frac{{{\partial ^2}u}}{{\partial {x^2}}} - \frac{{\partial u}}{{\partial t}}} \right) = F\left( {u,x,t,\varepsilon } \right)\) with the boundary conditions of the first kind is considered in a rectangle. The function F at the angular points is assumed to be quadratic. The full asymptotic approximation of the solution as ε → 0 is constructed, and its uniformity in the closed rectangle is substantiated.

In this work, we present a method for numerical approximation of fixed point operator, particularly for the mixed Volterra–Fredholm integro-differential equations. The main tool for error analysis is the Banach fixed point theorem. The advantage of this method is that it does not use numerical integration, we use the properties of rationalized Haar wavelets for approximate of integral. The cost of our algorithm increases accuracy and reduces the calculation, considerably. Some examples are provided toillustrate its high accuracy and numerical results are compared with other methods in the other papers.

A numerical method based on a second-order accurate Godunov-type scheme is described for solving the shallow water equations on unstructured triangular-quadrilateral meshes. The bottom surface is represented by a piecewise linear approximation with discontinuities, and a new approximate Riemann solver is used to treat the bottom jump. Flows with a dry sloping bottom are computed using a simplified method that admits negative depths and preserves the liquid mass and the equilibrium state. The accuracy and performance of the approach proposed for shallow water flow simulation are illustrated by computing one- and two-dimensional problems.

The situation when there are several different semimetrics on the set of objects in the recognition problem is considered. The problem of aggregating distances based on an unlabeled sample is stated and investigated. In other words, the problem of unsupervised reduction of the dimension of multiple metric descriptions is considered. This problem is reduced to the approximation of the original distances in the form of optimal matrix factorization subject to additional metric constraints. It is proposed to solve this problem exactly using the metric nonnegative matrix factorization. In terms of the problem statement and solution procedure, the metric data method is an analog of the principal component method for feature-oriented descriptions. It is proved that the addition of metric requirements does not decrease the quality of approximation. The operation of the method is demonstrated using toy and real-life examples.