Galerkin method for the linear-radiator integral equation

The article examines the general Galerkin method scheme for the singular integral equation in the problem of radiation from a finite-thickness linear radiator. Translated from Obratnye Zadachi Estestvoznaniya, Published by Moscow University, Moscow, 1997, pp. 150–158.     

Nonstationary interaction of supersonic inviscid gas flows

We consider the flow of supersonic homogeneous gas past a supersonic spherical source. This problem provides a gas-dynamic model of the interaction of interstellar wind with solar wind, and is thus also of independent interest. It is solved using an explicit through divergence scheme of third-order approximation. The analysis focuses on formation and stability of the structure of discontinuity surfaces and convergence to the stationary solution. The results are compared qualitatively and quantitatively with solutions obtained by other methods. Translated from Chislennye Metody v Matematicheskoi Fizike, Published by Moscow University, Moscow, 1996, pp. 125–128.     

High-frequency approximation of the green's function for the Helmholtz equation in an axial nonhomogeneous medium

A method is proposed for constructing the field of a point source in an axial two-dimensionally nonhomogeneous medium. The problem is reduced to an integral equation of the 2nd kind. If the frequency ω of the point source is sufficiently high, the equation can be solved by the method of successive approximations. The Neumann series converges in the usual sense for ω>>1 and it is an asymptotic series for ω→∞. The Neumann series also converges in the case of weak dependence of the medium on the coordinate φ. Translated from Obratnye Zadachi Estestvoznaniya, Published by Moscow University, Moscow, 1997, pp. 104–108.     

Supersonic longitudinal flow past a cylinder in the presence of a second coaxial cylinder in the wake

The compressible-gas complete Navier-Stokes equations are applied for a numerical analysis of the flow structure between two coaxial cylinders of equal radius set in tandem in a supersonic flow. The results for Re_∞=10⁴, M_∞=3 ω=0.75, Tw=1.08, when the distance between the coaxial cylinders is equal to two blunt-nose radii, show that the flow between the cylinders is subsonic and quasiperiodic. Translated from Obratnye Zadachi Estestvoznaniya, Published by Moscow University, Moscow, 1997, pp. 199–203.     

Complexity of finite sequences of zeros and ones and geometry of finite spaces of functions

The sequence 001 001 001 001 is simpler than 01 001 0111 001. We present a formal mathematical theory that gives an exact meaning to the preceding (intuitively clear) statement. Supported in part by the RFBR (Grant No. 05-01-00104). Read at the Moscow Mathematical Society meeting 22 November 2005, homepage: .     

Modified Navier-Stokes equations for finite-difference computation of viscous flows

The classical Navier-Stokes model is modified so as to increase its accuracy. A system of third-order differential equations is proposed describing viscous flow of liquids and gases. Translated from Obratnye Zadachi Estestvoznaniya, Published by Moscow University, Moscow, 1997, pp. 189–198.     

An algorithm for the constrained extremum problem

An algorithm is proposed to find the constrained minimum of a function. The algorithm uses differential equations that incorporate the problem constraints as singular manifolds. Translated from Obratnye Zadachi Estestvoznaniya, Published by Moscow University, Moscow, 1997, pp. 48–51.     

Applied aspects of the theory of continuation of solutions of wave-scattering boundary-value problems

The article compares two approaches to continuation of solutions of external boundary-value problems for the Helmholtz equation outside the original region in the solution space — analytical continuation and the generalized image method. The comparison focuses on naturalness and ease of use of the mathematical model for the construction of numerical algorithms. Translated from Chislennye Metody v Matematicheskoi Fizike, Published by Moscow University, Moscow, 1996, pp. 68–79.     

Continuous regularized proximal minimization method

A continuous regularization method based on the proximal method is proposed for minimization problems with an inexact objective function. Sufficient convergence conditions are given, and the regularizing operator is constructed. Translated from Chislennye Metody v Matematicheskoi Fizike, Published by Moscow University, Moscow, 1996. pp. 5–25.     

Harpedonaptae and abstract convexity

This is an overview of the origin and basic ideas of abstract convexity. The text was submitted by the author in English. This article bases on a talk on abstract convexity and cone-vexing abstraction at the International Workshop “Idempotent and Tropical Mathematics and Problems of Mathematical Physics,” Moscow, August 25–30, 2007.     

Bifurcation analysis of the mathematical model of the NO+CO/Pt(100) reaction

The article presents a detailed bifurcation analysis of steady and periodic states for a new mathematical model of the NO+CO/Pt(100) reaction. Various bifurcation diagrams are constructed in the planes of partial reagent pressures and surface temperature. Regions with oscillations and multiple steady solutions are investigated. Isolated branches of steady and periodic states are identified. An “explosive” bifurcation of the periodic solution leading to chaotic alternation of small-and large-amplitude oscillations is detected and analyzed for the first time. A good quantitative fit is demonstrated between modeling results and experimental data. Translated from Obratnye Zadachi Estestvoznaniya, Published by Moscow University, Moscow, 1997, pp. 52–78.     

Mathematical modeling of vertical electrical sounding of quasilayered media

The direct solution of the vertical electrical sounding problem is considered for three-dimensional quasilayered media. An efficient fast algorithm is obtained for computing the apparent resistivity curves in the two-dimensional case. A PC implementation of the algorithm is proposed. Translated from Obratnye Zadachi Estestvoznaniya, Published by Moscow University, Moscow, 1997, pp. 123–131.     

The attractor of the delayed functional-differential diffusion equation

We consider a class of functional-differential diffusion equations with delay and spatial rotation that arise in nonlinear optics. The existence of a compact global attractor is established for the corresponding discrete dynamic systems. Translated from Chislennye Metody v Matematicheskoi Fizike, Published by Moscow University, Moscow, 1996, pp. 147–152.     

Compressible gas flow inside a cylindrical channel with a source on the channel axis

The article solves the problem of compressible nonviscous gas flow inside a cylindrical channel in the presence of a source on the channel axis. The cases considered include a supersonic spherically symmetrical source and a hypersonic source modeling jet flow. The numerical solution is obtained by two methods: one method treats the flow singularities (shocks, etc.) in explicit form, while the other uses the “through calculation” procedure. The solution is applied to analyze the physical flow pattern and the distribution of gasdynamic parameters in the shock layer, in particular on the channel walls. Translated from Obratnye Zadachi Estestvoznaniya, Published by Moscow University, Moscow, 1997, pp. 169–180.     

An inverse problem for a system with a moving boundary

The problem of estimating the right-hand side of a nonlinear parabolic equation is considered. A finite-step algorithm based on the model positional control method and the finite element method is proposed. The algorithm is robust to informational noise and computational errors. Translated from Obratnye Zadachi Estestvoznaniya, Published by Moscow University, Moscow, 1997, pp. 23–33.     

Fredholm equations of the first kind

A method is described for solving the Fredholm integral equation of the 1st kind by passage to the moment L-problem. The problem is reduced to a linear programming problem. A bound of the method for a particular normed space is derived. Translated from Obratnye Zadachi Estestvoznaniya, Published by Moscow University, Moscow, 1997, pp. 34–38.     

Simulation of oscillation modes in a supersonic unrated jet impinging on a cavity

The flow of a supersonic underexpanded jet past a cavity is analyzed numerically. Through difference schemes approximate to first, second, and third order are used. The focus is on oscillation modes and their effect on back-flow. Deceased. Translated from Obratnye Zadachi Estestvoznaniya, Published by Moscow University, Moscow, 1997, pp. 181–188.     

Estimates of stability of characterization of additive types of distributions

An estimate of stability of characterization of distribution types is obtained for the case of additive types. Under some conditions, the estimate has the order ε^1/3L(ε), where L(ε) is a slowly varying function. Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, Russia, 1996, Part I.     

Application of the particle method to simulate one-dimensional bounded plasma with a distributed sources

We consider the behavior of a plasma bounded in the longitudinal direction by absorbing walls. The model contains charged particles (electrons and ions) moving in the direction of an external magnetic field with two velocity components: longitudinal and transverse. The charged particles are created in pairs by a distributed source. The working model is based on the electrostatic “particles in a cell” method augmented by Emmert's model for a volume source and a model of binary Coulomb particle collisions using the Monte Carlo method. Calculation results are reported for a model with electron-ion collisions and for a collisionless plasma model. Translated from Chislennye Metody v Matematicheskoi Fizike, Published by Moscow University, Moscow, 1996, pp. 100–109.     

Inverse coefficient problem for a nonlinear differential equation and an iterative solution method

The article considers the determination of the solution-dependent coefficient of a nonautonomous ordinary differetial equation with a parameter. Reduction of the inverse problem to a nonlinear operator equation is used to prove existence and uniqueness theorems and to propose an iterative solution method. Translated from Obratnye Zadachi Estestvoznaniya, Published by Moscow University, Moscow, 1997, pp. 5–17.