# No 2 (2011)

**Year:**2011**Articles:**23**URL:**https://journals.rudn.ru/miph/issue/view/530

## Articles

### Inverse Problem of Determination of Absorbtion Coefficient in Parabolic Equation in a Plane

#### Abstract

We consider existence and uniqueness of solution for the inverse problem of determination of the unknown coeﬃcient α(x) at the u(t,x) in nondivergent parabolic equation in a plane. The additional information is given by @op@prepare∫ 0Tu(t,x)χ(t)dt, with χ(t) being the weight function. It should be noted that the coeﬃcients of the equation depend both on time as well as spatial variable. For the problem concerned we obtain conditions suﬃcient for the existence and the uniqueness of generalized solution. We adduce the examples of the inverse problems satisfying the conditions imposed.

**Discrete and Continuous Models and Applied Computational Science**. 2011;(2):5-15

5-15

### The Convolution Type Equation on in Space of Functions those are Summed with Exponential Weights. Part 1

#### Abstract

In this paper (parts 1 and 2) the theory of one-sided invertibility of the convolution operator on R in space of functions those are summed with exponential weights is considered. In part 1 we present results on the boundedness of the convolution operator, the division theorem in the algebra of the analytic functions in the band and a Fredholm problem for Wiener-Hopf operator in studying space.

**Discrete and Continuous Models and Applied Computational Science**. 2011;(2):16-27

16-27

### On Solving Restoration Problem with Degenerated Diffusion by Separation Method

#### Abstract

The suffient conditions of restoration problems' solvability in a class of the Ito stochastic diﬀerential systems of the ﬁrst order (with random disturbances from a class of Wiener processes and the diﬀusion degenerated with regard to a part of variables) with the given properties of a movement, when a control is included into the coeﬃcient of drift, are obtained by separation method. The structure of the control parameter is deﬁned, ensuring suﬃcient conditions of the given integral manifold's existence of constructed equations' set in nonlinear and linear Problem statements.

**Discrete and Continuous Models and Applied Computational Science**. 2011;(2):28-34

28-34

### On Existence and Uniqueness of a Solution to a Singular Integrodifferential Equation

#### Abstract

We presented suﬃcient conditions of existence and uniqueness of solutions of the singular integrodiﬀerential equation of ﬁrst order in various couples of function spaces, giving corresponding motivation. The rate of convergence for the approximate solution depending on structural properties is also established.

**Discrete and Continuous Models and Applied Computational Science**. 2011;(2):35-43

35-43

### Discrete Inequalities of Hardy Type with Variable Limits of Summation. III

#### Abstract

It is fnished the study of the necessary and suﬃcient conditions of validity for discrete inequalities of Hardy type with variable limits of summation in the sequence spaces started in the papers @cite KH and @cite KH2.

**Discrete and Continuous Models and Applied Computational Science**. 2011;(2):44-50

44-50

### On the Dimensions of Counterexamples to Borsuk's Conjecture On Spheres of Small Radii

#### Abstract

In this article there are constructions of counterexamples to the Borsuk's conjecture deﬁned, which are sets laying on spheres of small radii. This work shows how the dimension of a counterexample depends on the radius of the covering sphere.

**Discrete and Continuous Models and Applied Computational Science**. 2011;(2):51-58

51-58

### Economics and Mathematical Modeling of Oligopoly Telecommunication Market

#### Abstract

This paper presents a model of M competing telecommunication companies. The telecommunication networks of companies have diﬀerent attributes which assumed ﬁx and the consumers have idiosyncratic tastes for these attributes. The networks are mandated to interconnect and the access charges are determined by companies cooperatively. M telecommunication network companies are engaged in a price competition to attract consumers. Each consumer selects a network and determines the consumption of the telecommunication services. The regulation policy of the telecommunication market is studied for this model.

**Discrete and Continuous Models and Applied Computational Science**. 2011;(2):59-70

59-70

### Quantum Well Models for Low-Dimensional Structures as Anharmonic Oscillators with Two and Three Minima

#### Abstract

By symbolic-numeric power series method the one-dimensional Shr̈odinger's equation with double- and triple-well potentials is solved. The energy spectra and wave function depending on the potential parameters are calculated. It is shown that magnitude of energy splitting is very sensitive to the form of the potential function.

**Discrete and Continuous Models and Applied Computational Science**. 2011;(2):71-82

71-82

### Continuous Analogue of Newton Method in Beam Dynamics Problems

#### Abstract

An algorithm of the continuous analogue of Newton method (CANM) is proposed for the solving of the boundary value problems of beam transport.
The eﬃciency of CANM has been practically shown on a number of the problems of beam dynamics leading to the solving of ordinary diﬀerential and integral equations.
The solving of the problem of determining the optimal (in sense of some criterions of quality) parameters Pi for charged particles transportation systems taking into account diﬀerent nonlinear eﬀects, is given. The results of the calculation of the consistent invisible straight section (insertion) of the accelerator obtained with the help of CANM are shown.

**Discrete and Continuous Models and Applied Computational Science**. 2011;(2):83-91

83-91

### Integration of Highly Oscillatory Functions

#### Abstract

The paper demonstrates approximate methods of integral calculation of highly oscillatory functions. The paper describes a quadrature method which adopts Chebyshev diﬀerential matrix to solve the ordinary diﬀerential equation (ODE) and thus obtain integral values. This method make the system of linear equations well-conditioned for general oscillatory integrals. Furthermore, even if the system of linear equations is ill-conditioned, regularization method can be adopted to solve it properly and eventually obtain accurate integral results.

**Discrete and Continuous Models and Applied Computational Science**. 2011;(2):92-97

92-97

### Solving Systems of Linear Differential Equations with Constant Coefficients

#### Abstract

The paper demonstrates the comparison of some diﬀerent algorithms for solving systems of ordinary diﬀerential equations with complex constant coeﬃcients describing the evolution of monochromatic linearly polarized plane electromagnetic waves in a stratiﬁed medium.

**Discrete and Continuous Models and Applied Computational Science**. 2011;(2):98-103

98-103

### Control of Motion along a Specified Curve and Inverse Problem of Dynamics

#### Abstract

The article deals with the problem of control over mechanical system motion along a speciﬁed curve in state space. The conditions of motion stability along a speciﬁed curve in coordinate space are determined. The constructing techniques of control forces, depending on the coordinates of mechanical system, are viewed in the article. Moreover, the solving of the problem of control over particles motions in the central ﬁeld of forces is performed.

**Discrete and Continuous Models and Applied Computational Science**. 2011;(2):104-110

104-110

### On Stability of Relative Equilibriums of a Rigid Body with a Vibrating Point of Support

#### Abstract

In this paper motions of a heavy rigid body, of which point of support performs vertical harmonical vibrations of a high frequency and a small amplitude, are studied. Using an approximate autonomous system of diﬀerential equations of the motion, an analysis of existence, bifurcations and stability of relative equilibrium is carried out, for which the mass center and the point of support of the body don't lie at the same vertical. It is shown that all of these relative equilibria are unstable.

**Discrete and Continuous Models and Applied Computational Science**. 2011;(2):111-122

111-122

### Expansion Method for Continuating at Extended Solution at Singular Points

#### Abstract

Stability of structures depends on the variation of the load factor on the continuation of their load path beyond a critical point. The computation of the continuation is known to be diﬃcult due to special properties of the stiﬀness matrices of structures in the vicinity of singular points. A new expansion method is presented that leads to a robust and accurate continuation algorithm. The method is applied to the stability analysis of a spherical dome for symmetric and asymmetric snow loads.

**Discrete and Continuous Models and Applied Computational Science**. 2011;(2):123-132

123-132

### Dielectric Spectroscopy of thin Films at Terahertz Frequencies

#### Abstract

The problem of thin ﬁlm dielectric spectroscopy at terahertz (THz) frequencies is under study in the paper. It has been stated that the technique employing surface plasmons (SP) excitation by the probing radiation on the metal substrate surface can be eﬀectively used for solving the problem. To adopt the SP spectroscopy technique to the THz range we have developed a number of methods and devices making possible to determine the SP's complex refractive index depending on the ﬁlm's optical properties. Some of the methods are based on interference of bulk and (or) surface waves, others - on the intensity measurements of the SP ﬁeld. In addition the methods developed enable one to perform the measurements for one pulse duration of the radiation.

**Discrete and Continuous Models and Applied Computational Science**. 2011;(2):133-147

133-147

### Darboux Transformations for the Generalized SchrЁodinger Equation

#### Abstract

The Darboux transformations of the n-th order is elaborated for a generalized Schr̈odinger equation with a position-dependent eﬀective mass and with a linearly energy-dependent potential. The Darboux transformations are given also in an integral form. A correspondence between the diﬀerential Darboux transformations and the integral ones has been established. The second-order Darboux transformations are analyzed both at diﬀerent energies and at the same transformation energy. The method is illustrated by several examples of constructing quantum potential wells with a given spectrum.

**Discrete and Continuous Models and Applied Computational Science**. 2011;(2):148-160

148-160

### On Sound Velocityin Two-Phase and Two-Component Medium

#### Abstract

We have obtained an expression for the velocity of propagation of small perturbations (sound velocity) in the medium consisting of ﬂuid vapour bubbles and metal particles. In addition, we have assumed that the system is in thermodynamical equilibrium, vapour bubbles and metal particles are distributed homogeneously, the metal is incompressible ( = 0), the bubble and particle sizes and the distance between them is much smaller than sound wave length.

**Discrete and Continuous Models and Applied Computational Science**. 2011;(2):161-164

161-164

### Chiral Model of Graphene

#### Abstract

The chiral model of graphene based on the SU(2) order parameter is suggested in the long-wave approximation, the ideal graphene plane being determined by the kink-like solution. Corrugation of the graphene surface is described in the form of ripple and rings. The approximate solution corresponding to an inﬁnite carbon nanotube is found.

**Discrete and Continuous Models and Applied Computational Science**. 2011;(2):165-168

165-168

### The System of Hamilton Equations for the Modes of the Electromagnetic Field in a Stratified Isotropic Medium

#### Abstract

The paper demonstrates the Hamiltonian nature of ordinary diﬀerential equations describing the evolution of monochromatic linearly polarized plane electromagnetic waves in a stratiﬁed medium. The possibility and necessity of using symplectic numerical methods for integrating the resulting system of Hamilton equations is established.

**Discrete and Continuous Models and Applied Computational Science**. 2011;(2):169-171

169-171

172-179

### Nashi avtory

**Discrete and Continuous Models and Applied Computational Science**. 2011;(2):180-181

180-181

### Pravila oformleniya statey

**Discrete and Continuous Models and Applied Computational Science**. 2011;(2):182-182

182-182

### O konferentsiyakh na fakul'tete Fiziko-matematicheskikh i estestvennykh naukv aprele 2011 g.

**Discrete and Continuous Models and Applied Computational Science**. 2011;(2):183-184

183-184