# No 3 (2014)

**Year:**2014**Articles:**23**URL:**https://journals.rudn.ru/miph/issue/view/503

## Articles

### Extension of scmapping on Duplicate of Space

#### Abstract

In this paper we consider topological duplications of so-paracompact spaces and extensions of sc-mappings on duplicate. The notions of so-paracompact space and sc-mapping (which generalize the notions of paracompact space and continuous mapping respectively) were introduced by the authors recently and based on so-sets (so-set is the union of the open set and the nowhere dense set). The aim of the work is the investigation of the properties of duplicates of mentioned spaces and mappings. It is proved that Alexandroff duplicate of soparacompact space is so-paracompact. It is proved also that mentioned duplicate is almost paracompact space. It was established that natural extension of sc-mapping on duplicate of space is so-mapping which possess of supplementary properties. It is proved that the natural extension of quasi-continuous mapping is quasi-continuous.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(3):5-10

5-10

### On Integrals of Ordinary Differential Equations Systems which are Representable in Finite Terms

#### Abstract

Existing theories on resolvability of nonlinear differential equations systems in a finite terms are generalization of Galois theory and for this reason the list of elementary operations is subject of the contract. In the Stockholm lectures (1897) Painleve gave on the example of the equations of the 1st and 2nd order property which is common for all equations, solvable in elementary, special and abelian functions: the general solutions of these equations depend on integration constants algebraically. Thus, if we record algebraic properties of the common decision, we can allocate a class of all-usable transcendental functions. This statement can be inscribed in a circle of the theory of Galois, i.e. we can construct the theory for the differential equations without fixing of this list. We consider an arbitrary system of ordinary differential equations g1(x1,. . ., x'1)=0,..., here g1,... are polynomials from x1,x'1 ... , which coefficients lie in a field k of functions of a variable t, for example in k = C(t). This system has solutions in an algebraically closed field K, for example in the field of Puiseux series. We will assume that ideal p =(f1,...) of ring K[x1,... ] is simple and that there is a differentiation D of the ring the rational functions on affine variety V (p/K), which kernel is a field of integrals of the system. Coefficients of integrals generate a field over k. We will designate its transcendence degree as r and prove that there are r-parametrical group of automorphisms for the field of integrals. This theorem will be used for calculation of integrals of these equations.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(3):11-16

11-16

### Zeros and Poles of the Functions with Weak Derivatives

#### Abstract

The classic results of Gergen J. J., Dressel F. G. are generalized to the class of the functions with weak derivatives. We suppose that these derivatives could be estimated by the proper functions multiplied by the weighted functions which have singularities at isolated boundary points. The crucial point of the study is the iteration process used for the evaluations of the functions represented by the potential operators. As a result of such iterations we succeed in lowering the degree of kernel singularities of the potential operators. The above mentioned method is based on representation formula of I.N. Vekua for the functions whose weak derivatives are summable over domains. The analytic functions participating in these representations could be considered as generalized constants. We study the classes of those functions whose generalized constants have finite numbers of poles and zeros. We prove theorems on behavior of the above mentioned functions in neighborhood of their zeros. Besides we study these functions in the neighborhood of the points where they haven’t finite limits. The main result of the paper is the theorem on the number of zeros and poles of the functions under consideration. This result is the generalization of theorem from the paper of Gergen J. J., Dressel F. G.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(3):17-27

17-27

### Looking for Families of Periodic Solutions of Ordinary Differential Equations Systems by Normal Form Method. Part I

#### Abstract

In this paper, we discuss the application of resonant normal form method to the search of periodic solutions families of autonomous systems of explicit ordinary differential equations with polynomial nonlinearities in the right parts. Further, using formulated by Prof. A. D. Bruno sufficient convergence condition for the normalizing transformation, we find local families of periodic solutions of systems of such ODE in the vicinity of stationary points. In this unified approach both Hamiltonian and not Hamiltonian systems are investigated. For reasons of volume the article is divided into two parts. In the first part we describe an algorithm of implementing the method of normal forms. Software packages created by the authors are briefly described separately. We have developed a RLISP language package for working in REDUCE system, and for MATHEMATICA system a package on the external language of this system. This packages allow us, in particular, to obtain formulas describing local (containing a fixed point) families of periodic solutions. The results of calculations are presented in the form of Fourier series segments of a given length with frequency and coefficients themselves calculated as parameter series segments. This representation corresponds to the special case of segments of Poisson series. It is important that using a single algorithm, one can study both two-dimensional and higher-order systems. The second part is devoted to fourth-order systems. The comparison of tabulation of formulas obtained with numerical solutions of the corresponding equations shows good quantitative agreement. The approach described can be used for modeling of physical and biological systems.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(3):28-45

28-45

### One-Step Stochastic Processes Simulation Software Package

#### Abstract

It is assumed that the introduction of stochastic in mathematical model makes it more adequate. But there is virtually no methods of coordinated (depended on structure of the system) stochastic introduction into deterministic models. Authors have improved the method of stochastic models construction for the class of one-step processes and illustrated by models of population dynamics. Population dynamics was chosen for study because its deterministic models were sufficiently well explored that allows to compare the results with already known ones. To optimize the models creation as much as possible some routine operations should be automated. In this case, the process of drawing up the model equations can be algorithmized and implemented in the computer algebra system. Furthermore, on the basis of these results a set of programs for numerical experiment can be obtained. The computer algebra system Axiom is used for analytical calculations implementation. To perform the numerical experiment FORTRAN and Julia languages are used. The Runge- Kutta method for stochastic differential equations is used as numerical method. The program complex for creating stochastic one-step processes models is constructed. Its application is illustrated by the predator-prey population dynamic system. Computer algebra systems are very convenient for the purposes of rapid prototyping in mathematical models design and analysis.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(3):46-59

46-59

### On the Modeling of Queueing Systems with Multiple Resources

#### Abstract

We consider queueing systems, in which customers occupy some resources that are released after customer departure. Arriving customers are lost if there is not enough free resources required for their servicing. In such systems for each customer it is necessary to record vector of occupied resources until its departure. This greatly complicates the stochastic processes describing the behavior of systems in time. Instead of systems of this type we propose to investigate their simplified analogy. Simplified system operates similarly to the original, except that the amount of resources released upon completion of service, is random and may differ from those that have been allocated to the customer. For given total amount of resources employed and the number of applications in the system, the amount of resources released at the completion of service, does not depend on the behavior of the system up to this point and has a distribution function, which can be easily computed using Bayes’ formula. Random processes describing the behavior of simplified systems are easier to analyze, because there is no need to memorize the volume of resources held by each customer. It is enough to record the total amount of occupied resources. The simulation results say that the characteristics of the original and simplified systems are very close.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(3):60-64

60-64

### On Sensitivity of Systems Reliability Characteristics to the Shape of Their Elements Life and Repair Time Distributions

#### Abstract

The paper deals with the problem of the systems ‹M2/GI/1› and ‹GI2/M/1› reliability characteristics sensitivity to the shape of their elements life and repair times distributions under restrictions on the availability of recovery. Partial differential equation for the time dependent and usual differential equations for the stationary micro-state probabilities of these systems are proposed. Explicit expressions for the micro-and macro-state stationary probabilities of these systems are given and they show their strong dependability on the shape of their elements life and repair times distributions. This dependence represents in terms of moment generation functions non-exponential distribution in the point of the exponential distribution parameters. Special software tool based on the MATLAB computer system has been developed for the numerical analysis of the system failure probability sensitivity to the shape of its elements life and recovery distributions and its comparison with the simplest Markov system. The numerical analysis shows that this dependence becomes negligible and vanishes for “fast” recovery (with recovery rate increasing). In particular, it has been shown that the failure probabilities of the systems ‹M2/GI/1› and ‹GI2/M/1› with Gamma and Weibull-Gnedenko distributions instead of the general ones quickly converge to zero with increasing recovery rate and coincide with the simplest Markov system ‹M2/M/1› for special value of the particular value of the parameter c =1.0.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(3):65-77

65-77

### Application of the Methods of Stochastic Micro Dynamics to the Research of Stability in the Systems of Economic Exchange

#### Abstract

The paper is devoted to the research of stability in the systems of economic exchange with the use of the stochastic micro dynamics methods (SMDM) based on the direct computer simulation of the processes progressing in the system. SMDM are derived from the molecular dynamics methods and are supposed to be used for the research of the systems which consist of a great number of particles of non-physical nature. For the simulation of economic system we have chosen agent simulation. Because of the stochastic component in the system which appears in the agent’s behaviour there was need to use Monte Carlo method to get reliable results of the simulation. The paper presents the formal model of the economic system which was the main test subject of the research work and results of its computer simulation with the different sets of input parameters. The results presented in the paper are supposed to be used for the future research of the multi-agent systems which components have non-physical origin.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(3):78-85

78-85

### Maximum Principle in a Problem of Maximization of the Income for Model of a Gas Deposit (Continued)

#### Abstract

This article is devoted to the study of the maximization of the accumulated income for the model of the gas deposit on a finite horizon, a detailed analysis of the obtained results and their comparison with the results of the previously posted this same problem on an infinite horizon. So far the same tasks, based on a model with interacting wells, were solved at a constant price for gas. In reality, however, the price for the goods quite often has a nonlinear dependence and depends on the volume of purchases. Therefore, the statement of the problem is modified by the inclusion in its description of the procurement function. A major tool in the search for the solution to the maximization of income on a finite horizon is the Pontryagin’s maximum principle under the condition of its existence. There are two areas, separated from each other parametric dependence. On each of the selected areas with the use of the method of “phase diagram” the optimal solution is being found. The optimal solution of the problem of maximization on a finite horizon is explicitly described. Joint analysis of the obtained solutions in the problems of maximization on finite and infinite horizon revealed that under certain conditions a part of wells is used inefficiently. Several ways to solve this problem are recommended.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(3):86-98

86-98

### On Generalized Mixed-Additive Regression Models with Spatially Structural Risk Factors

#### Abstract

An identifying of associated risk factors which enhance the risk of infection is the most intensively growing field of epidemiology. But too little investigations considered the spatial structure of the data, as well as possible nonlinear effects of the risk factors. We developed a bayesian spatial semi-parametric regression model for cholera epidemic data. Model estimation and inference is based on fully Bayesian approach via Markov Chain Monte Carlo (MCMC) simulations. The model is applied to cholera epidemic data from Ghana, Africa. Proximity to refuse dumps, density of refuse dumps, and proximity to potential cholera reservoirs were modeled as continuous functions; presence of slum settlers and population density were modeled as fixed effects, spatial references to the communities were modeled as structured and unstructured spatial effects. We found out that the risk of cholera is associated with slum settlements and high population density. The risk of cholera is equal and lower for communities with fewer refuse dumps, but variable and higher for communities with more refuse dumps. The risk is also lower for communities distant from refuse dumps and potential cholera reservoirs. The results also indicate distinct spatial variation in the risk of cholera infection.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(3):99-106

99-106

### The Principle of Feedback on the Quasi-Accelerations for Unstressed Stabilization in Finite Time of Given Man ifolds of Mechanical and Generalized Systems

#### Abstract

The procedure is described of application of the “principle of feedback on the quasi-accelerations” in the construction of auto-adjustment control vector for bringing the condition of mechanical and generalized systems without impact to a given manifold of phase state of systems in finite time, in full or partial uncertainty mass-inertial parameters of the system and disturbances acting on it. This process is called unstressed stabilization of system in a finite time. Varieties of the condition of systems are given by set of holonomic and nonholonomic soft links. A set of control vectors that provide a solution to this problem is obtained. Then from this set of vectors control vectors of minimal dimension and minimum Euclidean norm are allocated. The examples are shown of applying these results to solve problems of practical importance, such as process of control of unstressed docking of surface and underwater and spacecrafts objects, unstressed landing of landers to the moving platforms, and capture of fast moving objects, including “space debris”. In contrast to previous works of the author on the problems of control of mechanical systems, here, along with them, more general systems are also considered, including systems of other physical nature, such as the Helmholtz system, and a wide class of systems with variable masses, depending not only on generalized coordinates, but also on the generalized velocities. In addition, such systems may also include economic systems when considered as dynamic analogs of mechanical and generalized systems. It should be noted that in the above extension of the class of systems under study one have to reckon with the fact that the generalized matrix of quadratic form of the mass-inertial characteristics of the system may not be positive definite, unlike mechanical systems, but only to be nonsingular. This fact does not allow building a control without the use of elements of this matrix, it was possible in the case of mechanical systems. However, in the work a universal vestor could be built that does not depend on these elements, for any generalized systems, Multiplying that vector by this generalized matrix, the law of control of any given generalized system is determined. Thus, the results can be regarded as a significant contribution to the theory of auto-adjustment control of mechanical and generalized systems and their dynamic counterparts, when the goal of control is unstressed bringing of condition of system to given manifold, formed by program constraints, with incomplete information about the non-control forces and disturbances acting on system.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(3):107-114

107-114

### Dynamic Equation of Constrained Mechanical System

#### Abstract

This paper modifies an explicit dynamic equation of constrained mechanical system. Kinematic position of the system is defined by generalized coordinates, which are imposed on constraints. The equations of motion in the form of the Lagrange equations with undetermined multipliers are constructed based on d’Alambert-Lagrange’s principle. Dynamic equations are presented to the mind, resolved relative accelerations. Expressions for the undetermined multipliers are defined by considering the possible deviations from the constraints equations. For constraints stabilization additional variables used to estimate the deviations caused by errors in the initial conditions and the use of numerical methods. For approximation of ordinary differential equations solution, in particular, the nonlinear equations of first order, use explicit numerical methods. Linear equations of the constraints perturbation are constructed. The matrix of the coefficients of these equations is selected in the process of the dynamic equations numerical solution. Stability with respect to initial deviations from the constraints equations and stabilization of the numerical solution depend on the values of the elements of this matrix. As a result values for the matrix of coefficients corresponding to the solution of the dynamics equations by the method of Euler and fourth order Runge-Kutta method are defined. Suggested method for solving the problem of stabilization is used for modeling of the disk motion on a plane without slipping.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(3):115-124

115-124

### On One Numerical Method of Integrating the Dynamical Equations of Projectile Planar Flight Affected by Wind

#### Abstract

Common way to integrate the dynamical equations of projectile planar motion introduces two Cartesian coordinates x(t) and y(t) and attack angle ϑ(t), all depending on time t, and three coupled ordinary differential equations (ODE) each nominally of II-nd order. It leads to inevitable computational complexities and accuracy risks. The method proposed excludes the time variable and diminishes the number of functions to n = 2: the attack angle ϑ(b) and intercept a(b) of the tangent to the trajectory at the point with slope b = tan θ with the θ being the inclination angle. This approach based on Legendre transformation makes the integration more convenient and reliable in the studied case of quadratic in speed aerodynamic forces i.e. drag, lifting force, conservative and damping momenta and the wind affecting the flight. The effective dimensionality of new ODE system is diminished by 2 units and its transcendence is eliminated by simple substitution η = sin ϑ. Also the method enables to obtain easily and reliably the projectile trajectories in conditions of tail-, head-or side wind. Investigated are main ranges of aerodynamic parameters at which takes place different behavior of the attack angle ϑ vs slope b including quasi-stabilization and aperiodic auto-oscillations. In addition, it was revealed non-monotonous behavior of speed with two minima while projectile descending if launched at the angles θ0 close to 90 ∘ . The numerical method may implement into quality improvement of real combat or sporting projectiles such as arch arrow, lance, finned rocket etc.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(3):125-137

125-137

### Estimation of the Relativistic Phase-Shift Formula Applicability to Jupiter’s Satellite System

#### Abstract

In this work an estimate of the relativistic phase shift of space body satellite rotation observed from a remote planet is compared with the classical perturbation of the satellite orbit by other space bodies. The calculations are exemplified by Jupiter’s satellites. A satellite of the Amalthea group interacting with the Galilean satellites is chosen. The interaction of this satellite with the rest of its group is negligible as compared to that with external satellites, since the mass of any internal satellite is much less than that of external ones. A gravitational interaction of Jupiter’s satellite system has been considered within the weak-interaction approximation for inner satellites neglecting Galilean satellites’ action on the phase. Jupiter’s system is chosen since it has many satellites whose mutual interaction is rather strong due to small distances between them and their large mass, besides Jupiter is rather close to us, so it is possible to observe directly its satellites in a telescope and to check data empirically. A gravitational deviation of the chosen inner satellite is calculated to match against the value obtained from the relativistic phase shift formula. The relativistic shift between real and observable phases is given by a formula obtained by A. P. Yefremov in the framework of Quaternion theory. The formula for correction to the phase is a relativistic effect of time delay. The classical correction is estimated using celestial mechanics. An effect of the Galilean satellites on the inner satellites is considered. The phase correction is compared with the value predicted by Quaternion theory of relativity. In conclusion applicability of this formula has been discussed.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(3):138-144

138-144

### Instability of an Extraordinary Wave in a Hot Plasma

#### Abstract

In this paper the investigation of the laser wave parametric decay in the hot magnetized plasma is performed taking into account the relativistic electron mass. Strong external magnetic field affects essentially the efficiency of laser radiation energy input into plasma. The magnetic field of the wave modulates the external magnetic field which leads to the parametric acceleration of electrons in the crossing fields and to the amplification of the charge separation field. In this process up to 85% of the laser radiation energy transforms into the energy of plasma particles. The analysis of nonlinear dynamics of the extraordinary electromagnetic wave in the strong external magnetic field in the conditions of the parametric decay shows that the exponential increase in the amplitude of the secondary wave exited at half-frequency of the primary wave changes into a reverse process in which the energy returns to the primary wave and causes the large amplitude oscillations in plasma. Unlike the previous papers in this area the investigation considers the parametric decay in the plasma preliminary heated up to the relativistic temperature. The self-similar system of nonlinear equations in total derivatives which takes into account the relativistic heat electron mass is derived. The small perturbations of the heated plasma parameters are investigated using the dispersion equation which defines the phase and group velocities of the slow and fast extraordinary waves in the linear approximation. It is shown that unlike the cold plasma in the linear approximation the non-transparency band in the frequency region higher than upper-hybrid electron frequency disappears. Moreover, the asymptotes of the dispersion branches in the high frequency regions approach each other. In the final part of the paper the calculation of the parametric instability increment is performed. It reaches the maximum value when the exited wave frequency is equal exactly half the frequency of the laser pump wave. The analytical expression for the maximum increment is derived and its dependence on the electron thermal velocity is investigated.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(3):145-153

145-153

### Simulation of Gas Mixture Flows in Microchannels

#### Abstract

Problems of simulation of flows of the rarefied gas mixes in micro-channels of technical systems are considered. For decision of such problems a new approach is offered. This approach combines calculations on the quasi-gasdynamics (QGD) equations and molecular dynamic (MD) calculations. The QGD equations are used for calculation of main parameters of mix at macro-level. MD-calculations are used for correction of macro-parameters in the Knudsen layer. For approbation of a technique calculations of the expiration of binary nitrogen-hydrogen mix are carried out to the rarefied microspace. The stationary characteristics of a current received in calculations were compared to the parameters calculated within molecular and dynamic model, and also to results of natural experiments. Comparison showed that in case of the micron sizes of technical system QGD-modeling gives a qualitative consent with experiment and MD-model data. Receiving quantitative coincidence of results requires use of the realistic state equations of a mix component, for example, on a basis of the Virial decomposition coordinated with MD-model.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(3):154-163

154-163

### Approximate Extension of the Lorentz Symmetry up to Conformal in the Limit of Ultrahigh Energies

#### Abstract

The group-theoretical justification is presented for the original approach by Kirznits and Chechin which allows for the primary protons of ultra-high energy cosmic rays to overcome the energetic limit (about 50 EeV) of Greisen-Zatsepin-Kuzmin remaining in the scope of the usual ideas about the nature of the extra-galactic sources of the cosmic rays. But the experimental status of the GZK limit is at present not sufficiently definite due to the rareness of these events in this range of energies as well as due to the difficulty of their identification. Thus it seems reasonable to suggest one of the possible theoretical explanations of this limit without using any kind of so-called “new physics” (e.g., “cosmic strings” etc.). In this paper account is taken only for some natural extension of standard Lorentz kinematics as it formulated in the special relativity theory. It is shown that the explicit form of the factor deforming the Lorentz invariant in the energy-momentum space may be found on the grounds of the approximate transition from Lorentz symmetry to the conformal values of the Lorentz-factor of the order 10 10-10 11 . More technically, we replace the purely phenomenological approach of Kirznits and Chechin by more grounded regular expansion of special conformal transformation in terms of powers of inverse Lorenz-factor 1/y taking account only in the limiting case 1/y. In this way it occurs quite naturally that all the improved kinematics are capable to reproduce reasonably the observed data on the available sources of the extragalactic cosmic rays.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(3):164-170

164-170

### Investigation of Nonpotential Flow of Fluid in Porous Medium Taking into Account of Nonlinear Darcy Law and Variable Diffusion Coefficient

#### Abstract

We have considered the non-potential flow of the incompressible fluid in the porous medium taking into account nonlinear Darcy law and different types of the diffusion coefficient. The flow is supposed to be cylindrically-symmetric and stationary. The velocity has two components: ⃗ =(,0,). We have considered the flow when = 0 + (,),||≪ 0, ≪ 0,0 = const. The combination of the Euler equations reduces to the equation of second order, and continuity equation reduces to an equation of first order for (,) and (,). These equations are linear differential equations with solutions of the form (,)= ()(), = ()(). For () we have obtained the Bessel equation of zero order with solution √ in the form ()= −0( ), = const. From the relation between () and () we √ √ 1 ′ have obtained (): ()= = 1( ), = const. The system of equations for () and () is reduced to one equation of the third order for (). We have obtained the √︁ Φ 0 exact solution of this equation with fixed diffusion coefficient ()= ℎ +Φ1 where Φ0,Φ1, , , = const. A special case when constants in the equation are connected in the relation 0 =200(1 + 00 2) is fully considered. In this case for function () we have obtained the equation of second order. Exact solutions of this equation are obtained with three types of diffusion: () = 0, ()= 0, ()= 0 − , 0 = const, = const. We have established that for all solutions the component of the velocity (,) decreases exponentially with increasing of .

**Discrete and Continuous Models and Applied Computational Science**. 2014;(3):171-181

171-181

### NVESTIGATION OF THE STABILITY OF THE POTENTIAL FLUID FLOW IN A POROUS MEDIUM WITH VARIABLE TRANSVERSE DIFFUSION COEFFICIENT

#### Abstract

We have considered the potential fluid flow in porous medium taking into account variable diffusion coefficient in the tube of radius . The flow is supposed to be cylindrically-symmetric. The velocity has two components: ⃗ =(,0,). The Euler equation has in the right hand side the term which determines Darcy force: ⃗ = −⃗, where - inverse Darcy coefficient. Continuity equation has the term which describes transverse diffusion of flowing fluid. We have established that for Euler equations system the equality 2/ ≡ 2/ is fulfilled identically. It means that Euler equations system is compatible and integrable. For (,) we have obtained Bessel equation, for (,) - the equation with diffusion coefficient (). We have investigated the solution of the equation for (,) with three types of diffusion coefficient (). We have established that in all cases the equation has unstable solution.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(3):182-185

182-185

### Temperature Dependence of the Effective Refractive Index of LE11 and LM11-Modes in Optical Channel Sol-Gel Waveguides of Elevated Type

#### Abstract

The calculation of geometrical parameters of optical channel single-mode sol-gel waveguides (LE11-and LM11-Modes) of elevated type for operation near the critical regime is carried out at three values of refractive index of sol-gel material of the film. The features of the temperature dependences of the effective refractive index (ERI) of LE11-and LM11-modes by means of the method of the effective refractive index (MERI), based on the principles of shadowing field are calculated and explored. Unlike planar sol-gel waveguide, extremum of ERI was observed for modes of both types of polarization in case of channel waveguide of elevated type. The position of the extremum of temperature characteristic of ERI on the ratio of width to thickness of optical channel on the base of sol-gel material is explored. Dependences of temperature coefficient of the effective refractive index on temperature are obtained and the physical mechanism of behavior of the given curves is revealed. The comparison of temperature dependences of the ERI of channel waveguides of elevated type and planar waveguides near the critical regime confirmed the competitive influence of two factors - the negative thermooptical coefficient (TOC) of sol-gel material and thermal expansion of geometrical parameters of the optical channel (thickness and channel width), and also the dependence on partial power in sol-gel layer. It is shown that in the temperature range from 10 to 50 ∘C it is possible to create thermostable channel waveguides on the based of sol-gel films.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(3):186-193

186-193

### Properties of Titanium Dioxide Thin Films, Fabricated by Gel Methods

#### Abstract

Titanium dioxide films obtained by gel method are investigated. Optical properties of the fabricated films, such as thickness, refractive index and thermo-optic coefficient were studied by the methods of integrated optics which use waveguide propagation of radiation along the film. The parameters of the films fabricated by sol-gel and gel methods were compared. It was established that the pores in the films made by gel method contain smaller amounts of water and therefore have higher density. Refractive index of gel films was determined by the resonant angle of the waveguide excitation, calculated using the optical waveguide dispersion equations and amounted to 2.1-2.4. This value is higher than in the case of using sol-gel technology for fabrication of thin films (1.5-1.8). By the reflection and transmission spectra obtained using spectrophotometer, it was found that films produced under cetrain parameters of technological regime have anisotropic properties. It was established that the presence of anisotropy is due to the structure of the film in the form of a linear oligomer. The structure and morphology of the gel films was studied by electron microscopy. It is shown that the resulting films have a porous structure that allows their doping with substances allowing to create elements of integrated optics, such as lasers, amplifiers, etc.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(3):194-201

194-201

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**Discrete and Continuous Models and Applied Computational Science**. 2014;(3):202-203

202-203

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**Discrete and Continuous Models and Applied Computational Science**. 2014;(3):204-205

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