# Vol 27, No 2 (2019)

**Year:**2019**Articles:**6**URL:**https://journals.rudn.ru/miph/issue/view/1244**DOI:**https://doi.org/10.22363/2658-4670-2019-27-2

## Full Issue

## Mathematical Models and Methods in Economics

### Numerical Analysis of Ecology-Economic Model for Forest Fire Fighting in Baikal Region

#### Abstract

The paper is devoted to optimal control problem of forest fires fighting. Its prototype is the well-known Parks model. To investigate the model, we apply the modern programming language Julia, which is designed to mathematical calculations and numerical studies. We made an extensive computational experiment in this model and a numerical analysis of corresponding optimal control problems. The obtained results were examined both on the adequacy of the model, and on the possibility of using the Julia language and the included solvers of mathematical problems.

**Discrete and Continuous Models and Applied Computational Science**. 2019;27(2):

## Computational modeling and simulation

### On algebraic integrals of a differential equation

#### Abstract

We consider the problem of integrating a given differential equation in algebraic functions, which arose together with the integral calculus, but still is not completely resolved in finite form. The difficulties that modern systems of computer algebra face in solving it are examined using Maple as an example. Its solution according to the method of Lagutinski’s determinants and its implementation in the form of a Sagemath package are presented. Necessary conditions for the existence of an integral of contracting derivation are given. A derivation of the ring will be called contracting, if such basis B= {m1, m2, … } exists in which Dmi= cimi+o (mi). We prove that a contracting derivation of a polynomial ring admits a general integral only if among the indices c1, c2, … there are equal ones. This theorem is convenient for applying to the problem of finding an algebraic integral of Briot-Bouquet equation and differential equations with symbolic parameters. A number of necessary criteria for the existence of an integral are obtained, including those for differential equations of the Briot and Bouquet. New necessary conditions for the existence of a rational integral concerning a fixed singular point are given and realized in Sage.

**Discrete and Continuous Models and Applied Computational Science**. 2019;27(2):105-123

## Mathematical Modeling

### Seismic stability of oscillating building on kinematic supports

#### Abstract

The design of kinematic supports is considered, which allows to damp the oscillation energy of seismic waves during earthquakes. The building rests on supports that have the geometry of straight cylinders. When horizontal ground oscillations occur, the supports are deflected at a small angle . At the same time, their centre of gravity rises and tends to return to its original position under the action of two forces on each support: the weight of the building evenly distributed over all the supports, and the weight of the support itself. The first force is applied to the highest point of the support, the second one is applied to the centre of gravity of the support, so that the rotational moments of two forces act on the support. It should be noted that under very strong vibrations of the ground, the projection of the centre of gravity could move beyond the base of the support. In this case, the supports will begin to tip over. We confine ourselves to considering such deviations that the rotational moments of the forces of gravity still tend to return the supports to their initial state of equilibrium.

**Discrete and Continuous Models and Applied Computational Science**. 2019;27(2):124-132

### On the radiation losses during motion of an electron in the field of intense laser radiation

#### Abstract

Motion of the relativistic electron in the field of intense laser pulse of the arbitrary shape is considered. The pulse dimension is supposed to be of the order of the Gaussian laser beam dimension in the focal plane. It is supposed that the pulse is propagating along the external constant magnetic field. In the paraxial approximation the corrections of the first order to the vectors of the field of radiation as well as the force of the radiation friction are taken into account. Averaged relativistic equations of motion of electron are obtained with the help of averaging over the fast oscillations of the laser radiation. It is shown that with taking into account corrections of the first order to the field vectors an averaged force arises. This force is defined by pulsed character of radiation and proportional to the intensity but not to gradient of intensity. It is shown that radiation losses are of little importance in the transverse plane but may considerably act on the longitudinal motion of electron.

**Discrete and Continuous Models and Applied Computational Science**. 2019;27(2):133-142

### Characteristics of optical filters built on the basis of periodic relief reflective structures

#### Abstract

A scheme of a new type optical filter, built using a relief reflective periodic diffraction structure, which has a specific rectangular profile, is proposed. The input radiation beam is directed to the relief structure at a certain angle of incidence. The zero diffraction order beam is our output beam, which is separated from the other diffraction order beams with the help of a diaphragm. The incidence-reflection plane is parallel to the relief lines of the diffraction structure. The dependence of the output beam power on the angle of incidence and on the wavelength of the radiation is investigated. It is shown that the power transfer coefficient from the input to the output of the scheme substantially depends on the wavelength of the optical beam. The scheme can be used as an optical signal filter. The spectral characteristic of this type of filter has an oscillating character. The zero (minimum) values of the power transfer coefficient of radiation from the input to the output of the filter alternate with maximum values close to unity. The spectral characteristic of the filter is easy to change by changing the angle of incidence of the input beam to the relief reflecting structure. Filters of this type can be built for the ultraviolet, visible, and infrared range. Calculations of the dependence of the filter parameters on the relief depth and on the angle of incidence of the input optical beam to the relief structure are presented.

**Discrete and Continuous Models and Applied Computational Science**. 2019;27(2):143-153

### Numerical analysis of ecology-economic model for forest fire fighting in Baikal region

#### Abstract

Forest fires lead to the serious damage of ecological state and national economy of the country. This problem is especially relevant for Siberians. According to Greenpeace, Siberian forest fires in 2019 reached record levels in the entire history of observation in terms of burning area and the amount of carbon dioxide emitted into the atmosphere. It leads not only to a deterioration in the health of Siberians, but also to environmental problems of the region. Note that the large-scale fire-prevention measures entails enormous financial costs. Therefore, economical, ecological and mathematical modeling of the situations, arose in forest fires countering, becomes actual. The paper is devoted to optimal control problem of forest fires fighting. Its prototype is the well-known Parks model. To investigate the model, we apply the modern programming language Julia, which is designed to mathematical calculations and numerical studies. We made an extensive computational experiment in this model and a numerical analysis of corresponding optimal control problems. The obtained results were examined both on the adequacy of the model, and on the possibility of using the Julia language and the included solvers of mathematical problems.

**Discrete and Continuous Models and Applied Computational Science**. 2019;27(2):154-164