# Vol 24, No 3 (2018)

**Year:**2018**Articles:**7**URL:**https://journals.ssau.ru/est/issue/view/309

## Full Issue

## Статьи

### ON FRACTIONAL DIFFERENTIATION

#### Abstract

Due to the operation of fractional diﬀerentiation introduced with the help of Fourier integral, the results of calculating fractional derivatives for certain types of functions are given. Using the numerical method of integration, the values of fractional derivatives for arbitrary dimensionality *ε*, (where *ε* **is** any number greater than zero) are calculated. It is proved that for *integer* values of *ε* we obtain ordinary derivatives of the ﬁrst, second and more high orders. As an example it was considered heat conduction equation of Fourier, where spatial derivation was realized with the use of fractional derivatives. Its solution *is *given *by *Fourier integral. M*oreover*, it was shown that integral went into the required results in special case of the whole *ε* obtained in *n*-dimensional case, where *n*=1*,*2*...*, etc.

**Vestnik of Samara University. Natural Science Series**. 2018;24(3):7-13

### ON TOPOLOGICAL ALGEBRAS OF ANALYTIC FUNCTIONALS WITH A MULTIPLICATION DEFINED BY TRANSLATIONS

#### Abstract

We deﬁne a multiplication — convolution in the dual of a countable inductive limit *E *of weighted Fr´echet spaces of entire functions of several variables. This algebra is isomorphic to the commutant of the system of partial derivatives in the algebra of all continuous linear operators in *E*. In the constructed algebra of analytic functionals in two pure cases a topology is deﬁned. With this topology the mentioned algebra is topological and it is now topologically isomorphic to the considered commutant with its natural operator topology. It is proved that in this pure situations the present algebra has no zero divisors provided that polynomials are dense in *E*. We show that this condition is essential for the validity of the last statement.

**Vestnik of Samara University. Natural Science Series**. 2018;24(3):14-22

### BOUNDARY VALUE PROBLEM FOR THE ALLER — LYKOV MOISTURE TRANSPORT GENERALIZED EQUATION WITH CONCENTRATED HEAT CAPACITY

#### Abstract

The article considers the Aller — Lykov equation with a Riemann — Liouville fractional time derivative, boundary conditions of the third kind and with the concentrated speciﬁc heat capacity on the boundary of the domain. Similar conditions arise in the case with a material of a higher thermal conductivity when solving a temperature problem for restricted environment with a heater as a concentrated heat capacity. Analogous conditions also arise in practices for regulating the water-salt regime of soils, when desalination of the upper layer is achieved by draining of a surface of the ﬂooded for a while area. Using energy inequality methods, we obtained an a priori estimate in terms of the Riemann — Liouville fractional derivative, which revealed the uniqueness of the solution to the problem under consideration.

**Vestnik of Samara University. Natural Science Series**. 2018;24(3):23-29

### THE CAUCHY PROBLEM FOR THE HYPERBOLIC DIFFERENTIAL EQUATION OF THE THIRD ORDER

#### Abstract

In the article the Cauchy problem for the third order hyperbolic differential equation with nonmultiple characteristics is considered on the plane of two independent variables. The differential equation has tree nonmultiple characteristics and this equation is strongly hyperbolic equation. The regular solution of the Cauchy problem for the hyperbolic differential equation of the third order with the nonmultiple characteristics is constructed in an explicit form, the solution is obtained by the method of general solutions. The solution of the Cauchy problem enables describing the propagation of initial displacement, initial velocity and initial acceleration.

**Vestnik of Samara University. Natural Science Series**. 2018;24(3):30-34

### PHAPL WEB APPLICATION FOR AUTOMATIC BUILDING AND RESEARCH OF PHASE PORTRAITS ON PLANE

#### Abstract

The article aims to document PhaPl that’s a teaching web application to plot and research phase portraits of autonomous systems of 2 differential equations on a plane. The web application is very different compared with previously known programs: it has very easy graphical user interface and it gives clarity because it demonstrates all steps of solution. To get the full solution, it is enough to just enter a system to research. Initial conditions to plot phase trajectories are chosen automatically. Graphical representation of the phase plane is interactive. The web application is based on popular Free Software (SymPy, PyPy.js, MathJax, LZMA-JS). The web application is portable and works in web browsers that support JavaScript and canvas element of HTML5. The web application can be downloaded and used offline without connection to the Internet. The article describes advantages, disadvantages and peculiar properties of the web application. The software is deployed inPlekhanovRussianUniversityof Economics since 2018.

**Vestnik of Samara University. Natural Science Series**. 2018;24(3):41-52

### MODEL OF SELF-OSCILLATIONS WITHOUT HARMONICAS OF THE BASE FREQUENCY

#### Abstract

The nonlinearity of self-oscillatory system limiting amplitude of the generated signal is a source of the higher harmonicas of the base frequency. Harmonicas distort a form of self-oscillations and lower stability of their frequency. In the work the mathematical model of generation of self-oscillations, free from the highest harmonicas — strictly monochromatic self-oscillations is offered. The model is based on a method of equivalent (harmonious) linearization, popular in the applied theory of nonlinear oscillations. Numerical realization of the model in discrete time has allowed to formulate two algorithms of generation of monochromatic self-oscillations. One of them includes the procedure of numerical integration of a Cauchy problem for the system of ordinary differential equations. Another — reproduces processes in the discrete dynamic system designed on analog model-prototype. The monochromaticity of discrete self-oscillations is confirmed within the numerical experiment.

**Vestnik of Samara University. Natural Science Series**. 2018;24(3):53-59