Vol 24, No 3 (2018)

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. Moreover, 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.

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.

In this paper, we construct a solution of the inner-boundary problem with a nonlocal shift for the fractional diffusion equation in a rectangular region.

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.