Mathematical Model of the Stick-Slip Effect for Describing the "Drumbeat" Seismic Regime During the Eruption of the Kizimen Volcano in Kamchatka

During the eruption of the Kizimen volcano in 2010-2013. There was a uniform squeezing of the viscous lava flow. Simultaneously with its movement, earthquakes with an unusual quasi-periodicity were recorded, the "drumbeats" mode. In this work, we show that these earthquakes were generated by the movement of the flow front, which was observed for the first time in the practice of volcanological research. We represent the movement of the flow as an intermittent slip with the inclusion of the "stick-slip" mechanism with the initiation of a self-oscillating process. The plausibility of the phenomenological model at the qualitative level is confirmed by the mathematical model of a fractional nonlinear oscillator

Mathematical Model of a Wide Class Memory Oscillators

A mathematical model is proposed for describing a wide class of radiating or memory oscillators. As a basic equation in this model is an integro-dierential equation of Voltaire type with dierence kernels - memory functions, which were chosen by power functions. This choice is due, on the one hand, to broad applications of power law and fractal properties of processes in nature, and on the other hand it makes it possible to apply the mathematical apparatus of fractional calculus. Next, the model integro-differential equation was written in terms of derivatives of fractional Gerasimov - Caputo orders. Using approximations ofoperators of fractional orders, a non-local explicit nite-dierence scheme was compiled that gives a numerical solution to the proposed model. With the help of lemmas and theorems, the conditions for stability and convergence of the resulting scheme are formulated. Examples of the work of a numerical algorithm for some hereditary oscillators such as Duffing, Airy and others are given, their oscillograms and phase trajectories are constructed.В работе предложена математическая модель для описания широкого класса эредитарных осцилляторов или осцилляторов с памятью. В качестве базового уравнения в такой модели выступает интегро-дифференциальное уравнения вольтеровского типа с разностными ядрами - функциями памяти, которые были выбраны степенными функциями. Этот выбор, с одной стороны, обусловлен широкими приложениями степенных законов и фрактальными свойствами процессов в природе, а с другой, дает возможность применить математический аппарат дробного исчисления. Далее, в работе модельное интегро-дифференциальное уравнение было записано в терминах производных дробных порядков Герасимова - Капуто. Используя аппроксимации операторов дробных порядков, была составлена нелокальная явная конечно-разностная схема, которая дает численное решение предложенной модели. С помощью лемм и теорем сформулированы условия устойчивости и сходимости полученной схемы. Приведены примеры работы численного алгоритма для некоторых эредитарных осцилляторов, построены их осциллограммы и фазовые траектории

NUMERICAL ANALYSIS OF THE CAUCHY PROBLEM FOR A WIDE CLASS FRACTAL OSCILLATORS

The Cauchy problem for a wide class of fractal oscillators is considered in the paper and its numerical investigation is carried out using the theory of finite-difference schemes. Fractal oscillators characterize oscillatory processes with power memory or, in general, with heredity, and are described by means of integro-differential equations with difference kernels — memory functions. By choosing memory functions as power functions, integrodifferential equations are reduced to equations with derivatives of fractional orders. In this paper, using the approximation of the fractional derivatives of Gerasimov-Kaputo, a non-local explicit finite-difference scheme was developed, its stability and convergence are justified, estimates of the numerical accuracy of computational accuracy are presented. Examples of the work of the proposed explicit-finite scheme are given. It is shown that the order of computational accuracy tends to unity as the number of grid nodes increases and coincides with the order of approximation of the explicit finite difference scheme

STUDY POINTS OF REST HEREDITARITY DYNAMIC SYSTEMS VAN DER POL-DUFFING

Using numerical modeling, oscillograms and phase trajectories were constructed to study the limit cycles of a van der Pol Duffing nonlinear oscillatory system with a power memory. The simulation results showed that in the absence of a power memory (α = 2, β = 1) or the classical van der Pol Duffing dynamical system, there is a single stable limit cycle, i.e. Lienar theorem holds. In the case of viscous friction (α = 2, 0 < β < 1), there is a family of stable limit cycles of various shapes. In other cases, the limit cycle is destroyed in two scenarios: a Hopf bifurcation (limit cycle-limit point) or (limit cycle-aperiodic process). Further continuation of the research may be related to the construction of the spectrum of Lyapunov maximal exponents in order to identify chaotic oscillatory regimes for the considered hereditary dynamic system (HDS)

Solution of diffusion-advection equation of radon transport in many-layered geological media

Radon transport modeling in geological media is an important tool for solving problems and tasks of radioecology and geophysics. Comparison of radon field time series obtained by numerical and experimental methods is one of the most common and widely applicable ways to analyze the influence of state and variability of meteorological, electrical and actinometric parameters of atmosphere, cosmic weather factors, variations of deflected mode of geological medium on the level and variations of radon field. The solutions of stationary and non-stationary diffusion-advection equations of radon transport in many-layered geological media by numerical methods, notably by integro-interpolation method (balance method) are presented. The peculiarity of radon transport in many-layered media is taken into account in the developed numerical model. This peculiarity is connected with the transport equation coefficients which can change very rapidly at the border of two adjacent layers, i.e. they can be discontinuous at the borders of each layer that can be caused by parameters of soils greatly differing in value (density, porosity, radium content, diffusion and emanation coefficients). The present work is provided with an example of application of the developed numerical model for solving a practical problem on assessment of influence of deep seated uranium-containing rock on the value of radon volumetric activity at the depth of less-than or equal to 1 m. The article considers non-stationary numerical model calculations showing at what time moments the distribution curves of radon volumetric activity coincide with stationary regime of radon transport in geological media. The validity of the developed numerical solution has been confirmed by these calculations

MATHEMATICAL MODEL OF DYNAMICS OF SMALL ENTERPRISES WITH ACCOUNT OF MEMORY EFFECTS

The paper proposes a new mathematical model of the dynamics of small enterprises with the participation of foreign investment, which takes into account the memory effect and affects the rate of change in the value of production assets. This memory effect can be considered as a property of the economic environment, for example, the influence on the production of external factors, in which the cost of production assets will depend on its previous values. This non-local effect can be written in terms of a fractional order derivative. In this paper, we will assume that the order of the fractional derivative is a function of time. Therefore, we will solve the initial model equation using numerical methods of the theory of finite difference schemes. Further, in the work, visualization and interpretation of the calculation result was carried out