arXiv subjects

Roman Parovik

Publications and source records attributed to Roman Parovik.

5 recordsLinked to original sources

arXiv2021

Some Aspects of the Numerical Analysis of a Fractional Duffing Oscillator with a Fractional Variable Order Derivative of the Riemann-Liouville Type

In this paper, we consider some aspects of the numerical analysis of the mathematical model of fractional Duffing with a derivative of variable fractional order of the Riemann-Liouville type. Using numerical methods: an explicit finite-difference scheme based on the Grunwald-Letnikov and Adams-Bashford-Moulton approximations (predictor-corrector), the proposed numerical model is found. These methods have been verified with a test case. It is shown that the predictor-corrector method has a faster convergence than the method according to the explicit finite-difference scheme. For these schemes, using Runge's rule, estimates of the computational accuracy were made, which tended to unity with an increase in the number of calculated grid nodes.

math.NA↗

arXiv2021

A Computer Program for the Numerical Analysis of Economic Cycles Within the Framework of the Dubovsky Generalized Model

The article proposes a computer program for calculating economic crises according to the generalized mathematical model of S.V. Dubovsky. This model is represented by a system of ordinary nonlinear differential equations with fractional derivatives in the sense of Gerasimov-Caputo with initial conditions. Furthermore, according to a numerical algorithm based on an explicit nonlocal finite-difference scheme, oscillograms and phase trajectories were constructed. It is shown that changing the orders of fractional derivatives in the model can give rise to various modes, for example, damped modes with a steady-state amplitude. It is concluded that the orders of fractional derivatives are responsible for the intensity of the process.

math.NA↗

arXiv2021

Research of the hereditary dynamic Riccati system with modification fractional differential operator of Gerasimov-Caputo

In this paper, we study the Cauchy problem for the Riccati differential equation with constant coefficients and a modified Gerasimov-Caputo type fractional differential operator of variable order. Using Newton's numerical algorithm, calculation curves are constructed taking into account different values of the Cauchy problem parameters. The calculation results are compared with the previously obtained results. The computational accuracy of the numerical algorithm is investigated. It is shown using the Runge rule that the computational accuracy tends to the accuracy of the numerical method when increasing the nodes of the calculated grid.

math.NA↗

arXiv2021

Application of the Adams-Bashfort-Mowlton Method to the Numerical Study of Linear Fractional Oscillators Models

The paper presents a numerical analysis of the class of mathematical models of linear fractional oscillators, which is the Cauchy problem for a differential equation with derivatives of fractional orders in the sense of Gerasimov-Caputo. A method based on an explicit nonlocal finite-difference scheme (ENFDS) and the Adams-Bashfort-Moulton (ABM) method is considered a numerical analysis tool. An analysis of the errors of the methods is carried out, and it is shown that the ABM method is more accurate and converges faster to an exact solution than the ENFDS method.

math.NA↗

arXiv2021

Numerical Modeling of Kondratyev's Long Waves Taking into Account Heredity

The paper proposes a new mathematical model of economic cycles and crises, which generalizes the well-known model of Dubovsky S.V. The novelty of the proposed model lies in taking into account the effect of heredity (memory), as well as the introduction of harmonic functions responsible for the arrival of investments in fixed assets and new management technologies in innovation. The mathematical description is given using the Gerasimov-Caputo fractional derivatives, which are studied within the framework of the theory of fractional calculus. The mathematical model was investigated using the numerical method of Adams-Bashforth-Moulton (ABM), phase trajectories were constructed. It is shown that the proposed mathematical model can have both regular and chaotic regimes.

math.NA↗