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Mathematics
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Mathematical Modeling of Hereditarity Oscillatory Systems
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Mathematical Modeling of Hereditarity Oscillatory Systems

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematical Physics".

Deadline for manuscript submissions: closed (31 October 2020) | Viewed by 10794

Special Issue Editors

Dr. Roman Parovik

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Guest Editor
Physical Processes Modeling Laboratory, Institute of Cosmophysical Research and Radio Wave Propagation, Far Eastern Branch of the Russian Academy of Sciences, Kamchatskiy Kray, 684034 Paratunka, Russia
Interests: fractional calculus; fractional oscillators; fractional dynamics; numerical methods; mathematical modeling
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Arsen V. Pskhu

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Guest Editor
Institute of Applied Mathematics and Automation, Kabardino-Balkarian Scientific Center of Russian Academy of Sciences, 89-A Shortanov Street, 360000 Nalchik, Russia
Interests: fractional calculus; fractional differential equation; mathematical modeling
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

This Special Issue of the journal Mathematics is devoted to recent advancements in the study of the hereditary properties of oscillatory processes and systems, as well as their possible applications in mechanics, physics, biology, economics, and other sciences. The hereditary properties of oscillatory systems are manifested in the dependence of their current states on previous states. This memory property is usually described using integro-differential equations with difference kernels, which are called memory functions. In the particular case where the memory functions are power law, the integro-differential equations can be written in terms of derivatives of fractional orders, which are studied in the framework of fractional calculus. Therefore, oscillatory systems, taking into account the effect of “power memory”, are sometimes called fractional oscillators in the literature. As the works of recent years have shown, fractional oscillators can have important applied values, and their characteristics can be related to the properties of the medium through the orders of fractional derivatives included in the model equations.

Prof. Dr. Roman Parovik
Prof. Dr. Arsen V. Pskhu
Guest Editors

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Keywords

  • Oscillator
  • Fractional calculus
  • Heredity
  • Fractional derivative
  • Phase trajectories
  • Oscillograms

Published Papers (5 papers)

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Research

7 pages, 249 KiB  
Article
Ordinary Differential Equation with Left and Right Fractional Derivatives and Modeling of Oscillatory Systems
by Liana Eneeva, Arsen Pskhu and Sergo Rekhviashvili
Mathematics 2020, 8(12), 2122; https://doi.org/10.3390/math8122122 - 27 Nov 2020
Cited by 4 | Viewed by 1505
Abstract
We consider the principle of least action in the context of fractional calculus. Namely, we derive the fractional Euler–Lagrange equation and the general equation of motion with the composition of the left and right fractional derivatives defined on infinite intervals. In addition, we [...] Read more.
We consider the principle of least action in the context of fractional calculus. Namely, we derive the fractional Euler–Lagrange equation and the general equation of motion with the composition of the left and right fractional derivatives defined on infinite intervals. In addition, we construct an explicit representation of solutions to a model fractional oscillator equation containing the left and right Gerasimov–Caputo fractional derivatives with origins at plus and minus infinity. We derive a representation for the composition of the left and right derivatives with origins at plus and minus infinity in terms of the Riesz potential, and introduce special functions with which we give solutions to the model fractional oscillator equation with a complex coefficient. This approach can be useful for describing dissipative dynamical systems with the property of heredity. Full article
(This article belongs to the Special Issue Mathematical Modeling of Hereditarity Oscillatory Systems)
13 pages, 293 KiB  
Article
Fractional Diffusion–Wave Equation with Application in Electrodynamics
by Arsen Pskhu and Sergo Rekhviashvili
Mathematics 2020, 8(11), 2086; https://doi.org/10.3390/math8112086 - 22 Nov 2020
Cited by 10 | Viewed by 2047
Abstract
We consider a diffusion–wave equation with fractional derivative with respect to the time variable, defined on infinite interval, and with the starting point at minus infinity. For this equation, we solve an asympotic boundary value problem without initial conditions, construct a representation of [...] Read more.
We consider a diffusion–wave equation with fractional derivative with respect to the time variable, defined on infinite interval, and with the starting point at minus infinity. For this equation, we solve an asympotic boundary value problem without initial conditions, construct a representation of its solution, find out sufficient conditions providing solvability and solution uniqueness, and give some applications in fractional electrodynamics. Full article
(This article belongs to the Special Issue Mathematical Modeling of Hereditarity Oscillatory Systems)
16 pages, 1911 KiB  
Article
Hereditary Oscillator Associated with the Model of a Large-Scale αω-Dynamo
by Gleb Vodinchar
Mathematics 2020, 8(11), 2065; https://doi.org/10.3390/math8112065 - 19 Nov 2020
Cited by 4 | Viewed by 1555
Abstract
Cosmic magnetic fields possess complex time dynamics. They are characterized by abrupt polarity changes (reversals), fluctuations of fixed polarity, bursts and attenuations. These dynamic conditions can replace each other, including both regular and chaotic components. Memory in dynamo systems manifests itself in a [...] Read more.
Cosmic magnetic fields possess complex time dynamics. They are characterized by abrupt polarity changes (reversals), fluctuations of fixed polarity, bursts and attenuations. These dynamic conditions can replace each other, including both regular and chaotic components. Memory in dynamo systems manifests itself in a feedback mechanism when a strong magnetic field begins to change the properties of turbulent flows. A hereditary oscillator can be the simplest model of such complex oscillatory systems with memory. The article suggests the construction of such oscillator by means of two-mode approximation of magnetic field components in the αω-dynamo model. The hereditary member describes the suppression of a field turbulent generator by magnetic helicity and determines the shape of oscillator potential. The article describes the implicit difference scheme for numerical research of oscillator. It also describes the results of numerical simulation for two cases—instantaneous feedback and delay in feedback. The results of simulation are interpreted in terms of oscillator theory. It is shown that the observed dynamic regimes in the model go well with the change of potential shape. Full article
(This article belongs to the Special Issue Mathematical Modeling of Hereditarity Oscillatory Systems)
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14 pages, 477 KiB  
Article
Mathematical Model of Fractional Duffing Oscillator with Variable Memory
by Valentine Kim and Roman Parovik
Mathematics 2020, 8(11), 2063; https://doi.org/10.3390/math8112063 - 19 Nov 2020
Cited by 12 | Viewed by 2312
Abstract
The article investigates a mathematical model of the Duffing oscillator with a variable fractional order derivative of the Riemann–Liouville type. The study of the model is carried out using a numerical scheme based on the approximation of the fractional derivative of the Riemann–Liouville [...] Read more.
The article investigates a mathematical model of the Duffing oscillator with a variable fractional order derivative of the Riemann–Liouville type. The study of the model is carried out using a numerical scheme based on the approximation of the fractional derivative of the Riemann–Liouville type by a discrete analog—the fractional derivative of Grunwald–Letnikov. The adequacy of the numerical scheme is verified using specific examples. Using a numerical algorithm, oscillograms and phase trajectories are constructed depending on the values of the model parameters. Chaotic regimes of the Duffing fractional oscillator are investigated using the Wolf–Bennetin algorithm. The forced oscillations of the Duffing fractional oscillator are investigated using the harmonic balance method. Analytical formulas for the amplitude-frequency, phase-frequency characteristics, and also the quality factor are obtained. It is shown that the fractional Duffing oscillator possesses different modes: regular, chaotic, multi-periodic. The relationship between the order of the fractional derivative and the quality factor of the oscillatory system is established. Full article
(This article belongs to the Special Issue Mathematical Modeling of Hereditarity Oscillatory Systems)
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26 pages, 2828 KiB  
Article
Mathematical Modeling of Linear Fractional Oscillators
by Roman Parovik
Mathematics 2020, 8(11), 1879; https://doi.org/10.3390/math8111879 - 29 Oct 2020
Cited by 18 | Viewed by 2881
Abstract
In this work, based on Newton’s second law, taking into account heredity, an equation is derived for a linear hereditary oscillator (LHO). Then, by choosing a power-law memory function, the transition to a model equation with Gerasimov–Caputo fractional derivatives is carried out. For [...] Read more.
In this work, based on Newton’s second law, taking into account heredity, an equation is derived for a linear hereditary oscillator (LHO). Then, by choosing a power-law memory function, the transition to a model equation with Gerasimov–Caputo fractional derivatives is carried out. For the resulting model equation, local initial conditions are set (the Cauchy problem). Numerical methods for solving the Cauchy problem using an explicit non-local finite-difference scheme (ENFDS) and the Adams–Bashforth–Moulton (ABM) method are considered. An analysis of the errors of the methods is carried out on specific test examples. It is shown that the ABM method is more accurate and converges faster to an exact solution than the ENFDS method. Forced oscillations of linear fractional oscillators (LFO) are investigated. Using the ABM method, the amplitude–frequency characteristics (AFC) were constructed, which were compared with the AFC obtained by the analytical formula. The Q-factor of the LFO is investigated. It is shown that the orders of fractional derivatives are responsible for the intensity of energy dissipation in fractional vibrational systems. Specific mathematical models of LFOs are considered: a fractional analogue of the harmonic oscillator, fractional oscillators of Mathieu and Airy. Oscillograms and phase trajectories were constructed using the ABM method for various values of the parameters included in the model equation. The interpretation of the simulation results is carried out. Full article
(This article belongs to the Special Issue Mathematical Modeling of Hereditarity Oscillatory Systems)
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