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Mathematical Modeling of Linear Fractional Oscillators

Institute of Cosmophysical Research and Radio Wave Propagation, Far East Branch, Russian Academy of Sciences, Mirnaya, 7, Kamchatka Territory, 684034 Paratunka, Russia
Mathematics 2020, 8(11), 1879; https://doi.org/10.3390/math8111879
Received: 20 September 2020 / Revised: 22 October 2020 / Accepted: 26 October 2020 / Published: 29 October 2020
(This article belongs to the Special Issue Mathematical Modeling of Hereditarity Oscillatory Systems)
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. View Full-Text
Keywords: Adams–Bashforth–Moulton method; linear fractional oscillators; oscillograms; phase trajectories; Q-factor; amplitude–frequency response Adams–Bashforth–Moulton method; linear fractional oscillators; oscillograms; phase trajectories; Q-factor; amplitude–frequency response
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MDPI and ACS Style

Parovik, R. Mathematical Modeling of Linear Fractional Oscillators. Mathematics 2020, 8, 1879. https://doi.org/10.3390/math8111879

AMA Style

Parovik R. Mathematical Modeling of Linear Fractional Oscillators. Mathematics. 2020; 8(11):1879. https://doi.org/10.3390/math8111879

Chicago/Turabian Style

Parovik, Roman. 2020. "Mathematical Modeling of Linear Fractional Oscillators" Mathematics 8, no. 11: 1879. https://doi.org/10.3390/math8111879

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