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Mathematical modeling of linear oscillators hereditarity
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Монография посвящена вопросам математического моделирования эредитарных линейных колебательных систем, которые учитывают эффекты «памяти». Разработаны математические модели эредитарных линейных осцилляторов на основе формализма дробного интегро-дифференцирования, построены численные алгоритмы их решения с помощью конечно-разностных схем. На основе этих решений исследованы фазовые траектории линейных эредитарных осцилляторов. Данная монография может быть полезна студентам, магистрам, аспирантам и научным сотрудникам, которые изучают математические методы моделирования эредитарных процессов в средах с фрактальной структурой.
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... The operator fractional order (6) is also generalized by [11] to the case where n < α(t) < n + 1. ...
... An approximation of the derivative of a fractional variable order of the Gerasimov-Caputo type (6) in Equation (11) can be written as follows for t ∈ [t k , t k+1 ] as follows: ...
... The discrete Cauchy problem (13) approximates the original differential problem (11) with the order: max ...
The article discusses different schemes for the numerical solution of the fractional Riccati equation with variable coefficients and variable memory, where the fractional derivative is understood in the sense of Gerasimov-Caputo. For a nonlinear fractional equation, in the general case, theorems of approximation, stability, and convergence of a nonlocal implicit finite difference scheme (IFDS) are proved. For IFDS, it is shown that the scheme converges with the order corresponding to the estimate for approximating the Gerasimov-Caputo fractional operator. The IFDS scheme is solved by the modified Newton’s method (MNM), for which it is shown that the method is locally stable and converges with the first order of accuracy. In the case of the fractional Riccati equation, approximation, stability, and convergence theorems are proved for a nonlocal explicit finite difference scheme (EFDS). It is shown that EFDS conditionally converges with the first order of accuracy. On specific test examples, the computational accuracy of numerical methods was estimated according to Runge’s rule and compared with the exact solution. It is shown that the order of computational accuracy of numerical methods tends to the theoretical order of accuracy with increasing nodes of the computational grid.
... Hereditarity in an integro-differential equation is characterized by a memory function-its difference kernel. Depending on the type of memory function, we will obtain various classes of integro-differential equations to describe hereditary processes (see [7]). ...
... In [4,6,7] with the help of fractional calculus theory, a special case of the hereditary oscillatory process was investigated-a linear fractal oscillator; in some cases analytical solutions were obtained for it, phase trajectories and oscillograms were plotted. Such oscillatory systems may have other modes that are inherent in nonlinear oscillators. ...
... Indeed, by opening the sums in (5) and arranging the terms we come to (7). ...
In this paper we suggest an explicit finite-difference scheme for numerical simulation of the Cauchy problem with an integro-differential nonlinear equation that describes an oscillatory process with friction and memory (hereditarity), and with the corresponding local initial conditions. The problems of approximation, stability, and convergence of the proposed finite-difference scheme are investigated. The results of computer experiments that implement the proposed numerical scheme, confirming the theoretical estimates obtained in theorems, are given.
... Therefore, fractal dynamical systems are a special case of hereditary dynamical systems. In the simulation of fractal dynamical systems, it is convenient from the integro-differential equations to proceed to equations with derivatives of fractional orders [2]. The orders of the derivatives are related to the fractal dimension of the medium and, in the general case, can depend on the time [3], [4]. ...
... Further, in the work the calculated curves were constructed depending on various values of the control parameters taking into account the ratio (2) and without taking it into account. It was confirmed that the stability of the circuit is fulfilled under condition (2). Further investigation of the Cauchy problem (1) is connected with the consideration of the nonlinear right-hand side in the original equation. ...
... Therefore, fractal dynamical systems are a special case of hereditary dynamical systems. In the simulation of fractal dynamical systems, it is convenient from the integro-differential equations to proceed to equations with derivatives of fractional orders [2]. The orders of the derivatives are related to the fractal dimension of the medium and, in the general case, can depend on the time [3], [4]. ...
... Further, in the work the calculated curves were constructed depending on various values of the control parameters taking into account the ratio (2) and without taking it into account. It was confirmed that the stability of the circuit is fulfilled under condition (2). Further investigation of the Cauchy problem (1) is connected with the consideration of the nonlinear right-hand side in the original equation. ...
The mathematical model of the hereditary dynamic system is proposed, which describes free oscillations with allowance for the variable memory. Variable memory is reflected in the model in the form of derivatives of fractional order variables. Put the corresponding Cauchy problem, which was solved by the theory of finite-difference schemes. The constructed scheme was investigated for stability and convergence, the results of the research are formulated in the form of theorems.
... Due to the nonlinearity of the Cauchy problem (4), we will seek its solution using the numerical method of finite-difference schemes [31][32][33]. Consider a uniform mesh. ...
The article proposes a nonlocal explicit finite-difference scheme for the numerical solution of a nonlinear, ordinary differential equation with a derivative of a fractional variable order of the Gerasimov–Caputo type. The questions of approximation, convergence, and stability of this scheme are studied. It is shown that the nonlocal finite-difference scheme is conditionally stable and converges to the first order. Using the fractional Riccati equation as an example, the computational accuracy of the numerical method is analyzed. It is shown that with an increase in the nodes of the computational grid, the order of computational accuracy tends to unity, i.e., to the theoretical value of the order of accuracy.
... Then, Eq. (5) describes a linear hereditary Airy oscillator, which was considered in the author's papers [21,39] and has the following form. We choose the initial condition (6) for simplicity by homogeneous: ...
In this study, the model Riccati equation with variable coefficients as functions, as well as a derivative of a fractional variable order (VO) of the Gerasimov-Caputo type, is used to approximate the data for some physical processes with saturation. In particular, the proposed model is applied to the description of solar activity (SA), namely the number of sunspots observed over the past 25 years. It is also used to describe data from Johns Hopkins University on coronavirus infection COVID-19, in particular data on the Russian Federation and the Republic of Uzbekistan. Finally, it is used to study issues related to seismic activity, in particular, the description of data on the volumetric activity of Radon (RVA). The Riccati equation used in the mathematical model was numerically solved by constructing an implicit finite difference scheme (IFDS) and its implementation by the modified Newton method (MNM). The calculated curves obtained in the study are compared with known experimental data. It is shown that if the model parameters are chosen appropriately, the model curves will give results that correlate well with real experimental data. Moreover, with other parameters of the model, it is possible to make some prediction about the possible course of the considered processes.
The Riccati differential equation with a fractional derivative of variable order is considered. A derivative of variable fractional order in the original equation implies the hereditary property of the medium, i.e., the dependence of the current state of a dynamic system on its previous states. A software called Numerical Solution of a Fractional-Differential Riccati Equation (briefly NSFDRE) is created; it allows one to compute a numerical solution of the Cauchy problem for the Riccati differential equation with a derivative of variable fractional order. The numerical algorithm implemented in the software is based on the approximation of the variable-order derivative by finite differences and the subsequent solution of the corresponding nonlinear algebraic system. New distribution modes depending on the specific type of variable order of the fractional derivative were obtained. We also show that some distribution curves are specific for other hereditary dynamic systems.
Abstract. In this paper, oscillograms and phase trajectories are constructed using numerical
simulation to study the limiting cycles of a nonlinear FitzHugh-Nagumo oscillatory system with
power memory. The simulation results showed that in the absence of power memory (α=2, β=1) or the
classical dynamic FitzHugh-Nagumo system, there is a single stable limit cycle, i.e. the Lienard
theorem is fulfilled. In the case of viscous friction (α=2, 0< β<1), there is a family of stable
limit cycles of different shapes. In other cases, the limit cycle destroyed according to two
scenarios: 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 maximum
Lyapunov exponents for a purpose of identifying chaotic oscillatory
regimes for the considered hereditary dynamic system (HDS).
Work is devoted to mathematical modelling hereditarity oscillatory systems with the help of the mathematical apparatus of fractional calculus on the example of an Airy oscillator with friction. Model Airy equation was written in terms of Gerasimov - Caputo fractional derivatives. Next a finite-difference scheme to this generalized equation for numerical computation was proposed. The problems of approximation, stability and convergence of a numerical scheme are considered. The results of simulations are presented based on numerical solutions waveforms and phase trajectories depending on different values of the control parameters are built.
The paper presents a mathematical model of non-classical dynamic systems. A numerical method of difference schemes, depending on various parameters of the system were found numerical solutions of models. The phase trajectory.
We consider a nonlocal model of the wave process, which generalizes the classical model parmetricheskogo resonance Mathieu. It is proved that this model has a unique solution.
The paper deals with the explicit finite difference schemes for the fractional oscillator. The questions of approximation, stability and convergence of these schemes.
The paper presents a model of fractal parametric oscillator. Showing that the solution of such a model exists and is unique. A study of the solution with the aid of diagrams Stratton-Ince. The regions of instability, which can occur parametric resonance. It is suggested that this solution can be any signal, including acoustic.
Special Functions for Applied Scientists provides the required mathematical tools for researchers active in the physical sciences. The book presents a full suit of elementary functions for scholars at the PhD level and covers a wide-array of topics and begins by introducing elementary classical special functions. From there, differential equations and some applications into statistical distribution theory are examined. The fractional calculus chapter covers fractional integrals and fractional derivatives as well as their applications to reaction-diffusion problems in physics, input-output analysis, Mittag-Leffler stochastic processes and related topics. The authors then cover q-hypergeometric functions, Ramanujan's work and Lie groups. The latter half of this volume presents applications into stochastic processes, random variables, Mittag-Leffler processes, density estimation, order statistics, and problems in astrophysics. © 2008 Springer Science+Business Media, LLC. All rights reserved.
In this article, we use a finite difference technique to solve variable - order fractional integro - differential equations (VOFIDEs, for short). In these equations, the variable-order fractional integration(VOFI) and variable-order fractional derivative (VOFD) are described in the Riemann-Liouville's and Caputo's sense,respectively. Numerical experiments, consisting of two examples, are studied. The obtained numerical results reveal that the proposed finite difference technique is very effective and convenient for solving VOFIDEs.