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-differential equation of Voltaire type with difference 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 of operators of fractional orders, a non-local explicit finite-difference 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.

Keywords

mathematical model; Cauchy problem; heredity; derivative of fractional order; finite-difference scheme; stability; convergence; oscillograms; phase trajectory.

References

1. Volterra V. Sur les equations integro-differentielleset leurs applications. Acta Mathematica, 1912, vol. 35, no. 1, pp. 295-356. DOI: 10.1007/BF02418820
2. Mainardi F. Fractional Relaxation-Oscillation and Fractional Diffusion-Wave. Chaos, Soliton and Fractal, 1996, vol. 7, no. 9, pp. 1461-1477. DOI: 10.1016/0960-0779(95)00125-5
3. Petras I. Fractional-Order Nonlinear Systems. Modeling, Analysis and Simulation. Berlin, Heidelberg, Springer, 2011. DOI: 10.1007/978-3-642-18101-6
4. Xu Y., Agrawal O.P. Models and Numerical Solutions of Generalized Oscillator Equations. Journal of Vibration and Acoustics, 2014, vol. 136, Article ID 051005. DOI: 10.1115/1.4027241
5. Parovik R.I. Mathematical Modeling of Nonlocal Oscillatory Duffing System with Fractal Friction. Bulletin KRASEC. Physical and Mathematical Sciences, 2015, vol. 10, no. 1, pp. 16-21.
6. Parovik R.I. On a Credit Oscillatory System with the Inclusion of Stick-Slip. E3S Web of Conferences, 2016, vol. 11, Article ID 00018.
7. Parovik R.I. Mathematical Modelling of Hereditarity Airy Oscillator with Friction. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2017, vol. 10, no. 1, pp. 138-148. DOI: 10.14529/mmp170109
8. Kim V.A. Duffing Oscillator with External Harmonic Action and Variable Fractional Riemann - Liouville Derivative Character - Izing Viscous Friction. Bulletin KRASEC. Physical and Mathematical Sciences, 2016, vol. 13, no. 2, pp. 46-49.
9. Lipko O.D. Mathematical Model of Propagation of Nerve Impulses with Regard Hereditarity. Bulletin KRASEC. Physical and Mathematical Sciences, 2017, vol. 16, no. 1, pp. 33-43.
10. Kilbas A.A., Srivastava H.M., Trujillo J.J. Theory and Applications of Fractional Differential Equations. Amsterdam, Elsevier, 2006.
11. Schroeder M. Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. New York, W.H. Freeman, 1991.
12. Parovik R.I. Explicit Finite-Difference Scheme for the Numerical Solution of the Model Equation of Nonlinear Hereditary Oscillator with Variable-Order Fractional Derivatives. Archives of Control Sciences, 2016, vol. 26, no. 3, pp. 429-435. DOI: 10.1515/acsc-2016-0023
13. Xu Y., Erturk V.S. A Finite Difference Technique for Solving Variable-Order Fractional Integro-Differential Equations. Bulletin of the Iranian Mathematical Society, 2014, vol. 40, no. 3, pp. 699-712.