Geodesic
Mathematical modelling of hereditarity Airy oscillator with friction
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 10 (2017) no. 1, pp. 138-148 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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.
Keywords: Airy oscillator; hereditarity; Gerasimov–Caputo derivative; finite-difference scheme; the phase trajectory. 
@article{VYURU_2017_10_1_a8,
     author = {R. I. Parovik},
     title = {Mathematical modelling of hereditarity {Airy} oscillator with friction},
     journal = {Vestnik \^U\v{z}no-Uralʹskogo gosudarstvennogo universiteta. Seri\^a, Matemati\v{c}eskoe modelirovanie i programmirovanie},
     pages = {138--148},
     year = {2017},
     volume = {10},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VYURU_2017_10_1_a8/}
}
TY  - JOUR
AU  - R. I. Parovik
TI  - Mathematical modelling of hereditarity Airy oscillator with friction
JO  - Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie
PY  - 2017
SP  - 138
EP  - 148
VL  - 10
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/VYURU_2017_10_1_a8/
LA  - ru
ID  - VYURU_2017_10_1_a8
ER  - 
%0 Journal Article
%A R. I. Parovik
%T Mathematical modelling of hereditarity Airy oscillator with friction
%J Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie
%D 2017
%P 138-148
%V 10
%N 1
%U http://geodesic.mathdoc.fr/item/VYURU_2017_10_1_a8/
%G ru
%F VYURU_2017_10_1_a8
R. I. Parovik. Mathematical modelling of hereditarity Airy oscillator with friction. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 10 (2017) no. 1, pp. 138-148. http://geodesic.mathdoc.fr/item/VYURU_2017_10_1_a8/

[1] V. V. Uchaikin, Fractional Derivatives for Physicists and Engineers, v. I, Background and Theory, Higher Education Press, Beijing; Springer-Verlag, Berlin, 2013, 373 pp. | DOI | MR | Zbl

[2] Volterra V., Theory of Functionals and of Integral and Integro-Differential Equations, Dover, N.Y., 1959, 304 pp. | MR | Zbl

[3] G. B. Airy, “On the Intensity of Light in the Neighbourhood of a Caustic”, Transactions of the Cambridge Philosophical Society, 6 (1838), 379–402

[4] Honina S. N., Volotovskij S. G., “Mirror Laser Airy Beams”, Computer Optics, 34:2 (2014), 203–213 (in Russian)

[5] K. B. Oldham, J. Spanier, The Fractional Calculus. Theory and Applications of Differentiation and Integration to Arbitrary Order, Academic Press, London, 1974, 240 pp. | MR | Zbl

[6] K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differntial Equations, A Wiley-Interscience Publication, N.-Y., 1993, 384 pp. | MR

[7] Parovik R. I., Mathematical Modelling of Linear Oscillators Hereditarity, Kamchatskiy Gosudarstvennyy Universitet Imeni Vitusa Beringa, Petropavlovsk-Kamchatskiy, 2015, 178 pp.

[8] Parovik R. I., “Cauchy Problem of Generalized Airy Equation”, Reports Adyghe (Circassian) International Academy of Sciences, 16:3 (2014), 64–69 (in Russian)

[9] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006, 523 pp. | DOI | MR | Zbl

[10] F. Mainardi, “Fractional Relaxation-Oscillation and Fractional Diffusion-Wave Phenomena”, Chaos, Solitons Fractals, 7:9 (1996), 1461–1477 | DOI | MR | Zbl

[11] Afanas'ev V. V., Danilaev M. P., Pol'skij Yu. E., “Stabilization of Fractal Oscillator Inertial Effects”, Technical Physics Letters, 36:7 (2010), 1–6 (in Russian)

[12] I. Petras, Fractional-Order Nonlinear Systems. Modeling, Analysis and Simulation, Higher Education Press, Beijing; Springer-Verlag, Berlin, 2011, 218 pp. | DOI | Zbl

[13] M. S. Tavazoei, M. Haeri, “Chaotic Attractors in Incommensurate Fractional Order Systems”, Physica D: Nonlinear Phenomena, 237:20 (2008), 2628–2637 | DOI | MR | Zbl

[14] Y. A. Rossikhin, M. V. Shitikova, “Application of Fractional Calculus for Dynamic Problems of Solid Mechanics: Novel Trends and Recent Results”, Applied Mechanics Reviews, 63:1 (2010), 010801 | DOI

[15] Samarskij A. A., Gulin A. V., The Stability of Difference Schemes, Nauka, M., 1973, 415 pp.

[16] Parovik R. I., “Numerical Analysis Some Oscillation Equations with Fractional Order Derivatives”, Bulletin KRASEC. Physical and Mathematical Sciences, 9:2 (2014), 34–38 | DOI

[17] Y. Xu, V. Suat Ertürk, “A Finite Difference Technique for Solving Variable-Order Fractional Integro-Differential Equations”, Bulletin of the Iranian Mathematical Society, 40:3 (2014), 699–712 | MR | Zbl