Bulletin KRASEC. Phys. & Math. Sci, 2014, V. 9, №. 2, pp. 34-38. ISSN 2313-0156

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DOI: 10.18454/2313-0156-2014-9-2-34-38

MSC 35C05

NUMERICAL ANALYSIS SOME OSCILLATION EQUATIONS WITH FRACTIONAL ORDER DERIVATIVES

R.I. Parovik¹²

¹Institute of Cosmophysical Researches and Radio Wave Propagation Far-Eastern Branch, Russian Academy of Sciences, 684034, Kamchatskiy Kray, Paratunka, Mirnaya st., 7, Russia.

²Vitus Bering Kamchatka State University, 683031, Petropavlovsk-Kamchatsky, Pogranichnaya st., 4, Russia.

E-mail: romanparovik@gmail.com

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.

Key words: operator Gerasimov-Caputo, numerical solution, finite difference scheme, the phase trajectories.

References

  1. Nahushev A.M. Drobnoe ischislenie i ego prilozhenie [Fractional calculus and its application]. Moscow, Fizmatlit Publ., 2003. 272 p.
  2. Shogenov V.H., Ahkubekov A.A., Ahkubekov R.A. Metod drobnogo differencirovaniya v teorii brounovskogo dvizheniya [The method of fractional differentiation in the theory of Brownian motion]. Izvestie vuzov Severo-Kavkazskij region. Estestvennye nauki – Proceedings of the universities of the North Caucasus region. Natural sciences, 2004, no. 1, pp. 46-50.
  3. Shen S., Liu F. Error analysis of an explicit difference approximation for the fractional diffusion equation with insulated ends. ANZIAM J., 2005, 46(E), pp. 871-887.
  4. Mainardi F. Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos, Solitons and Fractals, 9(7), 1996, pp. 1461-1477.
  5. Mejlanov R.P., Yanpolov M.S. Osobennosti fazovoj traektorii fraktal’nogo oscillyatora [Features of the phase trajectory of fractal oscillator]. Pis’ma ZhTF – Letters to JTP, 2002, vol. 28, no. 1, pp. 67-73.
  6. Parovik R.I. Zadacha Koshi dlya nelokal’nogo uravneniya Mat’e [The Cauchy problem for the nonlocal Mathieu equation]. Doklady Adygskoj (Cherkesskoj) Mezhdunarodnoj Akademii Nauk – Reports Adyghe (Circassian) International Academy of Sciences, 2011, vol. 13, no. 2, pp. 90-98.
  7. Parovik R.I. Fractal Parametric Oscillator as a Model of a Nonlinear Oscillation System in Natural Mediums. Int. J. Communications, Network and System Sciences, 2013, vol. 6, pp. 134-138.
  8. Parovik R.I. Zadachi Koshi dlya obobschennogo uravneniya ‘Ejri [Cauchy problem for generalized Airy equation], Doklady Adygskoj (Cherkesskoj) Mezhdunarodnoj Akademii Nauk – Reports Adyghe (Circassian) International Academy of Sciences, 2014, vol. 16, no. 3, pp.64-69.

 

Original article submitted: 23.06.2014


ParParovik Roman Ivanovich – Ph.D. (Phys. & Math.), Dean of the Faculty of Physics and Mathematics Vitus Bering Kamchatka State University, Senior Researcher of Lab. Modeling of Physical Processes, Institute of Cosmophysical Researches and Radio Wave Propagation FEB RAS.

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