On a Finite-Difference Scheme for an Hereditary Oscillatory Equation


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.

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  1. 1.

    I. V. Drobysheva, “Mathematical modeling of nonlinear hereditary oscillators: Example of the Duffing oscillator with fractional derivatives in the Riemann–Liouville sense,” Vestn. KRAUNC. Fiz.-Mat. Nauki, No. 2 (13), 43–49 (2016).

  2. 2.

    R. O. Kenetova, “Features of the modeling of drive processes due to changes in the structures of the ethnic group,” Vestn. KRAUNC. Fiz.-Mat. Nauki, No. 4 (15), 24–29 (2016).

  3. 3.

    O. D. Lipko, “Mathematical model of the propagation of nerve impulses with hereditarity,” Vestn. KRAUNC. Fiz.-Mat. Nauki, No. 1 (17), 33–43 (2017).

  4. 4.

    M. Mainardi, “Fractional relaxation-occilation and fractional diffusion-wave phenomena,” Chaos, Solitons, and Fractals, 7, No. 9, 1461–1477 (1996).

    MathSciNet  Article  Google Scholar 

  5. 5.

    D. V. Makarov and R. I. Parovik, “Modeling of the economic cycles using the theory of fractional calculus,” J. Internet Banking Commerce, 21, No. S6 (2016).

  6. 6.

    A. M. Nakhushev, Fractional Calculus and Its Application [in Russian], Fizmatlit, Moscow (2003).

    Google Scholar 

  7. 7.

    R. I. Parovik, Mathematical Modeling of Linear Oscillators, Petropavlovsk-Kamchatsky (2015).

  8. 8.

    R. I. Parovik, “Finite-difference schemes for a fractal oscillator with variable fractional orders,” Vestn. KRAUNC. Fiz.-Mat. Nauki, No. 2 (11), 88–95 (2015).

  9. 9.

    R. I. Parovik, “Mathematical model of the fractal van der Pol oscillator,” Dokl. Adyg. (Cherkes.) Akad. Nauk, 17, No. 2, 57–62 (2015).

    Google Scholar 

  10. 10.

    R. I. Parovik, “Explicit finite-difference scheme for the numerical solution of the model equation of nonlinear hereditary oscillator with variable-order fractional derivatives,” Arch. Control Sci., 26, No. 3, 429–435 (2016).

    MathSciNet  Article  Google Scholar 

  11. 11.

    R. I. Parovik, “Fractional calculus in the theory of oscillatory systems,” Sovr. Naukoemk. Tekhnol., No. 1, 61–68 (2017).

  12. 12.

    R. I. Parovik, “Mathematical modeling of the hereditary Airy oscillator with friction,” Vestn. Yuzhno-Ural. Univ. Ser. Mat. Model., 10, No. 1, 138–148 (2017).

    MATH  Google Scholar 

  13. 13.

    I. Petras, Fractional-Order Nonlinear Systems. Modeling, Analysis and Simulation, Springer-Verlag, Berlin–Heidelberg (2011).

  14. 14.

    A. A. Potapov, Fractals in Radiophysics and Radiolocation. Sampling Topology [in Russian], Univ. Kniga, Moscow (2005).

    Google Scholar 

  15. 15.

    I. Pudlubny, “Fractional-order systems and PIλDδ-controllers,” IEEE Trans. Automat. Control, 11, No. 1, 208–214 (1999).

    MathSciNet  Article  Google Scholar 

  16. 16.

    M. Schroeder, Fractals, Chaos, Power Laws, W. H. Freeman and Company, New York (1991).

    Google Scholar 

  17. 17.

    A. Syta, G. Litak, S. Lenci, and M. Scheffler, “Chaotic vibrations of the Duffing system with fractional damping,” Chaos: Interdisc. J. Nonlin. Sci., 24, No. 1, 013107 (2014).

  18. 18.

    V. V. Uchaikin, Method of Fractional Derivatives [in Russian], Artishok, Ulyanovsk (2008).

    Google Scholar 

  19. 19.

    V. Volterra, “Sur les équations intégro-différentielles et leurs applications,” Acta Math., 35, No. 1, 295–356 (1912).

    MathSciNet  Article  Google Scholar 

  20. 20.

    Y. Xu and V. S. Erturk, “A finite difference technique for solving variable-order fractional integrodifferential equations,” Bull. Iran. Math. Soc., 40, No. 3, 699–712 (2014).

    MATH  Google Scholar 

  21. 21.

    V. V. Zaitsev, A. V. Karlov, and D. B. Nuravev, “Numerical analysis of self-oscillations of an active fractal oscillator,” Fiz. Voln. Protsess. Radiotekhn. Syst., 16, No. 2, 42–48 (2013).

    Google Scholar 

  22. 22.

    V. V. Zaitsev, A. V. Karlov, and G. P. Yarovoy, “Dynamics of self-oscillations of an active fractal oscillator,” Teor. Fiz., 14, 11–18 (2013).

    Google Scholar 

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Correspondence to R. I. Parovik.

Additional information

Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 154, Proceedings of the International Conference “Actual Problems of Applied Mathematics and Physics,” Kabardino-Balkaria, Nalchik, May 17–21, 2017, 2018.

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Parovik, R.I. On a Finite-Difference Scheme for an Hereditary Oscillatory Equation. J Math Sci 253, 547–557 (2021). https://doi.org/10.1007/s10958-021-05252-2

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Keywords and phrases

  • stability
  • convergence
  • explicit finite-difference scheme
  • hereditarity
  • integrodifferential equation
  • memory function
  • Runge rule
  • approximation

AMS Subject Classification

  • 37M05
  • 34A08